Artificial infectious disease optimization: A SEIQR epidemic dynamic model-based function optimization algorithm
Identifieur interne : 001495 ( Pmc/Corpus ); précédent : 001494; suivant : 001496Artificial infectious disease optimization: A SEIQR epidemic dynamic model-based function optimization algorithm
Auteurs : Guangqiu HuangSource :
- Swarm and Evolutionary Computation [ 2210-6502 ] ; 2015.
Abstract
To solve some complicated function optimization problems, an artificial infectious disease optimization algorithm based on the SEIQR epidemic model is constructed, it is called as the SEIQR algorithm, or SEIQRA in short. The algorithm supposes that some human individuals exist in an ecosystem; each individual is characterized by a number of features; an infectious disease (SARS) exists in the ecosystem and spreads among individuals, the disease attacks only a part of features of an individual. Each infected individual may pass through such states as susceptibility (S), exposure (E), infection (I), quarantine (Q) and recovery (R). State S, E, I, Q and R can automatically and dynamically divide all people in the ecosystem into five classes, it provides the diversity for SEIQRA; that people can be attacked by the infectious disease and then transfer it to other people can cause information exchange among people, information exchange can make a person to transit from one state to another; state transitions can be transformed into operators of SEIQRA; the algorithm has 13 legal state transitions, which corresponds to 13 operators; the transmission rules of the infectious disease among people is just the logic to control state transitions of individuals among S, E, I, Q and R, it is just the synergy of SEIQRA, the synergy can be transformed into the logic structure of the algorithm. The 13 operators in the algorithm provide a native opportunity to integrate many operations with different purposes; these operations include average, differential, expansion, chevy, reflection and crossover. The 13 operators are executed equi-probably; a stable heart rhythm of the algorithm is realized. Because the infectious disease can only attack a small part of organs of a person when it spreads among people, the part variables iteration strategy (PVI) can be ingeniously applied, thus enabling the algorithm to possess of high performance of computation, high suitability for solving some kinds of complicated optimization problems, especially high dimensional optimization problems. Results show that SEIQRA has characteristics of strong search capability and global convergence, and has a high convergence speed for some complicated functions optimization problems.
Url:
DOI: 10.1016/j.swevo.2015.09.007
PubMed: NONE
PubMed Central: 7104270
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PMC:7104270Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Artificial infectious disease optimization: A SEIQR epidemic dynamic model-based function optimization algorithm</title><author><name sortKey="Huang, Guangqiu" sort="Huang, Guangqiu" uniqKey="Huang G" first="Guangqiu" last="Huang">Guangqiu Huang</name></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmc">7104270</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7104270</idno><idno type="RBID">PMC:7104270</idno><idno type="doi">10.1016/j.swevo.2015.09.007</idno><idno type="pmid">NONE</idno><date when="2015">2015</date><idno type="wicri:Area/Pmc/Corpus">001495</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">001495</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Artificial infectious disease optimization: A SEIQR epidemic dynamic model-based function optimization algorithm</title><author><name sortKey="Huang, Guangqiu" sort="Huang, Guangqiu" uniqKey="Huang G" first="Guangqiu" last="Huang">Guangqiu Huang</name></author></analytic><series><title level="j">Swarm and Evolutionary Computation</title><idno type="ISSN">2210-6502</idno><idno type="eISSN">2210-6510</idno><imprint><date when="2015">2015</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>To solve some complicated function optimization problems, an artificial infectious disease optimization algorithm based on the SEIQR epidemic model is constructed, it is called as the SEIQR algorithm, or SEIQRA in short. The algorithm supposes that some human individuals exist in an ecosystem; each individual is characterized by a number of features; an infectious disease (SARS) exists in the ecosystem and spreads among individuals, the disease attacks only a part of features of an individual. Each infected individual may pass through such states as susceptibility (S), exposure (E), infection (I), quarantine (Q) and recovery (R). State S, E, I, Q and R can automatically and dynamically divide all people in the ecosystem into five classes, it provides the diversity for SEIQRA; that people can be attacked by the infectious disease and then transfer it to other people can cause information exchange among people, information exchange can make a person to transit from one state to another; state transitions can be transformed into operators of SEIQRA; the algorithm has 13 legal state transitions, which corresponds to 13 operators; the transmission rules of the infectious disease among people is just the logic to control state transitions of individuals among S, E, I, Q and R, it is just the synergy of SEIQRA, the synergy can be transformed into the logic structure of the algorithm. The 13 operators in the algorithm provide a native opportunity to integrate many operations with different purposes; these operations include average, differential, expansion, chevy, reflection and crossover. The 13 operators are executed equi-probably; a stable heart rhythm of the algorithm is realized. Because the infectious disease can only attack a small part of organs of a person when it spreads among people, the part variables iteration strategy (PVI) can be ingeniously applied, thus enabling the algorithm to possess of high performance of computation, high suitability for solving some kinds of complicated optimization problems, especially high dimensional optimization problems. Results show that SEIQRA has characteristics of strong search capability and global convergence, and has a high convergence speed for some complicated functions optimization problems.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Yang, Wei" uniqKey="Yang W">Wei Yang</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Buseh, Aaron G" uniqKey="Buseh A">Aaron G. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Swarm Evol Comput</journal-id><journal-id journal-id-type="iso-abbrev">Swarm Evol Comput</journal-id><journal-title-group><journal-title>Swarm and Evolutionary Computation</journal-title></journal-title-group><issn pub-type="ppub">2210-6502</issn><issn pub-type="epub">2210-6510</issn><publisher><publisher-name>Elsevier</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmc">7104270</article-id><article-id pub-id-type="publisher-id">S2210-6502(15)00074-7</article-id><article-id pub-id-type="doi">10.1016/j.swevo.2015.09.007</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>Artificial infectious disease optimization: A SEIQR epidemic dynamic model-based function optimization algorithm</article-title></title-group><contrib-group><contrib contrib-type="author" id="au0005"><name><surname>Huang</surname><given-names>Guangqiu</given-names></name></contrib></contrib-group><aff id="aff0005">School of Management, Xi’an University of Architecture and Technology, Xi’an 710055, China</aff><pub-date pub-type="pmc-release"><day>9</day><month>10</month><year>2015</year></pub-date><pmc-comment> PMC Release delay is 0 months and 0 days and was based on .</pmc-comment>
<pub-date pub-type="ppub"><month>4</month><year>2016</year></pub-date><pub-date pub-type="epub"><day>9</day><month>10</month><year>2015</year></pub-date><volume>27</volume><fpage>31</fpage><lpage>67</lpage><history><date date-type="received"><day>10</day><month>2</month><year>2015</year></date><date date-type="rev-recd"><day>5</day><month>8</month><year>2015</year></date><date date-type="accepted"><day>21</day><month>9</month><year>2015</year></date></history><permissions><copyright-statement>Copyright © 2015 Elsevier B.V. All rights reserved.</copyright-statement><copyright-year>2015</copyright-year><copyright-holder>Elsevier B.V.</copyright-holder><license><license-p>Since January 2020 Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre - including this research content - immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active.</license-p></license></permissions><abstract id="ab0010"><p>To solve some complicated function optimization problems, an artificial infectious disease optimization algorithm based on the SEIQR epidemic model is constructed, it is called as the SEIQR algorithm, or SEIQRA in short. The algorithm supposes that some human individuals exist in an ecosystem; each individual is characterized by a number of features; an infectious disease (SARS) exists in the ecosystem and spreads among individuals, the disease attacks only a part of features of an individual. Each infected individual may pass through such states as susceptibility (S), exposure (E), infection (I), quarantine (Q) and recovery (R). State S, E, I, Q and R can automatically and dynamically divide all people in the ecosystem into five classes, it provides the diversity for SEIQRA; that people can be attacked by the infectious disease and then transfer it to other people can cause information exchange among people, information exchange can make a person to transit from one state to another; state transitions can be transformed into operators of SEIQRA; the algorithm has 13 legal state transitions, which corresponds to 13 operators; the transmission rules of the infectious disease among people is just the logic to control state transitions of individuals among S, E, I, Q and R, it is just the synergy of SEIQRA, the synergy can be transformed into the logic structure of the algorithm. The 13 operators in the algorithm provide a native opportunity to integrate many operations with different purposes; these operations include average, differential, expansion, chevy, reflection and crossover. The 13 operators are executed equi-probably; a stable heart rhythm of the algorithm is realized. Because the infectious disease can only attack a small part of organs of a person when it spreads among people, the part variables iteration strategy (PVI) can be ingeniously applied, thus enabling the algorithm to possess of high performance of computation, high suitability for solving some kinds of complicated optimization problems, especially high dimensional optimization problems. Results show that SEIQRA has characteristics of strong search capability and global convergence, and has a high convergence speed for some complicated functions optimization problems.</p></abstract><kwd-group id="keys0005"><title>Keywords</title><kwd>Function optimization</kwd><kwd>Intelligent optimization computation</kwd><kwd>Epidemic dynamics</kwd><kwd>SEIQR epidemic model</kwd><kwd>Artificial infectious disease optimization</kwd></kwd-group></article-meta></front><body><sec id="s0005"><label>1</label><title>Introduction</title><p id="p0005">Infectious diseases can transmit from a person or any other species to another person or other species through various channels. Usually susceptible individuals can be infected with infectious diseases by directly contacting with infected individuals, body fluids and excreta of infected individuals, or materials polluted by infected individuals; infectious diseases can spread through air, water, food, skin contact, soil, blood, and etc <xref rid="bib1" ref-type="bibr">[1]</xref>. In nature, there exist a large number of infectious viruses, humans, animals and plants are subjected to their threat all the time. Behind almost every infectious disease, a sad story is always implied. Ebola, outbreak in early 2014, and still raging in Africa, has made thousands of Africans died <xref rid="bib2" ref-type="bibr">[2]</xref>, <xref rid="bib3" ref-type="bibr">[3]</xref>, <xref rid="bib4" ref-type="bibr">[4]</xref>; in early 2002, outbreak of SARS had made tens of thousands of Chinese people died, more Chinese people disabled <xref rid="bib5" ref-type="bibr">[5]</xref>, <xref rid="bib6" ref-type="bibr">[6]</xref>, <xref rid="bib7" ref-type="bibr">[7]</xref>, <xref rid="bib8" ref-type="bibr">[8]</xref>; H7N9 <xref rid="bib9" ref-type="bibr">[9]</xref>, <xref rid="bib10" ref-type="bibr">[10]</xref>, <xref rid="bib11" ref-type="bibr">[11]</xref>, <xref rid="bib12" ref-type="bibr">[12]</xref> or H5N6 <xref rid="bib13" ref-type="bibr">[13]</xref>, <xref rid="bib14" ref-type="bibr">[14]</xref>, which visits southern regions of China every year, makes a lot of poultry slaughtered.</p><p id="p0010">However, the purpose of this article is not to introduce a model to describe an infectious disease, but reveals an important application hiding in the spreading mechanism of each infectious disease, namely behind each infectious disease, even it hides an optimization algorithm which can solve some complicated function optimization problems! In other words, an infectious disease actually corresponds to an optimization algorithm. The task of this article is to reveal how this correspondence happens.</p><p id="p0015">Suppose the optimization problem we want to solve is as follows:<disp-formula id="eq0005"><label>(1)</label><mml:math id="M1" altimg="si0012.gif" overflow="scroll"><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mi>min</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi mathvariant="normal">s</mml:mi><mml:mi mathvariant="normal">.t</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo>∈</mml:mo><mml:mi>S</mml:mi><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula></p><p id="p0020">where <italic>R</italic><sup><italic>n</italic></sup> is a <italic>n</italic>-dimensional Euclidean space<italic>; <bold>X</bold></italic>=(<italic>x</italic><sub>1</sub>,<italic>x</italic><sub>2</sub>,…,<italic>x</italic><sub><italic>n</italic></sub>) is an <italic>n</italic>-dimensional decision vector; <italic>S</italic> is a search space; <italic>f</italic>(<italic><bold>X</bold></italic>) is an objective function; <italic>g</italic><sub><italic>i</italic></sub>(<italic><bold>X</bold></italic>)≥0 is the <italic>i</italic>th inequality constraint, <italic>i</italic>∈<italic>I</italic>, <italic>I</italic> is the set of inequality constraints.</p><p id="p0025">If <italic>f</italic>(<italic><bold>X</bold></italic>) is neither a concave function nor a convex function, or <italic>f</italic>(<italic><bold>X</bold></italic>) and <italic>g</italic><sub><italic>i</italic></sub>(<italic><bold>X</bold></italic>) are discontinuous or non-differentiable, or even their mathematical expressions don׳t know, then optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref> can only be solved by heuristic search methods <xref rid="bib15" ref-type="bibr">[15]</xref>, <xref rid="bib16" ref-type="bibr">[16]</xref>, <xref rid="bib17" ref-type="bibr">[17]</xref>, <xref rid="bib18" ref-type="bibr">[18]</xref>, <xref rid="bib19" ref-type="bibr">[19]</xref>, <xref rid="bib20" ref-type="bibr">[20]</xref>, one of which is the population-based intelligence optimization method <xref rid="bib21" ref-type="bibr">[21]</xref>, <xref rid="bib22" ref-type="bibr">[22]</xref>, <xref rid="bib23" ref-type="bibr">[23]</xref>, <xref rid="bib24" ref-type="bibr">[24]</xref>. For a population-based intelligence optimization algorithm, we always assume that for given <italic><bold>X</bold></italic>, <italic>f</italic>(<italic><bold>X</bold></italic>) and <italic>g</italic><sub><italic>i</italic></sub>(<italic><bold>X</bold></italic>) can be calculated, while <italic>f</italic>(<italic><bold>X</bold></italic>) and <italic>g</italic><sub><italic>i</italic></sub>(<italic><bold>X</bold></italic>) are always without any restrictions <xref rid="bib25" ref-type="bibr">[25]</xref>, <xref rid="bib26" ref-type="bibr">[26]</xref>, <xref rid="bib27" ref-type="bibr">[27]</xref>.</p><p id="p0030">Up to now, many population-based intelligence optimization algorithms have been developed, for example, genetic algorithm (GA) <xref rid="bib21" ref-type="bibr">[21]</xref>, <xref rid="bib22" ref-type="bibr">[22]</xref>, <xref rid="bib25" ref-type="bibr">[25]</xref>, <xref rid="bib28" ref-type="bibr">[28]</xref>, <xref rid="bib29" ref-type="bibr">[29]</xref>, <xref rid="bib30" ref-type="bibr">[30]</xref>, <xref rid="bib31" ref-type="bibr">[31]</xref>, <xref rid="bib32" ref-type="bibr">[32]</xref>, ant colony algorithm (ACA) <xref rid="bib33" ref-type="bibr">[33]</xref>, <xref rid="bib34" ref-type="bibr">[34]</xref>, particle swarm optimization (PSO) <xref rid="bib35" ref-type="bibr">[35]</xref>, <xref rid="bib36" ref-type="bibr">[36]</xref>, <xref rid="bib37" ref-type="bibr">[37]</xref>, <xref rid="bib38" ref-type="bibr">[38]</xref>, <xref rid="bib39" ref-type="bibr">[39]</xref>, <xref rid="bib40" ref-type="bibr">[40]</xref>, <xref rid="bib41" ref-type="bibr">[41]</xref>, biogeography-based optimization (BBO) <xref rid="bib42" ref-type="bibr">[42]</xref>, differential evolution (DE) <xref rid="bib43" ref-type="bibr">[43]</xref>, <xref rid="bib44" ref-type="bibr">[44]</xref>, <xref rid="bib45" ref-type="bibr">[45]</xref>, artificial bee colony (ABC) <xref rid="bib46" ref-type="bibr">[46]</xref>, <xref rid="bib47" ref-type="bibr">[47]</xref>, <xref rid="bib48" ref-type="bibr">[48]</xref>, <xref rid="bib49" ref-type="bibr">[49]</xref>, artificial immunity algorithm (AIA) <xref rid="bib50" ref-type="bibr">[50]</xref>, <xref rid="bib51" ref-type="bibr">[51]</xref>, <xref rid="bib52" ref-type="bibr">[52]</xref>, <xref rid="bib53" ref-type="bibr">[53]</xref>, <xref rid="bib54" ref-type="bibr">[54]</xref>, <xref rid="bib55" ref-type="bibr">[55]</xref>, <xref rid="bib56" ref-type="bibr">[56]</xref> and evolutionary strategy (ES) <xref rid="bib57" ref-type="bibr">[57]</xref>, <xref rid="bib58" ref-type="bibr">[58]</xref> and so on.</p><p id="p0035">Because a population-based intelligence optimization algorithm generally doesn׳t require special restrictions on objective function and constraints of an optimization problem, they have broad suitability and applicability <xref rid="bib35" ref-type="bibr">[35]</xref>. A common feature of these algorithms is that evolutionary scene is very simple and corresponding operators are very few <xref rid="bib35" ref-type="bibr">[35]</xref>.</p><p id="p0040">NFL <xref rid="bib59" ref-type="bibr">[59]</xref>, <xref rid="bib60" ref-type="bibr">[60]</xref> has pointed out, there is not an algorithm that can solve all optimization problems within finite time, but there is an algorithm that can solve some classes of optimization problems, for example, the simplex method can solve all linear programming problems <xref rid="bib61" ref-type="bibr">[61]</xref>. Though operators contained in a population-based intelligence optimization algorithm are very simple, they are widely researched and applied in the wake of the corresponding algorithm <xref rid="bib35" ref-type="bibr">[35]</xref>.</p><p id="p0045">Each algorithm can solve some kinds of optimization problems. If cores of these algorithms are extracted and combined into some new operators, then these new operators may have better suitability and wider application. For example, suppose we extract <italic>A</italic> cores from <italic>A</italic> algorithms, each core is called as an operation, if we select randomly <italic>a</italic> operations from the <italic>A</italic> cores to combine a new operator, then we can obtain <inline-formula><mml:math id="M2" altimg="si0013.gif" overflow="scroll"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>A</mml:mi><mml:mi>a</mml:mi></mml:msubsup></mml:math></inline-formula> new operators. Obviously, the suitability of a new algorithm that possesses of the <inline-formula><mml:math id="M3" altimg="si0013.gif" overflow="scroll"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>A</mml:mi><mml:mi>a</mml:mi></mml:msubsup></mml:math></inline-formula> new operators may be better than that of anyone of the <italic>A</italic> algorithms; applicable scope of the new algorithm may be wider than that of anyone of the <italic>A</italic> algorithms.</p><p id="p0050">The above-mentioned strategy is called as integration in the article.</p><p id="p0055">When a population-based intelligence optimization algorithm makes iteration, all variables in an individual, which is the biological explanation of an alternative solution of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>, take part in computation simultaneously, we call the iteration strategy as all variables iteration (AVI). AVI means all dimensions of an optimization problem take part in computation simultaneously. The algorithms applying AVI include: GA, PSO, AFSA <xref rid="bib62" ref-type="bibr">[62]</xref>, BAT <xref rid="bib63" ref-type="bibr">[63]</xref>, Cuckoo Search (CS) <xref rid="bib64" ref-type="bibr">[64]</xref>, <xref rid="bib65" ref-type="bibr">[65]</xref>, <xref rid="bib66" ref-type="bibr">[66]</xref>, Glowworm-inspired Agent Swarms <xref rid="bib67" ref-type="bibr">[67]</xref>, <xref rid="bib68" ref-type="bibr">[68]</xref>, AIA and its variations <xref rid="bib50" ref-type="bibr">[50]</xref>, <xref rid="bib51" ref-type="bibr">[51]</xref>, <xref rid="bib52" ref-type="bibr">[52]</xref>, <xref rid="bib53" ref-type="bibr">[53]</xref>, <xref rid="bib54" ref-type="bibr">[54]</xref>, <xref rid="bib55" ref-type="bibr">[55]</xref>, <xref rid="bib56" ref-type="bibr">[56]</xref>, ES <xref rid="bib57" ref-type="bibr">[57]</xref>, <xref rid="bib58" ref-type="bibr">[58]</xref>, and so on. Because all variables of each alternative solution take part in computation during iteration, a population-based intelligence algorithm applying AVI may consume more CPU time, its efficiency of computation may be low relatively. Therefore it is not suitable to solve high dimensional optimization problems.</p><p id="p0060">When a population-based intelligence optimization algorithm makes iteration, only a very small part of variables in an individual take part in computation simultaneously, we call the iteration strategy as part variables iteration (PVI). PVI means only a few of dimensions of an optimization problem take part in computation simultaneously. The algorithms applying PVI include: DE, BBO, ABC and so on. Because only a small part of variables of an alternative solution during iteration take part in computation, a population-based intelligence algorithm applying PVI may consume a little of CPU time, its efficiency of computation may be high comparatively, consequently it is suitable to solve some high dimensional optimization problems.</p><p id="p0065">Therefore, developing a PVI-based population-based intelligence algorithm may be a good developing direction.</p><p id="p0070">If a population-based intelligence optimization algorithm has its heart rhythm, then the algorithm may behave like a live animal. If the heart rhythm of the algorithm is stable, then it means that all operators contained in the algorithm are executed equi-probably; if the heart rhythm throbs (heart palpitation), then it means that a special operator is executed with higher probability under special conditions, a targeted exploration may be realized.</p><p id="p0075">In a population-based intelligence optimization algorithm, many individuals work together, it is the basic property that a population-based intelligence optimization algorithm differs from a traditional optimization algorithm. In the article, we call the property as synergy, but we assign it much wider implications: synergy may be cooperation, competition, interaction, role changing, state transition and so on <xref rid="bib35" ref-type="bibr">[35]</xref>, <xref rid="bib86" ref-type="bibr">[86]</xref>, <xref rid="bib87" ref-type="bibr">[87]</xref>, <xref rid="bib88" ref-type="bibr">[88]</xref>.</p><p id="p0080">Information exchange among individuals is always carried out during iteration in a population-based intelligence optimization algorithm. By information exchange a new search strategy is formed. If information exchange of an algorithm is very sufficient, then the algorithm’s performance may be good <xref rid="bib35" ref-type="bibr">[35]</xref>, <xref rid="bib46" ref-type="bibr">[46]</xref>, <xref rid="bib57" ref-type="bibr">[57]</xref>, <xref rid="bib62" ref-type="bibr">[62]</xref>, <xref rid="bib63" ref-type="bibr">[63]</xref>, <xref rid="bib64" ref-type="bibr">[64]</xref>, <xref rid="bib87" ref-type="bibr">[87]</xref>, <xref rid="bib88" ref-type="bibr">[88]</xref>.</p><p id="p0085">Diversity of individuals in a population-based intelligence optimization algorithm can make individuals to evolve vividly, and then reducing the probability that evolution drops into local pitfalls <xref rid="bib69" ref-type="bibr">[69]</xref>, <xref rid="bib70" ref-type="bibr">[70]</xref>, <xref rid="bib87" ref-type="bibr">[87]</xref>, <xref rid="bib88" ref-type="bibr">[88]</xref>.</p><p id="p0090">Evolution of population with the survival of the fittest is always the basis of a population-based intelligence optimization algorithm. If there exists evolution in a population, each individual has instincts to make itself to grow better or become stronger <xref rid="bib25" ref-type="bibr">[25]</xref>, <xref rid="bib71" ref-type="bibr">[71]</xref>, <xref rid="bib72" ref-type="bibr">[72]</xref>, <xref rid="bib73" ref-type="bibr">[73]</xref>, <xref rid="bib74" ref-type="bibr">[74]</xref>, then it means that the algorithm can converge to global optima of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref> with higher probability <xref rid="bib26" ref-type="bibr">[26]</xref>, <xref rid="bib35" ref-type="bibr">[35]</xref>, <xref rid="bib51" ref-type="bibr">[51]</xref>, <xref rid="bib61" ref-type="bibr">[61]</xref>, <xref rid="bib83" ref-type="bibr">[83]</xref>, <xref rid="bib86" ref-type="bibr">[86]</xref>, <xref rid="bib87" ref-type="bibr">[87]</xref>, <xref rid="bib88" ref-type="bibr">[88]</xref>.</p><p id="p0095">Synergy can be transferred into the logic structure of a population-based intelligent optimization algorithm; information exchange can be described into operators of the algorithm; diversity can make search to develop along different directions; while the evolution and instincts of each individual will enable each individual to evolve toward better fitness so as to arrive at global optima at higher probability <xref rid="bib26" ref-type="bibr">[26]</xref>, <xref rid="bib35" ref-type="bibr">[35]</xref>, <xref rid="bib51" ref-type="bibr">[51]</xref>, <xref rid="bib61" ref-type="bibr">[61]</xref>, <xref rid="bib74" ref-type="bibr">[74]</xref>, <xref rid="bib83" ref-type="bibr">[83]</xref>, <xref rid="bib86" ref-type="bibr">[86]</xref>, <xref rid="bib87" ref-type="bibr">[87]</xref>, <xref rid="bib88" ref-type="bibr">[88]</xref>.</p><p id="p0100">In nature, is there a scenario which can reflect the above-mentioned 8 properties simultaneously, namely integration, PVI, heart rhythm and heart palpitation, synergy, information exchange, diversity, and evolution? The answer is YES.</p><p id="p0105">In the article, we introduce a new population-based intelligence optimization algorithm by telling the following sad story.</p><p id="p0110">There are many people in an ecosystem; people always like to eat wild animals. One day, SARS (severe acute respiratory syndromes, SARS <xref rid="bib5" ref-type="bibr">[5]</xref>, <xref rid="bib6" ref-type="bibr">[6]</xref>, <xref rid="bib7" ref-type="bibr">[7]</xref>, <xref rid="bib8" ref-type="bibr">[8]</xref>) broke out suddenly within the ecosystem. At first, the SARS virus attacked some people because these people ate some wild animals that had infected with the SARS virus; hence these people were exposed, but did not come on.</p><p id="p0115">Because SARS can spread by air, and people have the habitual nature of living together in the ecosystem, so the exposed people can easily transmit their diseases to other people. After a period of latency, the exposed people became ill. Because people did not know SRAS at that time, it provided a good opportunity for the virus to spread among people easily and widely. Through closely contacting with the exposed or sick people, many people got infected. After several months, thousands and thousands of people had been infected with SARS, many people died, many people became maimed. A huge fear spread in the ecosystem, all kinds of social and economic activities stopped unavoidably.</p><p id="p0120">From the sad story described above, we can find that at some time, some people were not infected with SARS, they are healthy (susceptible) (we call it state S); some people are exposed (we call it state E) because they are just infected with SARS but do not come on after making closely contacting with some exposed or sick people that had been infected with SARS or ate some wild animals infected with SARS; some people became ill (we call it state I) after they had been exposed for a period of time; some people were quarantined (we call it state Q) once they were found to have infected with SARS and became ill; some people were recovered (we call it state R) after they were exposed, infected, quarantined or vaccinated by medical treatment. Each person can only stay at one state at one time, this is to say, all people in the ecosystem were divided into five classes S, E, I, Q and R automatically and dynamically.</p><p id="p0125">The aforementioned story can be described by an epidemic dynamic model call the SEIQR epidemic model <xref rid="bib75" ref-type="bibr">[75]</xref>, <xref rid="bib76" ref-type="bibr">[76]</xref>, <xref rid="bib77" ref-type="bibr">[77]</xref>, <xref rid="bib78" ref-type="bibr">[78]</xref>, <xref rid="bib79" ref-type="bibr">[79]</xref>, which is established base on the famous Kermack–Mckendrick bin model <xref rid="bib80" ref-type="bibr">[80]</xref>, <xref rid="bib81" ref-type="bibr">[81]</xref>. The SARS-based SEIQR model describes the SARS virus attacks human population; any individual in the population may pass through five states: susceptible (S), exposed (E), infected (I), quarantined (Q) and recovered (R).</p><p id="p0130">State S, E, I, Q and R can automatically and dynamically divide all people in the ecosystem into five classes. It is just the diversity the SEIQR epidemic model can provide for us. That people can be attacked by SARS and then transfer their diseases to other people can cause information exchange among people. Information exchange can make a person to transit from one state to another; state transitions can be transformed into operators of the algorithm we introduce in the article. The rules of SARS transmission among people is just the logic to control state transitions of individuals among S, E, I, Q and R, it is just the synergy the SEIQR epidemic model can provide for us. The synergy can be transformed into the logic structure of the algorithm.</p><p id="p0135">State transitions among state S, E, I, Q and R can produce 13 legal state transitions, which will be described in <xref rid="s0035" ref-type="sec">Section 2.4</xref>, if these legal state transitions are triggered evenly, then the 13 operators the 13 legal state transitions correspond to are executed equi-probably, a stable heart rhythm is realized. The 13 operators contained in the algorithm provide a native opportunity to integrate many operations with different purposes, and then integration is realized.</p><p id="p0140">Because SARS can only attack a small part of organs of a person when it spreads among people, an organ corresponds with a variable of an alternative solution of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>, PVI can be ingeniously applied, thus enabling the proposed algorithm to possess of high performance of computation, high suitability for solving some kinds of complicated optimization problems, especially high dimensional optimization problems.</p><p id="p0145">Based on the aforementioned discussions, we summarize the following key points:<list list-type="simple" id="li0005"><list-item id="o0005"><label>(1)</label><p id="p0150"><italic>Synergy</italic>. It is realized by the rules of SARS transmission which are described by the Kermack–Mckendrick bin model.</p></list-item><list-item id="o0010"><label>(2)</label><p id="p0155"><italic>Information exchange</italic>. When SARS spreads among people, exposed or infected people transmit their virus to other people, making their states to transfer among S, E, I, Q and R, and then information exchange is realized automatically and dynamically.</p></list-item><list-item id="o0015"><label>(3)</label><p id="p0160"><italic>Diversity</italic>. Individuals in the SEIQR epidemic model are automatically and dynamically divided into 5 classes, namely class S, E, I, Q and R, individuals in each class can be considered into a subclass of population, and then diversity is realized naturally.</p></list-item><list-item id="o0020"><label>(4)</label><p id="p0165"><italic>Evolution</italic>. Each individual in the ecosystem has instincts to survive, which makes itself to grow better or become stronger.</p></list-item><list-item id="o0025"><label>(5)</label><p id="p0170"><italic>Integration</italic>. There are 13 operators in the proposed algorithm, thus it provides a good opportunity to integrate some cores of some optimization algorithms together.</p></list-item><list-item id="o0030"><label>(6)</label><p id="p0175"><italic>Heart rhythm</italic>. 13 legal state transitions can lead the corresponding 13 operators to execute equi-probably; the curve of a state transition is very similar to that of a person’s heart rhythm. If a special operator is found to have high performance when SEIQRA solves a problem, then the operator will be executed with higher probability, a targeted exploration is realized, it is just the heart palpitation.</p></list-item><list-item id="o0035"><label>(7)</label><p id="p0180"><italic>PVI</italic>. When SARS attacks people, only a very small part of organs are infected, through test we find that only 1/1000–1/10 of all variables take part in computation, the proposed algorithm can obtain high performance, thus a native reduction of dimensionality is realized.</p></list-item></list></p><p id="p0185">The SEIQR epidemic model built based on the Kermack–Mckendrick bin model <xref rid="bib32" ref-type="bibr">[32]</xref>, <xref rid="bib33" ref-type="bibr">[33]</xref> is an nonlinear mathematical model that describes dynamic behaviors of individuals in random state transition among susceptibility (S), exposure (E), infection (I), quarantine (Q) and recovery (R) under the action of an infectious disease, the model considers an infectious disease not from the perspective of pathological knowledge of the infectious disease, but from the description of spread of the infectious disease, analysis of quantitative change rules of infected individuals and revelation of developmental state of the infectious disease according to general mechanism of transmission of the infectious disease and through the process of quantitative relation <xref rid="bib33" ref-type="bibr">[33]</xref>, <xref rid="bib34" ref-type="bibr">[34]</xref>, <xref rid="bib35" ref-type="bibr">[35]</xref>, which means that state transitions among state S, E, I, Q and R can be described by mathematical equations based on the Kermack–Mckendrick bin model. That an infectious disease spreads among individuals enables interaction among individuals to be reflected incisively and vividly, the biological meaning of the interaction can be clearly illustrated; what an infectious disease attack is a few of organs of an individual, the phenomenon, when mapped to the situation of searching optimum solutions of an optimization problem, is that the number of variables to be processed at each time deals with only a very small part of all variables. Therefore, the biological meaning of the variables treatment strategy in the algorithm is very clear.</p><p id="p0190">Since the SEIQR epidemic model can appropriately describe epidemic rules of infectious diseases among individuals, which to a large extent, are beneficial to help depict information exchange among many individuals; while an individual is just a biologic explanation of an alternation solution of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>; information exchange among many individuals means information exchange among many alternative solutions of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>. Therefore the model has unique advantages in solving some complicated function optimization problems. Based on the SEIQR epidemic model in which individuals interact each other, this paper presents a new algorithm for function optimization, namely the SEIQR algorithm, or SEIQRA in short. The paper focuses on solving the following 6 problems:<list list-type="simple" id="li0010"><list-item id="o0040"><label>(1)</label><p id="p0195">How to transfer a SEIQR epidemic model into a SEIQR function optimization algorithm that can solve some complicated optimization problems.</p></list-item><list-item id="o0045"><label>(2)</label><p id="p0200">How to make the operators in SEIQRA fully reflect the ideas of the SEIQR epidemic model.</p></list-item><list-item id="o0050"><label>(3)</label><p id="p0205">How to illustrate the convergence of SEIQRA.</p></list-item><list-item id="o0055"><label>(4)</label><p id="p0210">How to determine the suitable setting of parameters in SEIQRA and analyze the stability of parameter setting in SEIQRA.</p></list-item><list-item id="o0060"><label>(5)</label><p id="p0215">How to trace dynamic behaviors of the operators in SEIQRA.</p></list-item><list-item id="o0065"><label>(6)</label><p id="p0220">How to analyze the exploration and exploitation ability of SEIQRA.</p></list-item></list></p><p id="p0225">The structure of the article is illustrated in
<xref rid="f0005" ref-type="fig">Fig. 1</xref>. In nature, there are many stories that epidemic diseases spread among animals and/or humans, these stories can be described by epidemic dynamic models based on the Kermack and Mckendrick Assumptions [<xref rid="bib80" ref-type="bibr">[80]</xref>, <xref rid="bib81" ref-type="bibr">[81]</xref>]. Besides SARS, the SEIQR epidemic model can describe many other infectious diseases; each infectious disease can be transferred into a population-based intelligence optimization algorithm. SEIQRA proposed in the paper provides a reference to convert these stories of infectious diseases into population-based intelligence optimization algorithms.<fig id="f0005"><label>Fig. 1</label><caption><p>The structure of the article.</p></caption><alt-text id="at0005">Fig. 1</alt-text><graphic xlink:href="gr1"></graphic></fig></p></sec><sec id="s0010"><label>2</label><title>The algorithm design based on the SEIQR epidemic model</title><p id="p0230">To enable SEIRQA to adapt many kinds of optimization problems, the objective function of optimization problems <xref rid="eq0005" ref-type="disp-formula">(1)</xref> is rewritten as follows:<disp-formula id="eq0010"><label>(2)</label><mml:math id="M4" altimg="si0014.gif" overflow="scroll"><mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>∀</mml:mo><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mo>{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mi>I</mml:mi><mml:mo>}</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo>;</mml:mo><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo>∈</mml:mo><mml:mi>S</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>max</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mi mathvariant="normal">otherwise</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula>where <italic>F</italic><sub>max</sub> is a very large real number used to punish the alternative solutions that does not satisfy the constraints of optimization problems <xref rid="eq0005" ref-type="disp-formula">(1)</xref>.</p><sec id="s0015"><label>2.1</label><title>SARS</title><p id="p0235">The SEIQR epidemic model <xref rid="bib75" ref-type="bibr">[75]</xref>, <xref rid="bib76" ref-type="bibr">[76]</xref>, <xref rid="bib77" ref-type="bibr">[77]</xref>, <xref rid="bib78" ref-type="bibr">[78]</xref>, <xref rid="bib79" ref-type="bibr">[79]</xref> can be used to describe diseases like Ebola, SARS, H7N9, H5N6, Dengue and so on. Here we use the model to describe SARS, SARS is an acute respiratory infectious disease caused by the SARS corona virus (SARS-CoV), WHO named it as the severe acute respiratory syndrome (SARS) <xref rid="bib5" ref-type="bibr">[5]</xref>. The main way to spread this disease is flying saliva transmission within short distance or close contact with patients’ respiratory secretions. Fever, headache, muscle aches, fatigue, dry cough phlegm of patients are main clinical manifestations; respiratory distress may appear in severe cases. The disease has stronger infectivity, significant gathering phenomena happen in families and hospitals. Global initial cases occurred in Foshan, Guangdong Province of China, in November 2002, and quickly formed a popular trend.</p><p id="p0240">The SARS virus is discharged out of body through respiratory tract secretions such as oral saliva, sneezing and so on, and spreads through the air; peaks of infection are in the autumn, winter and early spring of a year. The latency period of SARS is 3–5 days; after 10–14 days the disease outbursts, poisoning symptoms such as fever and fatigue worsen, and frequent cough, short breathing or difficulty breathing happens; only with slightly activity, asthma and heart palpitations may occur, patients are forced to stay in bed. The secondary infections of respiratory tract are prone to happen at this period. After 2–3 weeks, heat fades, other signs and symptoms relieve and even disappear; Absorption and recovery of pulmonary inflammatory changes are relatively slow, they need 2 weeks to be absorbed completely back to normal level after body temperature returns to normal.</p></sec><sec id="s0020"><label>2.2</label><title>SEIQR epidemic model</title><p id="p0245">When people establish epidemic dynamic models for some infectious diseases, the bin modeling method represented by Kermack and McKendrick [<xref rid="bib80" ref-type="bibr">[80]</xref>, <xref rid="bib81" ref-type="bibr">[81]</xref>] is always used. Based on the method, the SEIQR epidemic model gives the following assumptions <xref rid="bib75" ref-type="bibr">[75]</xref>, <xref rid="bib76" ref-type="bibr">[76]</xref>, <xref rid="bib77" ref-type="bibr">[77]</xref>, <xref rid="bib78" ref-type="bibr">[78]</xref>, <xref rid="bib79" ref-type="bibr">[79]</xref>:<list list-type="simple" id="li0015"><list-item id="o0070"><label>(1)</label><p id="p0250">An human population in an ecosystem is divided into 5 bins (classes): susceptible individuals form class S; the exposed individuals (All those have closely contacted with carriers in the ecosystem, but not come on yet, they are potential onsets) class E; infected but not isolated individuals class I, isolated individuals class Q; recovered individuals that possess of short term of immunity class R; the number of individuals at time <italic>t</italic> in class S, E, I, Q and R is expressed by <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>).</p></list-item><list-item id="o0075"><label>(2)</label><p id="p0255">Infected individuals are forcibly isolated once they are found to be infected; the quarantine rate is <italic>δ</italic>.</p></list-item><list-item id="o0080"><label>(3)</label><p id="p0260">Only susceptible individuals are pulsatingly vaccinated, the vaccination rate is <italic>p</italic>, 0<<italic>p</italic><1, the cycle of pulsating vaccination is <italic>T</italic>.</p></list-item><list-item id="o0085"><label>(4)</label><p id="p0265">The standard transmission rate <italic>βS</italic>(<italic>t</italic>)<italic>I</italic>(<italic>t</italic>) is applied, <italic>β</italic> is the contacting coefficient, 0<<italic>β</italic><1.</p></list-item><list-item id="o0090"><label>(5)</label><p id="p0270">The latency period of the infectious disease is <italic>τ</italic>, recovered individuals can obtain immunity. But after a period of time <italic>λ</italic>, the recovered individuals may lose immunity and transfer into susceptible individuals again, the losing rate of immunity is <italic>ρ</italic>, 0<<italic>ρ</italic><1.</p></list-item></list></p><p id="p0275">According to the aforementioned assumptions, the flowchart of the SEIQR epidemic model with time lag and quarantine is shown in <xref rid="f0010" ref-type="fig">Fig. 2</xref>(a) <xref rid="bib1" ref-type="bibr">[1]</xref>, <xref rid="bib75" ref-type="bibr">[75]</xref>, <xref rid="bib76" ref-type="bibr">[76]</xref>, <xref rid="bib77" ref-type="bibr">[77]</xref>, <xref rid="bib78" ref-type="bibr">[78]</xref>, <xref rid="bib79" ref-type="bibr">[79]</xref>.</p><p id="p0280">In
<xref rid="f0010" ref-type="fig">Fig. 2</xref>(a), <italic>A</italic> expresses the constant immigration of individuals; <italic>μ</italic> the normal mortality rate of individuals; <italic>α</italic><sub>1</sub>, <italic>α</italic><sub>2</sub> and <italic>α</italic><sub>3</sub> the mortality rate of infected, quarantined and exposed individuals because of illness respectively; <italic>γ</italic><sub>1</sub>, <italic>γ</italic><sub>2</sub> and <italic>γ</italic><sub>3</sub> the recovery coefficient of exposed, infected and quarantined individuals respectively. Under the impact of pulsating immunization, the corresponding epidemic model is as follows:<disp-formula id="eq0015"><label>(3)</label><mml:math id="M5" altimg="si0015.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>ρ</mml:mi><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mi>τ</mml:mi></mml:mrow></mml:msup><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mfrac><mml:mrow><mml:mi 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stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:mi>δ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>δ</mml:mi><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo 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id="eq0020"><label>(4)</label><mml:math id="M6" altimg="si0016.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo 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stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>n</mml:mi><mml:mi>T</mml:mi></mml:math></disp-formula><fig id="f0010"><label>Fig. 2</label><caption><p>The flowchart of the SEIQR epidemic model: (a) before simplification and (b) after simplification.</p></caption><alt-text id="at0010">Fig. 2</alt-text><graphic xlink:href="gr2"></graphic></fig></p><p id="p0285">Suppose the number of individuals is <italic>N</italic>(<italic>t</italic>), then <italic>N</italic>(<italic>t</italic>)=<italic>S</italic>(<italic>t</italic>)+<italic>E</italic>(<italic>t</italic>)+<italic>I</italic>(<italic>t</italic>)+<italic>Q</italic>(<italic>t</italic>)+<italic>R</italic>(<italic>t</italic>). For the sake of briefness and quick computation, we do not consider the normal mortality rate of individuals and the mortality rate of infected, quarantined and exposed individuals because of illness, namely let <italic>μ</italic>=0, <italic>α</italic><sub>1</sub>=0, <italic>α</italic><sub>2</sub>=0, <italic>α</italic><sub>3</sub>=0, <italic>A</italic>=0, then <italic>N</italic>(<italic>t</italic>) is constant <italic>N</italic>, namely <italic>N</italic>(<italic>t</italic>)=<italic>N</italic>, <italic>N</italic> is the number of individuals in an ecosystem. The flowchart of the simplified SEIQR epidemic model is shown in <xref rid="f0010" ref-type="fig">Fig. 2</xref>(b), its corresponding epidemic dynamic model is simplified into the following form:<disp-formula id="eq0025"><label>(5)</label><mml:math id="M7" altimg="si0017.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo 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stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>δ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>δ</mml:mi><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>ρ</mml:mi><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mi>t</mml:mi><mml:mo>≠</mml:mo><mml:mi>n</mml:mi><mml:mi>T</mml:mi></mml:math></disp-formula></p><p id="p0290">In formula <xref rid="eq0020" ref-type="disp-formula">(4)</xref>, <xref rid="eq0025" ref-type="disp-formula">(5)</xref>, there are population-level state variables <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>), there is no individual information in such a description. The method that population-level state variables <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>) are transferred into individual-level state variables <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>) (<italic>i</italic>=1, 2, …, <italic>N</italic>) is as follows:</p><p id="p0295">For population-level state variables <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>), we have<disp-formula id="eq0030"><label>(6)</label><mml:math id="M8" altimg="si0018.gif" overflow="scroll"><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>I</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>Q</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>R</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math></disp-formula></p><p id="p0300">Formula <xref rid="eq0030" ref-type="disp-formula">(6)</xref> means that there are <italic>S</italic>(<italic>t</italic>) susceptible individuals, <italic>E</italic>(<italic>t</italic>) vaccinated individuals, <italic>I</italic>(<italic>t</italic>) infected individuals, <italic>Q</italic>(<italic>t</italic>) quarantined individuals, <italic>R</italic>(<italic>t</italic>) recovered individuals among <italic>N</italic> individuals. <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>) are integer number, they are very difficult to process in solving differential equations.</p><p id="p0305">We rewrite formula <xref rid="eq0030" ref-type="disp-formula">(6)</xref> as follows:<disp-formula id="eq0035"><label>(7)</label><mml:math id="M9" altimg="si0019.gif" overflow="scroll"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></disp-formula></p><p id="p0310">Formula <xref rid="eq0035" ref-type="disp-formula">(7)</xref> means that if that the total number of individuals in the ecosystem is 1 unit, then <inline-formula><mml:math id="M10" altimg="si0020.gif" overflow="scroll"><mml:mfrac><mml:mrow><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M11" altimg="si0021.gif" overflow="scroll"><mml:mfrac><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M12" altimg="si0022.gif" overflow="scroll"><mml:mfrac><mml:mrow><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula>, <inline-formula><mml:math id="M13" altimg="si0023.gif" overflow="scroll"><mml:mfrac><mml:mrow><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula> and <inline-formula><mml:math id="M14" altimg="si0024.gif" overflow="scroll"><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula> are the percentage of susceptible, exposed, infected, quarantined and recovered individuals respectively, they are continuous real number, they are very convenient to process in solving differential equations.</p><p id="p0315">If we redefine state variables <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>) as the percentage of susceptible, exposed, infected, quarantined and recovered individuals in an ecosystem respectively, then we can rewrite formula <xref rid="eq0035" ref-type="disp-formula">(7)</xref> as follows:<disp-formula id="eq0040"><label>(8)</label><mml:math id="M15" altimg="si0025.gif" overflow="scroll"><mml:mrow><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></disp-formula></p><p id="p0320">Formula <xref rid="eq0040" ref-type="disp-formula">(8)</xref> means that suppose that the total number of individuals in the ecosystem is 1 unit, then <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>) represent the proportion of individuals in class S, E, I, Q and R respectively. So, for each individual in the ecosystem, <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>) represent the probability that the individual belongs to class S, E, I, Q and R respectively, or the probability that the individual stays at state S, E, I, Q and R respectively.</p><p id="p0325">Therefore formula <xref rid="eq0040" ref-type="disp-formula">(8)</xref> can be understood into the individual-level state equation:<disp-formula id="eq0045"><label>(9)</label><mml:math id="M16" altimg="si0026.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mspace width=".25em"></mml:mspace><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mi>i</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:mi>can</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:mi>be</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:mi>any</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:mi>individual</mml:mi><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0330">where <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>) are the probability that an individual stays at state S, E, I, Q and R or belongs to class S, E, I, Q and R.</p><p id="p0335">A similar analysis can be made from formula <xref rid="eq0015" ref-type="disp-formula">(3)</xref> to formula <xref rid="eq0025" ref-type="disp-formula">(5)</xref>.</p><p id="p0340">At last, which state an individual will stay at? It is defined by the probability distribution of <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>). For example, if <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>) is the maximum one among <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>), then the probability of the individual staying at state R is highest.</p><p id="p0345">Using formula <xref rid="eq0040" ref-type="disp-formula">(8)</xref>, <xref rid="eq0045" ref-type="disp-formula">(9)</xref>, <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>) are changed into continuous real number, population-level state variables <italic>S</italic>(<italic>t</italic>), <italic>E</italic>(<italic>t</italic>), <italic>I</italic>(<italic>t</italic>), <italic>Q</italic>(<italic>t</italic>) and <italic>R</italic>(<italic>t</italic>) are transferred into individual-level state variables <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>), they can be easily processed in mathematical computation, especially in solving differential equations, the method is widely used in Computational Biology and Population Dynamics.</p><p id="p0350">Based on the aforementioned analysis, we can apply formula <xref rid="eq0020" ref-type="disp-formula">(4)</xref> and <xref rid="eq0025" ref-type="disp-formula">(5)</xref> to any individual in an ecosystem, say individual <italic>i</italic>, formula <xref rid="eq0020" ref-type="disp-formula">(4)</xref> and <xref rid="eq0025" ref-type="disp-formula">(5)</xref> can be rewritten into the following discrete iterating form, considering that <italic>β</italic>, <italic>γ</italic><sub>1</sub>, <italic>γ</italic><sub>2</sub>, <italic>γ</italic><sub>3</sub>, <italic>δ</italic> and <italic>p</italic> vary with time:<disp-formula id="eq0050"><label>(10)</label><mml:math id="M17" altimg="si0027.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mi>t</mml:mi><mml:mo>≠</mml:mo><mml:mi>n</mml:mi><mml:mi>T</mml:mi></mml:math></disp-formula><disp-formula id="eq0055"><label>(11)</label><mml:math id="M18" altimg="si0028.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>n</mml:mi><mml:mi>T</mml:mi></mml:math></disp-formula></p><p id="p0355"><italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>+1) can be expressed as follows:<disp-formula id="eq0060"><mml:math id="M19" altimg="si0029.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula></p><p id="p0360">This is because:<disp-formula id="eq0065"><label>(12)</label><mml:math id="M20" altimg="si0030.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula></p><p id="p0365">In formula <xref rid="eq0065" ref-type="disp-formula">(12)</xref>, we use <inline-formula><mml:math id="M21" altimg="si0031.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>. At time <italic>t</italic>, in formula <xref rid="eq0050" ref-type="disp-formula">(10)</xref>, <xref rid="eq0055" ref-type="disp-formula">(11)</xref>, all parameters take their values by <italic>β</italic><sup><italic>t</italic></sup>=<italic>Rnd</italic>(0, <italic>β</italic>), <inline-formula><mml:math id="M22" altimg="si0032.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>=<italic>Rnd</italic>(0, <italic>γ</italic><sub>1</sub>), <inline-formula><mml:math id="M23" altimg="si0033.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>=<italic>Rnd</italic>(0, <italic>γ</italic><sub>2</sub>), <inline-formula><mml:math id="M24" altimg="si0034.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>=<italic>Rnd</italic>(0, <italic>γ</italic><sub>3</sub>), <italic>p</italic><sup><italic>t</italic></sup>=<italic>Rnd</italic>(0, <italic>p</italic>), <italic>δ</italic><sup><italic>t</italic></sup>=<italic>Rnd</italic>(0, <italic>δ</italic>), <italic>ρ</italic><sup><italic>t</italic></sup>=<italic>Rnd(</italic>0, <italic>ρ</italic>). <italic>Rnd</italic>(<italic>a</italic>, <italic>b</italic>) means to generate a random real number with even distribution within interval [<italic>a</italic>, <italic>b</italic>].</p></sec><sec id="s0025"><label>2.3</label><title>Scenario design of the algorithm</title><p id="p0370">Suppose there are <italic>N</italic> human individuals in an ecosystem, these individuals are numbered with 1, 2, …, <italic>N</italic>, let <italic>P</italic>={1, 2, …, <italic>N</italic>}. Each individual is characterized by <italic>n</italic> features, namely for individual <italic>i</italic>, its features are (<italic>x</italic><sub><italic>i</italic>1</sub>, <italic>x</italic><sub><italic>i</italic>2</sub>, …, <italic>x</italic><sub><italic>in</italic></sub>). There exists an infective disease called SARS in the ecosystem, which spreads among some individuals. Individuals will be infected with SARS through closely contacting with flying saliva or respiratory secretions of carriers. The virus just attacks a small part of features of an individual. In order to protect human individuals from SARS, susceptible individuals can be vaccinated. The vaccinated individuals will not be infected during a certain period of time, nor do they disseminate SARS to any other individuals. But the immunity of individuals is temporary, and it will lose effectiveness after a period of time. Then the human individuals without immunity will be infected with the infective disease again. The rules of SARS spreading among individuals are as follows:<list list-type="simple" id="li0020"><list-item id="o0095"><label>(1)</label><p id="p0375">If an individuals who stays at susceptibility state S makes close contact with some individuals who have been exposed or infected with SARS, then it will catch the infectious disease. Individuals who are just infected with SARS do not come on at once; their diseases transfer firstly into the latency period, which is called exposure state E. The individuals staying within the latency period can transmit their diseases to other individuals also; these individuals at exposure state E can be cured if they accept medical treatment.</p></list-item><list-item id="o0100"><label>(2)</label><p id="p0380">After a period of time, some individuals who have stayed at exposure state E become ill, they transfer into infected state I, the individuals stayed in infection state I can transmit their virus to other individuals when they make close contact with them; these individuals at infection state I can be cured if they accept medical treatment.</p></list-item><list-item id="o0105"><label>(3)</label><p id="p0385">Once an infected individual is found, it will be quarantined at once, namely it moves into quarantine state Q; an infected individual who stays at quarantine state Q will stay at the state until the individual is cured. An individual who stays at state Q cannot spread its disease to any other individuals.</p></list-item><list-item id="o0110"><label>(4)</label><p id="p0390">If individuals who stay at quarantine state Q are cured, they will enter into recovery state R and obtain a certain period of immunity. Individuals who stay at state R will be not infected within a period of time, so they cannot transmit the infectious disease to other individuals at this period of time.</p></list-item><list-item id="o0115"><label>(5)</label><p id="p0395">The immunity of individuals who stay at recovery state R may be lost after a period of time, the individuals who lose their immunity will enter into susceptibility state S again; when these individuals contact closely with the exposed or infected individuals, they will become exposed again.</p></list-item><list-item id="o0120"><label>(6)</label><p id="p0400">In order to prevent SARS to spread casually and widely, some susceptible individuals may be vaccinated cyclically, which is called pulsating vaccination.</p></list-item></list></p><p id="p0405">Step (1)–(6) are derived from <xref rid="f0010" ref-type="fig">Fig. 2</xref>(b), namely Step (1) means state transition S→E and E→R; Step (2) E→I and I→R; Step (3) I→Q; Step (4) Q→R; Step (5) R→S; Step (6) S→R. The other 5 state transitions S→S, E→E, I→I, Q→Q and R→R are no necessary to be included into Step (1)–(6).</p><p id="p0410">There are 8 state transitions in Step (1)–(6), 5 state transitions which keep their states unchanged are S→S, E→E, I→I, Q→Q and R→R. Therefore there are 8+5=13 state transitions, as listed in
<xref rid="t0005" ref-type="table">Table 1</xref>.<table-wrap position="float" id="t0005"><label>Table 1</label><caption><p>Legal state transition of the SEIQR epidemic model.</p></caption><alt-text id="at0105">Table 1</alt-text><table frame="hsides" rules="groups"><thead><tr><th>State at time <italic>t</italic>−1</th><th>State at time <italic>t</italic></th><th>Type of state transition</th><th>Operator</th></tr></thead><tbody><tr><td>S</td><td>S</td><td>S→S</td><td>S–S</td></tr><tr><td>S</td><td>E</td><td>S→E</td><td>S–E</td></tr><tr><td>S</td><td>R</td><td>S→R</td><td>S–R</td></tr><tr><td>E</td><td>E</td><td>E→E</td><td>E–E</td></tr><tr><td>E</td><td>I</td><td>E→I</td><td>E–I</td></tr><tr><td>E</td><td>R</td><td>E→R</td><td>E–R</td></tr><tr><td>I</td><td>I</td><td>I→I</td><td>I–I</td></tr><tr><td>I</td><td>Q</td><td>I→Q</td><td>I–Q</td></tr><tr><td>I</td><td>R</td><td>I→R</td><td>I–R</td></tr><tr><td>Q</td><td>Q</td><td>Q→Q</td><td>Q–Q</td></tr><tr><td>Q</td><td>R</td><td>Q→R</td><td>Q–R</td></tr><tr><td>R</td><td>R</td><td>R→R</td><td>R–R</td></tr><tr><td>R</td><td>S</td><td>R→S</td><td>R–S</td></tr></tbody></table></table-wrap></p><p id="p0415">The above-mentioned scenario is mapped onto the process of searching the global optimum solutions of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>, its meaning is as follows.</p><p id="p0420">The search space of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref> corresponds to the ecological system, an individual in the ecological system corresponds to an alternative solution of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>, the set of alternative solutions <italic>N</italic> individuals correspond to is <italic>X</italic>={<italic><bold>X</bold></italic><sub>1</sub>, <italic><bold>X</bold></italic><sub>2</sub>, …, <italic><bold>X</bold></italic><sub><italic>N</italic></sub>}. A feature of individual <italic>i</italic> (<italic>i</italic>=1, 2, …, <italic>N</italic>) corresponds to a variable of alternative solution <italic><bold>X</bold></italic><sub><italic>i</italic></sub> of the optimization problem, namely feature <italic>j</italic> of individual <italic>i</italic> corresponds to variable <italic>x</italic><sub><italic>ij</italic></sub> of alternative solution <italic><bold>X</bold></italic><sub><italic>i</italic></sub>, so the number of features of individual <italic>i</italic> is equal to the number of variables of alternative solution <italic><bold>X</bold></italic><sub><italic>i</italic></sub>, they are both <italic>n</italic>. Therefore, individual <italic>i</italic> and alternative solution <italic><bold>X</bold></italic><sub><italic>i</italic></sub> are equivalent.</p><p id="p0425">The physical strength of an individual is represented by individual physique index (IPI), which exactly corresponds to the objective function value of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>. Good alternative solutions correspond to individuals with higher IPI values which represent these individuals with strong physique, while bad alternative solutions correspond to individuals with lower IPI values which represent these individuals with weak physique. For optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>, IPI of individual <italic>i</italic> is calculated as follows:<disp-formula id="eq0070"><label>(13)</label><mml:math id="M25" altimg="si0035.gif" overflow="scroll"><mml:mrow><mml:mi>I</mml:mi><mml:mi>P</mml:mi><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mi mathvariant="normal">if</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant="normal">if</mml:mi><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo><</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width=".25em"></mml:mspace><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>…</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math></disp-formula></p><p id="p0430">At time <italic>t</italic>, randomly generate <italic>β</italic><sup><italic>t</italic></sup>, <inline-formula><mml:math id="M26" altimg="si0032.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M27" altimg="si0033.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M28" altimg="si0034.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>p</italic><sup><italic>t</italic></sup>, <italic>δ</italic><sup><italic>t</italic></sup>, <italic>ρ</italic><sup><italic>t</italic></sup> of the ecosystem, formula <xref rid="eq0050" ref-type="disp-formula">(10)</xref>, <xref rid="eq0055" ref-type="disp-formula">(11)</xref> is used to calculate susceptible probability <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), exposure probability <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), infection probability <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), quarantine probability <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and recovery probability <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>) of individual <italic>i</italic>. At time <italic>t</italic>, which state individual <italic>i</italic> stays at among state S, E, I, Q and R is determined by the probability distribution of <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>). Legal state transitions of any individual listed in <xref rid="t0005" ref-type="table">Table 1</xref> meet the situation depicted in <xref rid="f0010" ref-type="fig">Fig. 2</xref>(b).</p><p id="p0435">For state S, E, I, Q and R, there are 5×5=25 possible state transitions among individuals, but legal state transitions that satisfy <xref rid="f0010" ref-type="fig">Fig. 2</xref>(b) are 13, as shown in <xref rid="t0005" ref-type="table">Table 1</xref>. Except the 13 legal state transitions in <xref rid="t0005" ref-type="table">Table 1</xref>, other state transitions are illegal. The 13 legal state transitions can be described by 13 operators, as shown in <xref rid="t0005" ref-type="table">Table 1</xref>. From formula <xref rid="eq0050" ref-type="disp-formula">(10)</xref>, <xref rid="eq0055" ref-type="disp-formula">(11)</xref> we can know that susceptible probability <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), exposure probability <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), infection probability <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), quarantine probability <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and recovery probability <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>) of individual <italic>i</italic> vary with time, so the growth state of individual <italic>i</italic> will transfer randomly among state S, E, I, Q and R. This state transition, when mapped onto the solution space of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>, means that each alternative solution in the solution space moves from one location to another, then the random search in the solution space is realized.</p><p id="p0440">During the random search process, if IPI of individual <italic>i</italic> at time <italic>t</italic> is higher than that at time <italic>t</italic>−1, individual <italic>i</italic> will continue to grow, which means that individual <italic>i</italic> is getting closer to global optimum solutions. On the contrary, if IPI of individual <italic>i</italic> at time <italic>t</italic> is less than or equal to that at time <italic>t</italic>−1, individual <italic>i</italic> will stop growing, which means that individual <italic>i</italic> stays at the position of time <italic>t</italic>−1 and does not move. The random search strategy is called “each-step-is-not-bad” <xref rid="bib83" ref-type="bibr">[83]</xref>, <xref rid="bib86" ref-type="bibr">[86]</xref>, <xref rid="bib87" ref-type="bibr">[87]</xref>, <xref rid="bib88" ref-type="bibr">[88]</xref>, it allows the algorithm to converge.</p><p id="p0445">According to the scenario design of SEIQRA, the implementation process of SEIQRA is described as follows.</p><sec id="s0030"><label>2.3.1</label><title>Logical structure of algorithm SEIQRA</title><p id="p0450"><list list-type="simple" id="li0025"><list-item id="o0125"><label>(1)</label><p id="p0455">At time <italic>t</italic>, <italic>β</italic><sup><italic>t</italic></sup>, <inline-formula><mml:math id="M29" altimg="si0032.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M30" altimg="si0033.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M31" altimg="si0034.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>p</italic><sup><italic>t</italic></sup>, <italic>δ</italic><sup><italic>t</italic></sup> of a population are generated randomly. At the beginning, <italic>t</italic>=0.</p></list-item><list-item id="o0130"><label>(2)</label><p id="p0460">Use formula (10) and (11) to calculate the susceptibility probability <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), exposure probability <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), infection probability <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), quarantine probability <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and recovery probability <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>) of individual <italic>i</italic> at time <italic>t</italic> respectively.</p></list-item><list-item id="o0135"><label>(3)</label><p id="p0465">Determine which state is selected according to probability distribution {<italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>)} of individual <italic>i</italic>. There are five cases as follows:<list list-type="simple" id="li0030"><list-item id="o0140"><label>(31)</label><p id="p0470">If state S is selected, then it means that at time <italic>t</italic> individual <italic>i</italic> stays at state S. At this time we judge at which state individual <italic>i</italic> stayed at time <italic>t</italic>−1.There are two legal cases:</p></list-item></list></p></list-item></list><list list-type="simple" id="li0035"><list-item id="o0145"><label>(a)</label><p id="p0475">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state S, then we can know that now individual <italic>i</italic> stays still at the susceptible state S, namely S→S. In this case we use the S–S operator to calculate the growth state <inline-formula><mml:math id="M32" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item><list-item id="o0150"><label>(b)</label><p id="p0480">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state R, then we can know that now individual <italic>i</italic> loses its immunity and transfers into susceptible state S, namely R→S. In this case we use the R–S operator to calculate the growth state <inline-formula><mml:math id="M33" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.<list list-type="simple" id="li0040"><list-item id="o0155"><label>(32)</label><p id="p0485">If state E is selected, then it means that at time <italic>t</italic> individual <italic>i</italic> stays at state E. At this time we judge at which state individual <italic>i</italic> stayed at time <italic>t</italic>−1.There are two legal cases:</p></list-item></list></p></list-item></list><list list-type="simple" id="li0045"><list-item id="o0160"><label>(a)</label><p id="p0490">If at time <italic>t</italic>−1 individual <italic>i</italic> stays at state E, then we can know that now individual <italic>i</italic> still stays at the exposed state E, namely E→E. In this case we use the E–E operator to calculate the growth state <inline-formula><mml:math id="M34" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item><list-item id="o0165"><label>(b)</label><p id="p0495">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state S, then we can know that now individual <italic>i</italic> is infected with SARS but does not come on, namely S→E. In this case we use the S–E operator to calculate the growth state <inline-formula><mml:math id="M35" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item></list><list list-type="simple" id="li0050"><list-item id="o0170"><label>(33)</label><p id="p0500">If state I is selected, then it means that at time <italic>t</italic> individual <italic>i</italic> stays at state I. At this time we judge at which state individual <italic>i</italic> stayed at time <italic>t</italic>−1. There are two legal cases:</p></list-item></list><list list-type="simple" id="li0055"><list-item id="o0175"><label>(a)</label><p id="p0505">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state E, then we can know that now individual <italic>i</italic> becomes ill after it is exposed (transferring from state E into state I), namely E→I. In this case we use the E–I operator to calculate the growth state <inline-formula><mml:math id="M36" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item><list-item id="o0180"><label>(b)</label><p id="p0510">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state I, then we can know that now individual <italic>i</italic> continues to be sick, namely I→I. In this case we use the I–I operator to calculate the growth state <inline-formula><mml:math id="M37" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item></list><list list-type="simple" id="li0060"><list-item id="o0185"><label>(34)</label><p id="p0515">If state Q is selected, then it means that at time <italic>t</italic> individual <italic>i</italic> stays at state Q. At this time we judge at which state individual <italic>i</italic> stayed at time <italic>t</italic>−1.There are two legal cases:</p></list-item></list><list list-type="simple" id="li0065"><list-item id="o0190"><label>(a)</label><p id="p0520">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state I, then we can know that now individual <italic>i</italic> is quarantined after it is infected, namely I→Q. In this case we use the I–Q operator to calculate the growth state <inline-formula><mml:math id="M38" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item><list-item id="o0195"><label>(b)</label><p id="p0525">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state Q, then we can know that now individual <italic>i</italic> is still in quarantine state, namely Q→Q. In this case we use the Q–Q operator to calculate the growth state <inline-formula><mml:math id="M39" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item></list><list list-type="simple" id="li0070"><list-item id="o0200"><label>(35)</label><p id="p0530">If state R is selected, then it means that at time <italic>t</italic> individual <italic>i</italic> stays at state R. At this time we judge at which state individual <italic>i</italic> stayed at time <italic>t</italic>−1.There are five legal cases:</p></list-item></list><list list-type="simple" id="li0075"><list-item id="o0205"><label>(a)</label><p id="p0535">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state S, then we can know that now individual <italic>i</italic> is vaccinated, and it cannot be infected within a time of time, namely S→R. In this case we use the S–R operator to calculate the growth state <inline-formula><mml:math id="M40" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item><list-item id="o0210"><label>(b)</label><p id="p0540">If at time <italic>t</italic>−1 individual <italic>i</italic> stays at state E, then we can know that now individual <italic>i</italic> is cured after it is exposed, namely E→R. In this case we use the E–R operator to calculate the growth state <inline-formula><mml:math id="M41" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item><list-item id="o0215"><label>(c)</label><p id="p0545">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state I, then we can know that now individual <italic>i</italic> is cured after it is infected, namely I→R. In this case we use the I–R operator to calculate the growth state <inline-formula><mml:math id="M42" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item><list-item id="o0220"><label>(d)</label><p id="p0550">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state Q, then we can know that now individual <italic>i</italic> is cured and obtains a certain period of immunity simultaneously after it is isolated, namely Q→R. In this case we use the Q–R operator to calculate the growth state <inline-formula><mml:math id="M43" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item><list-item id="o0225"><label>(e)</label><p id="p0555">If at time <italic>t</italic>−1 individual <italic>i</italic> stayed at state R, then we can know that now individual <italic>i</italic> does not lose its immunity and still stays at state R, namely R→R. In this case we use the R–R operator to calculate the growth state <inline-formula><mml:math id="M44" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>.</p></list-item></list><list list-type="simple" id="li0080"><list-item id="o0230"><label>(4)</label><p id="p0560">If at time <italic>t</italic> the state transition of individual <italic>i</italic> is illegal, then individual <italic>i</italic> keeps its original state unchanged.</p></list-item><list-item id="o0235"><label>(5)</label><p id="p0565">Repeat Step (2) to (4) until each individual is processed, then go to Step (6).</p></list-item><list-item id="o0240"><label>(6)</label><p id="p0570">Let <italic>t</italic>=<italic>t</italic>+1.Repeat Step (1) to (5) until the global optimum solution is found.</p></list-item></list></p></sec></sec><sec id="s0035"><label>2.4</label><title>Design of ecological operators</title><p id="p0575">At time <italic>t</italic>, the initial values of <italic>N</italic> individuals in the ecosystem are <inline-formula><mml:math id="M45" altimg="si0037.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>.</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, <italic>i</italic>=1,2, …, <italic><bold>N</bold></italic>, we give the design method of the 13 ecological operators listed in <xref rid="t0005" ref-type="table">Table 1</xref>.</p><p id="p0580">In GA <xref rid="bib21" ref-type="bibr">[21]</xref>, <xref rid="bib22" ref-type="bibr">[22]</xref>, <xref rid="bib25" ref-type="bibr">[25]</xref>, <xref rid="bib28" ref-type="bibr">[28]</xref>, <xref rid="bib29" ref-type="bibr">[29]</xref>, <xref rid="bib30" ref-type="bibr">[30]</xref>, <xref rid="bib31" ref-type="bibr">[31]</xref>, <xref rid="bib32" ref-type="bibr">[32]</xref>, when two chromosomes carry out crossover operation, male and female chromosome are randomly selected from many chromosomes; position of crossover in each chromosome is randomly selected also. The idea that results from genetics, but do not fully abide by genetics can greatly simplify the design of GA, the method is widely used in population-based intelligence optimization algorithms.</p><p id="p0585">Similarly, in SEIQRA, we assume that<list list-type="simple" id="li0085"><list-item id="o0245"><label>(1)</label><p id="p0590">SARS can only attack a small part of features of an individual each time; these features are randomly selected from all features of the individual. Once an individual is attacked by SARS, its all features possess automatically of the SARS virus.</p></list-item><list-item id="o0250"><label>(2)</label><p id="p0595">When an individual stays at certain state, its all features possess automatically of the corresponding properties acclimatizing to the state. For example, when an individual stays at state S, its all features can be attacked by SARS; when the individual stays at state E, its all features contain the SARS virus, but do not come on; when the individual stays at state I, its all features are infected with the SARS virus, and become ill; when the individual stays at state Q, its all features are immunized, but do not recover; when the individual stays at state R, its all features are not only immunized, but also are cured.</p></list-item></list></p><sec id="s0040"><label>2.4.1</label><title>The biological meaning of transition types</title><p id="p0600">Individual <italic>i</italic> can be described with the following structure:<disp-formula id="eq0075"><mml:math id="M46" altimg="si0038.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">〈</mml:mo><mml:mrow><mml:mi>S</mml:mi><mml:mi>E</mml:mi><mml:mi>I</mml:mi><mml:mi>Q</mml:mi><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">t</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">〉</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:math></disp-formula>Where <italic>SEIQR</italic><sub><italic>i</italic></sub>(<italic>t</italic>) expresses the state of individual <italic>i</italic> at time <italic>t</italic>, <inline-formula><mml:math id="M47" altimg="si0039.gif" overflow="scroll"><mml:mrow><mml:mi>S</mml:mi><mml:mi>E</mml:mi><mml:mi>I</mml:mi><mml:mi>Q</mml:mi><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">E</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">I</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Q</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>; <inline-formula><mml:math id="M48" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> expresses the state value of features of individual <italic>i</italic> at time <italic>t</italic>.</p><p id="p0605">Based on the <xref rid="f0010" ref-type="fig">Fig. 2</xref>(b), we can find the general method to explain each state transition and update <italic>SEIQR</italic><sub><italic>i</italic></sub>(<italic>t</italic>) and <inline-formula><mml:math id="M49" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> of individual <italic>i</italic>, and define the biological meaning of transition types. By decomposing <xref rid="f0010" ref-type="fig">Fig. 2</xref>(b), we obtain the following two cases:<list list-type="simple" id="li0090"><list-item id="o0255"><label>(1)</label><p id="p0610">At time <italic>t</italic>, an individual can transfer from state <italic>A</italic> to state <italic>B</italic>, as shown in
<xref rid="f0015" ref-type="fig">Fig. 3</xref>(a), where <italic>A</italic> and <italic>B</italic><inline-formula><mml:math id="M50" altimg="si0011.gif" overflow="scroll"><mml:mo>∈</mml:mo></mml:math></inline-formula>{S, E, I, Q, R}, but <italic>A</italic>≠<italic>B</italic>. A large number of state transitions belong to the case, for example, S→E, S→R, E→I, E→R, I→Q, I→R, Q→R and R→S.<fig id="f0015"><label>Fig. 3</label><caption><p>Two kinds of state transition of an individual at time <italic>t</italic>.</p></caption><alt-text id="at0015">Fig. 3</alt-text><graphic xlink:href="gr3"></graphic></fig></p><p id="p0615">When an individual transfers from state <italic>A</italic> to state <italic>B</italic>, or an individual at class <italic>A</italic> becomes an individual at class <italic>B</italic>, we will let the composite state values of some features of several individuals who stay at state <italic>B</italic> endow to the corresponding features of the individual who stays at state <italic>A</italic>, namely we enable some features of the individual to possess of similar state values of the corresponding features of some individuals who have stayed at state <italic>B</italic>. This method realizes that an individual transfers from class <italic>A</italic> to class <italic>B</italic> really. For example, for state transition S→E, we let the average state values of some features of several exposed or infected individuals who have stayed at state E or state I endow to the corresponding features of a susceptible individual who stays at state S, then we can make the individual to get exposed, namely state transition S→E is achieved. This strategy means that some exposed or infected individuals transfer their own something with virus to a susceptible individual, making it to get exposed.</p></list-item><list-item id="o0260"><label>(2)</label><p id="p0620">At time <italic>t</italic>, when an individual stays at state <italic>A</italic>, <italic>A</italic><inline-formula><mml:math id="M51" altimg="si0011.gif" overflow="scroll"><mml:mo>∈</mml:mo></mml:math></inline-formula>{S, E, I, Q, R}, any state transition does not take place, it is equivalent to <italic>A</italic>→<italic>A</italic>, as shown in <xref rid="f0015" ref-type="fig">Fig. 3</xref>(b). Each node in <xref rid="f0010" ref-type="fig">Fig. 2</xref>(b) actually implies the case shown in <xref rid="f0015" ref-type="fig">Fig. 3</xref>(b), for example, S→S, E→E, I→I, Q→Q and R→R.</p></list-item></list></p><p id="p0625">When an individual stays at state <italic>A</italic>, in order to enable the individual to evolve towards good direction, but its state remains the same, we will let the composite state values of some features of several strong individuals whose IPI values are higher than that of the individual transfer to the corresponding features of the individual, namely strong individuals with higher IPI transfer strong feature information to weak individuals with lower IPI so as to make these weak individuals to grow for the better direction. The reason is that when individuals stay at susceptible, exposed, infected, isolated or recovery state, they always makes their own physique to be improved by keeping in good health, health care, training or other ways; when individuals stay at state of illness, they always make their own physique to be enhanced through medical treatment and nutritional supplement so as to achieve the goal of conquer disease.</p><p id="p0630">After state transitions are explicitly expressed according to <xref rid="f0015" ref-type="fig">Fig. 3</xref>, the flowchart shown in <xref rid="f0010" ref-type="fig">Fig. 2</xref>(b) can be illustrated as
<xref rid="f0020" ref-type="fig">Fig. 4</xref>.<fig id="f0020"><label>Fig. 4</label><caption><p>State transitions of the SEIQR epidemic model.</p></caption><alt-text id="at0020">Fig. 4</alt-text><graphic xlink:href="gr4"></graphic></fig></p><p id="p0635">For state transition <italic>A</italic>→<italic>B</italic>, we randomly select <italic>L</italic> individuals from those who have stayed at state <italic>B</italic>, let the composite state value of a randomly selected feature <italic>j</italic> of the <italic>L</italic> individuals be transferred to the corresponding feature <italic>j</italic> of current individual <italic>i</italic> who stays at state <italic>A</italic>, then the individual obtains the similar properties with the individuals staying at state <italic>B</italic>. Thus, state transition <italic>A</italic>→<italic>B</italic> is realized, namely<disp-formula id="eq0080"><mml:math id="M52" altimg="si0040.gif" overflow="scroll"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mover><mml:mo>⟹</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mo>→</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:mover><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></disp-formula></p><p id="p0640">Where <italic>i</italic><sub>1</sub>, <italic>i</italic><sub>2</sub>, …, <italic>i</italic><sub><italic>L</italic></sub> are the numbers of the <italic>L</italic> randomly selected individuals who have stayed at state <italic>B</italic>; function <italic>f</italic> expresses the method of combining the state value of feature <italic>j</italic> of the <italic>L</italic> individuals, it produces a new state value of feature <italic>j</italic>, the new state value will be transferred to the corresponding feature <italic>j</italic> of current individual <italic>i</italic> staying at state <italic>A</italic>; <italic>f</italic> is equivalent to a mixer, and can be defined according to <xref rid="s0045" ref-type="sec">Section 2.4.2</xref>.</p><p id="p0645">For state transition <italic>A</italic>→<italic>A</italic>, we randomly select <italic>L</italic> strong individuals from those who have stayed at state <italic>A</italic> but their IPI values are higher than that of current individual <italic>i</italic>; let the composite state value of a randomly selected feature <italic>j</italic> of the <italic>L</italic> strong individuals be transferred to the corresponding feature <italic>j</italic> of current individual <italic>i</italic> who stays at state <italic>A</italic>, then the individual may become stronger also, but its state does not transfer. Thus, state transition <italic>A</italic>→<italic>A</italic> is realized. Namely<disp-formula id="eq0085"><mml:math id="M53" altimg="si0041.gif" overflow="scroll"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mover><mml:mo>⟹</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mo>→</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:mover><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></disp-formula></p><p id="p0650">where <italic>s</italic><sub>1</sub>, <italic>s</italic><sub>2</sub>, …, <italic>s</italic><sub><italic>L</italic></sub> are the numbers of the <italic>L</italic> strong individuals, but IPI values of the <italic>L</italic> strong individuals are higher than that of individual <italic>i</italic>.</p></sec><sec id="s0045"><label>2.4.2</label><title>Method of operators design</title><p id="p0655">Function <italic>f</italic> mentioned in <xref rid="s0040" ref-type="sec">Section 2.4.1</xref> is defined as average, differential, expansion, chevy, reflection and crossover operation.</p><sec id="s0050"><label>2.4.2.1</label><title>Design of operations</title><p id="p0660">At time <italic>t</italic>−1, suppose the set <italic>C</italic><sup><italic>t</italic>−1</sup> of individuals that satisfy given requirements, the current individual is <italic>i</italic>, its affected feature is <italic>j</italic>, then the design methods of average, differential, expansion, chevy, reflection and crossover operation are as follows:<list list-type="simple" id="li0095"><list-item id="o0265"><label>(1)</label><p id="p0665">Average Operation <italic>AVG</italic>(<italic>C</italic><sup><italic>t</italic>−1</sup>, <italic>i</italic>, <italic>j</italic>). Randomly select <italic>L</italic> individuals from <italic>C</italic><sup><italic>t</italic>−1</sup>, let the state values of feature <italic>j</italic> of the <italic>L</italic> individuals are averaged to produce a new compound value by use of average operation, the new value is transferred to the corresponding feature <italic>j</italic> of individual <italic>i</italic> at time <italic>t</italic>, enabling individual <italic>i</italic> to have similar properties with the individuals in <italic>C</italic><sup><italic>t</italic>−1</sup>, namely<disp-formula id="eq0090"><mml:math id="M54" altimg="si0042.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:munder><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:math></disp-formula>where, <italic>i</italic><sub>1</sub>, <italic>i</italic><sub>2</sub>, …, <italic>i</italic><sub><italic>L</italic></sub> are the numbers of the <italic>L</italic> individuals selected randomly from <italic>C</italic><sup><italic>t</italic>−1</sup>; <inline-formula><mml:math id="M55" altimg="si0043.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">V</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold-italic">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold-italic">t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M56" altimg="si0044.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="italic">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M57" altimg="si0045.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> and <inline-formula><mml:math id="M58" altimg="si0046.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula> are the state values of feature <italic>j</italic> of individual <italic>i</italic> at time <italic>t</italic> and <italic>t</italic>−1 respectively.</p></list-item><list-item id="o0270"><label>(2)</label><p id="p0670">Differential Operation <italic>DE</italic>(<italic>C</italic><sup><italic>t</italic>−1</sup>, <italic>i</italic>, <italic>j</italic>). Randomly select 3 individuals from <italic>C</italic><sup><italic>t</italic>−1</sup>, let the state values of feature <italic>j</italic> of the 3 individuals are differentiated to produce a new compound value by use of differential operation, the new value is transferred to the corresponding feature <italic>j</italic> of individual <italic>i</italic> at time <italic>t</italic>, enabling individual <italic>i</italic> to have similar properties with the individuals in <italic>C</italic><sup><italic>t</italic>−1</sup>, namely<disp-formula id="eq0095"><mml:math id="M59" altimg="si0047.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>0.5</mml:mn><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0675">If <inline-formula><mml:math id="M60" altimg="si0048.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>, then 3 individuals <italic>i</italic><sub>1</sub>, <italic>i</italic><sub>2</sub> and <italic>i</italic><sub>3</sub> are randomly selected from <italic>C</italic><sup><italic>t</italic>−1</sup>, namely <inline-formula><mml:math id="M61" altimg="si0049.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>; if <inline-formula><mml:math id="M62" altimg="si0050.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo><</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>, then 3 individuals <italic>i</italic><sub>1</sub>, <italic>i</italic><sub>2</sub> and <italic>i</italic><sub>3</sub> are randomly selected from <italic>N</italic> individuals, namely <inline-formula><mml:math id="M63" altimg="si0051.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math></inline-formula>. Obviously, the differential operation is the core of DE <xref rid="bib43" ref-type="bibr">[43]</xref>, <xref rid="bib44" ref-type="bibr">[44]</xref>, <xref rid="bib45" ref-type="bibr">[45]</xref>.</p></list-item><list-item id="o0275"><label>(3)</label><p id="p0680">Expansion Operation <italic>EPN</italic>(<italic>C</italic><sup><italic>t</italic>−1</sup>, <italic>i</italic>, <italic>j</italic>). Randomly select an individual from <italic>C</italic><sup><italic>t</italic>−1</sup>, let the state value of feature <italic>j</italic> of the individual is expanded to produce a new compound value by use of expansion operation, the new value is transferred to the corresponding feature <italic>j</italic> of individual <italic>i</italic> at time <italic>t</italic>, enabling individual <italic>i</italic> to have similar properties with the individual in <italic>C</italic><sup><italic>t</italic>−1</sup>, namely<disp-formula id="eq0100"><mml:math id="M64" altimg="si0052.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">gBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:math></disp-formula>where if <inline-formula><mml:math id="M65" altimg="si0053.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, then individual <italic>k</italic> is randomly selected from <italic>C</italic><sup><italic>t</italic>−1</sup>, namely <inline-formula><mml:math id="M66" altimg="si0054.gif" overflow="scroll"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>; if <inline-formula><mml:math id="M67" altimg="si0055.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, then let <italic>k</italic>=<italic>i</italic><sub>gBest</sub>, <italic>i</italic><sub>gBest</sub> is the number of the best individual up to time <italic>t</italic>−1; <italic>r</italic><sub>1</sub>=<italic>Rnd</italic>(−1,1). Obviously, the expansion operation is the core of ABC <xref rid="bib46" ref-type="bibr">[46]</xref>, <xref rid="bib47" ref-type="bibr">[47]</xref>, <xref rid="bib48" ref-type="bibr">[48]</xref>, <xref rid="bib49" ref-type="bibr">[49]</xref>.</p></list-item><list-item id="o0280"><label>(4)</label><p id="p0685">Chevy Operation <italic>CHV</italic>(<italic>C</italic><sup><italic>t</italic>−1</sup>, <italic>i</italic>, <italic>j</italic>). Find the best individual <italic>i</italic><sub>lBest</sub> in <italic>C</italic><sup><italic>t</italic>−1</sup>, let the state value of feature <italic>j</italic> of the best individual is chevyed to produce a new compound value by use of chevy operation, the new value is transferred to the corresponding feature <italic>j</italic> of individual <italic>i</italic> at time <italic>t</italic>, enabling individual <italic>i</italic> to have similar properties with the individual in <italic>C</italic><sup><italic>t</italic>−1</sup>, namely<disp-formula id="eq0105"><mml:math id="M68" altimg="si0056.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">gBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">gBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="M69" altimg="si0057.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">lBest</mml:mi></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>; <italic>r</italic><sub>2</sub>=<italic>Rnd</italic>(0.4,0.6), <italic>r</italic><sub>3</sub>=1.3<italic>Rnd</italic>(0,1), <italic>r</italic><sub>4</sub>=1.8<italic>Rnd</italic>(0,1). Obviously, the chevy operation is the core of PSO <xref rid="bib35" ref-type="bibr">[35]</xref>, <xref rid="bib36" ref-type="bibr">[36]</xref>, <xref rid="bib37" ref-type="bibr">[37]</xref>, <xref rid="bib38" ref-type="bibr">[38]</xref>, <xref rid="bib39" ref-type="bibr">[39]</xref>, <xref rid="bib40" ref-type="bibr">[40]</xref>, <xref rid="bib41" ref-type="bibr">[41]</xref>.</p></list-item><list-item id="o0285"><label>(5)</label><p id="p0690">Reflection Operation <italic>RFL</italic>(<italic>C</italic><sup><italic>t</italic>−1</sup>, <italic>i</italic>, <italic>j</italic>). Randomly select 2 individuals from <italic>C</italic><sup><italic>t</italic>−1</sup>, let the state values of feature <italic>j</italic> of the 2 individual are reflected to produce a new compound value by use of reflection operation, the new value is transferred to the corresponding feature <italic>j</italic> of individual <italic>i</italic> at time <italic>t</italic>, enabling individual <italic>i</italic> to have similar properties with the individuals in <italic>C</italic><sup><italic>t</italic>−1</sup>, namely<disp-formula id="eq0110"><mml:math id="M70" altimg="si0058.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">gBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="1em"></mml:mspace><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0695">If |<italic>C</italic><sup><italic>t</italic>−1</sup>|>1, then <italic>i</italic><sub>1</sub> and <italic>i</italic><sub>2</sub> are randomly selected from <italic>C</italic><sup><italic>t</italic>−1</sup>, namely <inline-formula><mml:math id="M71" altimg="si0059.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>; if |<italic>C</italic><sup><italic>t</italic>−1</sup>|=1, then <italic>i</italic><sub>1</sub> is randomly selected from <italic>C</italic><sup><italic>t</italic>−1</sup>, namely <inline-formula><mml:math id="M72" altimg="si0060.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, <italic>i</italic><sub>2</sub>=<italic>i</italic><sub>gBest</sub>; if |<italic>C</italic><sup><italic>t</italic>−1</sup>|=0, then <inline-formula><mml:math id="M73" altimg="si0061.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">gBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>.</p></list-item><list-item id="o0290"><label>(6)</label><p id="p0700">Crossover Operation <italic>CRS</italic>(<italic>C</italic><sup><italic>t</italic>−1</sup>, <italic>i</italic>, <italic>j</italic>). Randomly select 2 individuals from <italic>C</italic><sup><italic>t</italic>−1</sup>, let the state values of feature <italic>j</italic> of the 2 individual mate to produce a new compound value by use of crossover operation, the new value is transferred to the corresponding feature <italic>j</italic> of individual <italic>i</italic> at time <italic>t</italic>, enabling individual <italic>i</italic> to have similar properties with the individuals in <italic>C</italic><sup><italic>t</italic>−1</sup>, namely</p></list-item></list><disp-formula id="eq0115"><mml:math id="M74" altimg="si0062.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mi>R</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo><</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mi>R</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo minsize="28pt">}</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">gBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mi>a</mml:mi><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mspace width=".25em"></mml:mspace></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">gBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mspace width=".25em"></mml:mspace></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0705">If |<italic>C</italic><sup><italic>t</italic>−1</sup>|>1, then <italic>i</italic><sub>1</sub> and <italic>i</italic><sub>2</sub> are randomly selected from <italic>C</italic><sup><italic>t</italic>−1</sup>, namely <inline-formula><mml:math id="M75" altimg="si0059.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>; if |<italic>C</italic><sup><italic>t</italic>−1</sup>|=1, then <italic>i</italic><sub>1</sub> is randomly selected from <italic>C</italic><sup><italic>t</italic>−1</sup>, namely <inline-formula><mml:math id="M76" altimg="si0060.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, <italic>i</italic><sub>2</sub>=<italic>i</italic><sub>gBest</sub>; <italic>i</italic>f |<italic>C</italic><sup><italic>t</italic>−1</sup>|=0, then <inline-formula><mml:math id="M77" altimg="si0061.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">gBest</mml:mi></mml:mrow></mml:msub><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>; <italic>a</italic>=<italic>Rnd</italic>(−0.5,1.5). Obviously, the crossover operation is the core of RC-GA <xref rid="bib32" ref-type="bibr">[32]</xref>.</p></sec><sec id="s0055"><label>2.4.2.2</label><title>Design of operators</title><p id="p0710">At time <italic>t</italic>−1, randomly select <italic>L</italic> individuals from the susceptible, exposed, infected, quarantined and recovered individuals respectively to form the following sets:</p><p id="p0715">The set of susceptible individuals:<disp-formula id="eq0120"><mml:math id="M78" altimg="si0063.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0720">The set of exposed individuals:<disp-formula id="eq0125"><mml:math id="M79" altimg="si0064.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0725">The set of infected individuals:<disp-formula id="eq0130"><mml:math id="M80" altimg="si0065.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0730">The set of quarantined individuals:<disp-formula id="eq0135"><mml:math id="M81" altimg="si0066.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0735">The set of recovered individuals:<disp-formula id="eq0140"><mml:math id="M82" altimg="si0067.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0740">At time <italic>t</italic>−1, randomly select <italic>L</italic> individuals whose IPI values are higher than that of current individual <italic>i</italic> from the susceptible, exposed, infected, quarantined and recovered individuals respectively to form the following sets:</p><p id="p0745">The set of strong susceptible individuals:<disp-formula id="eq0145"><mml:math id="M83" altimg="si0068.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PS</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PS</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PS</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PS</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0750">The set of strong exposed individuals:<disp-formula id="eq0150"><mml:math id="M84" altimg="si0069.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PE</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PE</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PE</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PE</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0755">The set of strong infected individuals:<disp-formula id="eq0155"><mml:math id="M85" altimg="si0070.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PI</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PI</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PI</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PI</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0760">The set of strong quarantined individuals:<disp-formula id="eq0160"><mml:math id="M86" altimg="si0071.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PQ</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PQ</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PQ</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PQ</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0765">The set of strong recovered individuals:<disp-formula id="eq0165"><mml:math id="M87" altimg="si0072.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PR</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PR</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PR</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PR</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">}</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0770">For each operator, its standard structure is as follows:<table-wrap position="float" id="t0105"><alt-text id="at0205">Table</alt-text><table frame="hsides" rules="groups"><tbody><tr><td><bold>Operator</bold><italic>X</italic></td></tr><tr><td><bold> IF</bold><italic>Rnd</italic>(0,1)<0.5 <bold>THEN</bold></td></tr><tr><td><bold> </bold>Execute operation <italic>A</italic>;</td></tr><tr><td><bold> ELSE</bold></td></tr><tr><td><bold> </bold>Execute operation <italic>B</italic>;</td></tr><tr><td><bold> END IF</bold></td></tr><tr><td><bold> END</bold></td></tr></tbody></table></table-wrap></p><p id="p0775">For each state transition listed in <xref rid="t0005" ref-type="table">Table 1</xref>, its corresponding operator and the match of operation <italic>A</italic> and <italic>B</italic> are shown in
<xref rid="t0010" ref-type="table">Table 2</xref>.<table-wrap position="float" id="t0010"><label>Table 2</label><caption><p>Design of operators in SEIQRA.</p></caption><alt-text id="at0110">Table 2</alt-text><table frame="hsides" rules="groups"><thead><tr><th>Type of state transition</th><th>Operator</th><th>Match of operation <italic>A</italic> and <italic>B</italic></th><th>Function of operation <italic>A</italic> and <italic>B</italic></th></tr></thead><tbody><tr><td>S→S</td><td>S–S</td><td>Average operation/differential operation</td><td><italic>AVG</italic>(<inline-formula><mml:math id="M88" altimg="si0001.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PS</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>DE</italic>(<inline-formula><mml:math id="M89" altimg="si0001.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PS</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>S→E</td><td>S–E</td><td>Average operation/expansion operation</td><td><italic>AVG</italic>(<inline-formula><mml:math id="M90" altimg="si0002.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>∪</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>EPN</italic>(<inline-formula><mml:math id="M91" altimg="si0002.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>∪</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>S→R</td><td>S–R</td><td>Differential operation/expansion operation</td><td><italic>DE</italic>(<inline-formula><mml:math id="M92" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>EPN</italic>(<inline-formula><mml:math id="M93" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>CHV</italic>(<inline-formula><mml:math id="M94" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>RFL</italic>(<inline-formula><mml:math id="M95" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>CRS</italic>(<inline-formula><mml:math id="M96" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>AVG</italic>(<inline-formula><mml:math id="M97" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>E→E</td><td>E–E</td><td>Differential operation/chevy operation</td><td><italic>DE</italic>(<inline-formula><mml:math id="M98" altimg="si0004.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PE</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>CHV</italic>(<inline-formula><mml:math id="M99" altimg="si0004.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PE</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>E→I</td><td>E–I</td><td>Expansion operation/chevy operation</td><td><italic>EPN</italic>(<inline-formula><mml:math id="M100" altimg="si0005.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>CHV</italic>(<inline-formula><mml:math id="M101" altimg="si0005.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>E→R</td><td>E–R</td><td>Expansion operation/reflection operation</td><td><italic>EPN</italic>(<inline-formula><mml:math id="M102" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>RFL</italic>(<inline-formula><mml:math id="M103" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>I→I</td><td>I–I</td><td>Chevy operation/reflection operation</td><td><italic>CHV</italic>(<inline-formula><mml:math id="M104" altimg="si0006.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PI</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>RFL</italic>(<inline-formula><mml:math id="M105" altimg="si0006.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PI</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>I→Q</td><td>I–Q</td><td>Chevy operation/crossover operation</td><td><italic>CHV</italic>(<inline-formula><mml:math id="M106" altimg="si0007.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>CRS</italic>(<inline-formula><mml:math id="M107" altimg="si0007.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>I→R</td><td>I–R</td><td>Reflection operation/ crossover operation</td><td><italic>RFL</italic>(<inline-formula><mml:math id="M108" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>CRS</italic>(<inline-formula><mml:math id="M109" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>Q→Q</td><td>Q–Q</td><td>Reflection operation/average operation</td><td><italic>RFL</italic>(<inline-formula><mml:math id="M110" altimg="si0008.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PQ</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>AVG</italic>(<inline-formula><mml:math id="M111" altimg="si0008.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PQ</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>Q→R</td><td>Q–R</td><td>Crossover operation /differential operation</td><td><italic>CRS</italic>(<inline-formula><mml:math id="M112" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>DE</italic>(<inline-formula><mml:math id="M113" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>R→R</td><td>R–R</td><td>Crossover operation/average operation</td><td><italic>CRS</italic>(<inline-formula><mml:math id="M114" altimg="si0009.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PR</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>AVG</italic>(<inline-formula><mml:math id="M115" altimg="si0009.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PR</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr><tr><td>R→S</td><td>R–S</td><td>Differential operation/crossover operation</td><td><italic>DE</italic>(<inline-formula><mml:math id="M116" altimg="si0010.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)/<italic>EPN</italic>(<inline-formula><mml:math id="M117" altimg="si0010.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>)</td></tr></tbody></table></table-wrap><table-wrap position="float" id="t0015"><label>Table 3</label><caption><p>The strategy of taking values of the parameters in SEIQRA.</p></caption><alt-text id="at0115">Table 3</alt-text><table frame="hsides" rules="groups"><thead><tr><th><bold>Name of parameter</bold></th><th><bold>Strategy of initialization</bold></th></tr></thead><tbody><tr><td>The maximum periods: <italic>G</italic></td><td><italic>G</italic> is used to prevent the iteration from dropping into infinite looping when the convergence condition is not satisfied, generally <italic>G</italic>=8000–300,000.</td></tr><tr><td>The lowest error: <italic>ε</italic></td><td><italic>ε</italic>>0, the smaller the <italic>ε</italic> is, the higher the precision of the found optimum solution will be, but the longer the calculation time will be, generally the scope of value is <italic>ε</italic>=10<sup>-5</sup>–10<sup>−</sup><sup>10</sup>.</td></tr><tr><td>The number of variables: <italic>n</italic></td><td><italic>n</italic> is defined by the actual optimization problem to be solved.</td></tr><tr><td>The number of work individuals: <italic>N</italic></td><td>Although <italic>N</italic> taking larger value can expand the search scope, the overall time complexity of the algorithm is proportional to <italic>N</italic>, therefore <italic>N</italic> cannot be made too large, the precision of the parameter value is without too high, simply on the basis of the specific optimization problem and the speed of your computer, and range of value for <italic>N</italic> is 100–2000.</td></tr><tr><td>The number of individuals that join in information exchange: <italic>L</italic></td><td><italic>L</italic>≥1, the parameter is not sensitive, generally <italic>L</italic>=6</td></tr><tr><td>The biggest probability that a feature of an individual is attacked by an infectious disease: <italic>E</italic><sub>0</sub></td><td>0<<italic>E</italic><sub>0</sub><1, at every step of evolution, only a part of features are allowed to be attacked by an infectious disease. if <italic>n</italic> is smaller, then <italic>E</italic><sub>0</sub>=1/200–1/20; if <italic>n</italic> is bigger, then <italic>E</italic><sub>0</sub>=1/500–1/1000</td></tr><tr><td>Latency period: <italic>τ</italic></td><td>Always set <italic>τ</italic>=1</td></tr><tr><td>Immunity period: <italic>λ</italic></td><td>Always set <italic>λ</italic>=1</td></tr><tr><td>Vaccination cycle: <italic>T</italic></td><td>Always set <italic>T</italic>=4</td></tr><tr><td>The upper limit of the SEIQR epidemic model’s parameters <italic>β</italic>, <italic>γ</italic><sub>1</sub>, <italic>γ</italic><sub>2</sub>, <italic>γ</italic><sub>3</sub>, <italic>p</italic>, <italic>δ</italic>: <italic>d</italic><sub>0</sub></td><td>0<<italic>d</italic><sub>0</sub><1, always set <italic>d</italic><sub>0</sub>=0.1</td></tr><tr><td>The initialization number of individuals: <italic>N</italic><sub>0</sub></td><td><italic>N</italic><sub>0</sub>≥10,000</td></tr><tr><td>The allowable maximum generations the global optimization solution keeps unchanged: <italic>t</italic><sub>max</sub></td><td><italic>t</italic><sub>max</sub>=10–100</td></tr><tr><td>The shrinking factor: <italic>ρ</italic></td><td>0<<italic>ρ</italic><1, generally <italic>ρ</italic>=0.8</td></tr><tr><td>The lower bound of interval shrinkage: <italic>LU</italic><sub>0</sub></td><td><italic>LU</italic><sub>0</sub>>0, generally <italic>LU</italic><sub>0</sub>=<italic>ε</italic></td></tr><tr><td>The ratio of individuals being replaced: <italic>r</italic></td><td><italic>r</italic>=0–1, generally <italic>r</italic>=0.5</td></tr><tr><td>The parameters of the SEIQR epidemic model: <italic>β</italic><sub>0</sub>, <italic>γ</italic><sub>1</sub>, <italic>γ</italic><sub>2</sub>, <italic>γ</italic><sub>3</sub>, <italic>p</italic>, <italic>δ</italic></td><td><italic>β</italic><sub>0</sub>, <italic>γ</italic><sub>1</sub>, <italic>γ</italic><sub>2</sub>, <italic>γ</italic><sub>3</sub>, <italic>p</italic> and <italic>δ</italic>=<italic>Rnd</italic>(0,<italic>d</italic><sub>0</sub>), <italic>d</italic><sub>0</sub>=0.1</td></tr></tbody></table></table-wrap></p><p id="p0780">By careful arrangement, each operation in <xref rid="t0010" ref-type="table">Table 2</xref> has the same probability 1/6 to be executed. The meaning of elements in <xref rid="t0010" ref-type="table">Table 2</xref> is explained as follows:</p><p id="p0785">For state transition S→S, its corresponding operator is S–S, the structure of the operator is as follows:<table-wrap position="float" id="t0110"><alt-text id="at0210">Table</alt-text><table frame="hsides" rules="groups"><tbody><tr><td><bold>Operator</bold> S–S</td></tr><tr><td><bold> IF</bold><italic>Rnd</italic>(0,1)<0.5 <bold>THEN</bold></td></tr><tr><td><bold> </bold>Execute average operation: <italic>AVG</italic>(<inline-formula><mml:math id="M118" altimg="si0001.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PS</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>);</td></tr><tr><td><bold> ELSE</bold></td></tr><tr><td><bold> </bold>Execute differential operation: <italic>DE</italic>(<inline-formula><mml:math id="M119" altimg="si0001.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">PS</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>);</td></tr><tr><td><bold> END IF</bold></td></tr><tr><td><bold> END</bold></td></tr></tbody></table></table-wrap></p><p id="p0790">For state transition S→R, its corresponding operator is S–R, the structure of the operator is as follows:<table-wrap position="float" id="t0115"><alt-text id="at0215">Table</alt-text><table frame="hsides" rules="groups"><tbody><tr><td><bold>Operator</bold> S–R</td></tr><tr><td><bold> IF</bold><italic>Rnd</italic>(0,1)<1/6 <bold>THEN</bold></td></tr><tr><td><bold> </bold>Execute average operation: <italic>DE</italic>(<inline-formula><mml:math id="M120" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>);</td></tr><tr><td><bold> ELSE IF</bold><italic>Rnd</italic>(0,1)<2/6 <bold>THEN</bold></td></tr><tr><td><bold> </bold>Execute differential operation: <italic>EPN</italic>(<inline-formula><mml:math id="M121" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>);</td></tr><tr><td><bold> ELSE IF</bold><italic>Rnd</italic>(0,1)<3/6 <bold>THEN</bold></td></tr><tr><td><bold> </bold>Execute differential operation: <italic>CHV</italic>(<inline-formula><mml:math id="M122" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>);</td></tr><tr><td><bold> ELSE IF</bold><italic>Rnd</italic>(0,1)<4/6 <bold>THEN</bold></td></tr><tr><td><bold> </bold>Execute differential operation: <italic>RFL</italic>(<inline-formula><mml:math id="M123" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>);</td></tr><tr><td><bold> ELSE IF</bold><italic>Rnd</italic>(0,1)<5/6 <bold>THEN</bold></td></tr><tr><td><bold> </bold>Execute differential operation: <italic>CRS</italic>(<inline-formula><mml:math id="M124" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>);</td></tr><tr><td><bold> ELSE</bold></td></tr><tr><td><bold> </bold>Execute differential operation: <italic>AVG</italic>(<inline-formula><mml:math id="M125" altimg="si0003.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>i</italic>, <italic>j</italic>);</td></tr><tr><td><bold> END IF</bold></td></tr><tr><td><bold>END</bold></td></tr></tbody></table></table-wrap></p></sec></sec><sec id="s0060"><label>2.4.3</label><title>Design of other operators</title><sec id="s0065"><label>2.4.3.1</label><title>Operator growth</title><p id="p0795">For an individual, compare its new generation with its current generation, replace the current generation with the new generation if the latter are better than the former; otherwise keep the former unchanged. For optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>, operator growth can be described as follows:<disp-formula id="eq0170"><label>(14)</label><mml:math id="M126" altimg="si0073.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">V</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mi mathvariant="normal">If</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:mi>I</mml:mi><mml:mi>P</mml:mi><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">V</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:mi>I</mml:mi><mml:mi>P</mml:mi><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mi mathvariant="normal">otherwise</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>…</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math></disp-formula></p><p id="p0800">Where function IPI(<inline-formula><mml:math id="M127" altimg="si0074.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">V</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>) and IPI(<inline-formula><mml:math id="M128" altimg="si0036.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>) are calculated according to formula <xref rid="eq0070" ref-type="disp-formula">(13)</xref>. The operator growth possesses of the property of “each-step-is-not-bad”, ensuring the algorithm to converge globally <xref rid="bib83" ref-type="bibr">[83]</xref>, <xref rid="bib86" ref-type="bibr">[86]</xref>, <xref rid="bib87" ref-type="bibr">[87]</xref>, <xref rid="bib88" ref-type="bibr">[88]</xref>.</p></sec><sec id="s0070"><label>2.4.3.2</label><title>Operator INIT</title><p id="p0805">Suppose that the dimensionality of the search space of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref> is <italic>n</italic>, the search interval for each variable is [<italic>l</italic><sub><italic>i</italic></sub>, <italic>u</italic><sub><italic>i</italic></sub>], <italic>i</italic>=1, 2, …, <italic>n</italic>, the constructing operator INIT of the orthogonal table L<sub><italic>N</italic></sub>(<italic>N</italic><sup><italic>n</italic></sup>) that uses the orthogonal Latin squares generating method <xref rid="bib82" ref-type="bibr">[82]</xref> to produce <italic>N</italic> initial solutions is as follows:</p><sec id="s0075"><label>2.4.3.2.1</label><title>Operator INIT</title><p id="p0810"><list list-type="simple" id="li0100"><list-item id="u0295"><p id="p0815">Step 1: Calculate discrete points for each variable <italic>y</italic><sub><italic>ij</italic></sub>:<disp-formula id="eq0175"><mml:math id="M129" altimg="si0075.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>…</mml:mi><mml:mn>..</mml:mn><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>;</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>…</mml:mi><mml:mn>..</mml:mn><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math></disp-formula></p></list-item><list-item id="u0300"><p id="p0820">Step 2: Use the orthogonal Latin squares generating method to produce initial solution <italic>x</italic><sub><italic>ij</italic></sub>:</p></list-item></list><disp-formula id="eq0180"><mml:math id="M130" altimg="si0076.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>…</mml:mi><mml:mn>..</mml:mn><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>…</mml:mi><mml:mn>..</mml:mn><mml:mo>.</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></disp-formula>where <italic>k</italic>=(<italic>i</italic>+<italic>j</italic>-1+<italic>m</italic>) mod <italic>N</italic>; if <italic>k</italic>=0, then <italic>k</italic>=<italic>N</italic>; if <italic>j</italic>=1 then <italic>m</italic>=0, else <italic>m</italic>=<italic>n</italic><sub>0</sub>; <italic>n</italic><sub>0</sub> is a random number selected randomly from [1, <italic>N</italic>], namely <italic>n</italic><sub>0</sub>=<italic>Rnd</italic>(1, <italic>N</italic>).</p><p id="p0825"><italic>N</italic> initial solutions <italic><bold>X</bold></italic><sub><italic>i</italic></sub>=(<italic>x</italic><sub><italic>i</italic>1</sub>, <italic>x</italic><sub><italic>i</italic>2</sub>, …, <italic>x</italic><sub><italic>in</italic></sub>), <italic>i</italic>=1, 2, …, <italic>N</italic>, determined by operator INIT have good balance dispersion and neat comparability.</p></sec></sec><sec id="s0080"><label>2.4.3.3</label><title>Operator REINIT</title><p id="p0830">When an individual makes search, if the current position of the individual keeps unchanged for a long time, we think that the individual has stayed into sticky state. Two situations may cause sticky state: one is that an individual drops into a local optimum solution; the other is that current position of an individual cannot be updated by the information thrown out by other individuals.</p><p id="p0835">When all individuals drop into stick state, search will stop automatically; global optimization solution will keep unchanged forever. On the other way, when some problems have high condition numbers, enhancing precision of global optima is always difficult when individuals easily drop into sticky state. At this time, a reinitialization–redistribution of individuals is necessary.</p><p id="p0840">Suppose from time <italic>t</italic><sub>0</sub> to time <italic>t</italic>, global optimization solution keeps unchanged, we say that evolution has dropped into stick state. Reinitialization–redistribution of individuals will make search to escape from stick state or get high precision global optima. The operator of reinitialization–redistribution of individuals (Operator REINIT) is designed as follows:</p><sec id="s0085"><label>2.4.3.3.1</label><title>Operator REINIT</title><p id="p0845">If <italic>t</italic>−<italic>t</italic><sub>0</sub>><italic>t</italic><sub>max</sub>, where <italic>t</italic><sub>max</sub> is the allowable maximum periods that global optimization solution keeps unchanged, then execute the following operations:<list list-type="simple" id="li0105"><list-item id="u0305"><p id="p0850">Step 1: (Decrease the interval of each decision variable): Decrease the interval <italic>LU</italic><sub><italic>i</italic></sub> of decision variable <italic>x</italic><sub><italic>i</italic></sub> (<italic>i</italic>=1, 2, …, <italic>n</italic>) by <italic>LU</italic><sub><italic>i</italic></sub>=<italic>ρLU</italic><sub><italic>i</italic></sub>, where <italic>ρ</italic> is the shrinking factor of the interval of decision variable <italic>x</italic><sub><italic>i</italic></sub>, 0<<italic>ρ</italic><1; <italic>LU</italic><sub><italic>i</italic></sub> is its change limit. At start, <inline-formula><mml:math id="M131" altimg="si0077.gif" overflow="scroll"><mml:mrow><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:math></inline-formula>; afterwards, if <inline-formula><mml:math id="M132" altimg="si0078.gif" overflow="scroll"><mml:mrow><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo><</mml:mo><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, then <inline-formula><mml:math id="M133" altimg="si0077.gif" overflow="scroll"><mml:mrow><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:math></inline-formula>, <italic>LU</italic><sub>0</sub> is the lower bound of interval shrinkage; the smaller <italic>LU</italic><sub>0</sub> is, the narrower the scope of search is, and the higher the precision of optimum solutions will be, but the scope of search may not cover global optimum solutions.</p></list-item><list-item id="u0310"><p id="p0855">Step 2: (Operation reinitialization: Reinitialize a very big number of individuals): Take each variable <inline-formula><mml:math id="M134" altimg="si0079.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula> (<italic>i</italic>=1, 2, …, <italic>n</italic>) of the current global solution <inline-formula><mml:math id="M135" altimg="si0080.gif" overflow="scroll"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mo mathvariant="bold-italic">*</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> as center, namely for <inline-formula><mml:math id="M136" altimg="si0079.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula> (<italic>i</italic>=1, 2, …, <italic>n</italic>), its change extent is within <inline-formula><mml:math id="M137" altimg="si0081.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow></mml:math></inline-formula>, then take <inline-formula><mml:math id="M138" altimg="si0081.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow></mml:math></inline-formula> as the interval of variable <italic>x</italic><sub><italic>i</italic></sub> of individual <italic><bold>X</bold></italic><sub><italic>i</italic></sub>, use the above-mentioned operator INIT to initialize all <italic>N</italic><sub>0</sub> individuals; <italic>N</italic><sub>0</sub> can assign a very larger number, for example, <italic>N</italic><sub>0</sub>=2000 or even much bigger number.</p></list-item><list-item id="u0315"><p id="p0860">Step 3: (Operation redistribution: Redistribute a small number of individuals): Find the new best individual <inline-formula><mml:math id="M139" altimg="si0082.gif" overflow="scroll"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> from the just initialized <italic>N</italic><sub>0</sub> Individuals; then redistribute <italic>N</italic> individuals within <inline-formula><mml:math id="M140" altimg="si0083.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow></mml:math></inline-formula> by the above-mentioned algorithm INIT, where <italic>N</italic> is the number of work individuals used by SEIQRA, <italic>N</italic>«<italic>N</italic><sub>0</sub>, for example <italic>N</italic>=50–200.</p></list-item><list-item id="u0320"><p id="p0865">Step 4: (Replace the individuals with lower IPI values according to ratio <italic>r</italic>): Replace the current individuals whose IPI values are very low with the newly-generated individuals whose IPI values are very high, the ratio of replacement is <italic>r</italic>. For example, when <italic>r</italic>=0.5, it means that 50% of the current individuals, whose IPI values are very low among all the current individuals, are replaced with 50% of the newly-generated individuals, whose IPI values are very high among all the newly-generated individuals.</p></list-item></list></p><p id="p0870">Reinitialization of a very big number of individuals is used to find the approximate position of global optima; in the vicinity of the approximate position of global optima, redistribution of a small number of work individuals is easy to find the global optima quickly and accurately by the algorithm SEIQRA.</p><p id="p0875">Conclusively, operation reinitialization is good at enhancing precision of an optimum solution, while operation redistribution is good at breaking sticky state of an individual.</p></sec></sec></sec></sec><sec id="s0090"><label>2.5</label><title>Construction method of SEIQRA</title><p id="p0880">Based on the logical structure of Algorithm SEIQRA mentioned in <xref rid="s0025" ref-type="sec">Section 2.3</xref>, we can design a simple structure of SEIQRA as follows:</p><p id="p0885"><bold>Algorithm SEIQRA</bold><table-wrap position="float" id="t0120"><alt-text id="at0220">Table</alt-text><table frame="hsides" rules="groups"><tbody><tr><td>Initialize all the parameters SEIQRA involves as specified in <xref rid="t0015" ref-type="table">Table 3</xref>;</td></tr><tr><td><bold>FOR</bold><italic>t</italic>=1 <bold>TO</bold><italic>G</italic> //Where <italic>t</italic> is used to count generations or periods of evolution; <italic>G</italic> is the maximum generations or periods of evolution.</td></tr><tr><td><bold> FOR</bold> each individual <italic>i</italic></td></tr><tr><td><list list-type="simple" id="li0110"><list-item id="u0325"><p id="p0890"><bold>Step 1:</bold> Calculate the state transition probability of the individual by state transition Eqs. <xref rid="eq0050" ref-type="disp-formula">(10)</xref>, <xref rid="eq0055" ref-type="disp-formula">(11)</xref>;</p></list-item><list-item id="u0330"><p id="p0895"><bold>Step 2:</bold> Determine which state <italic>s</italic> the individual may transfer to at time <italic>t</italic> based on its state transition probability computed by Step1;</p></list-item><list-item id="u0335"><p id="p0900"><bold>Step 3:</bold> If state transition from time <italic>t</italic>−1 to time <italic>t</italic> is legal, then carry out information exchange by transferring some features of some randomly-selected individuals at time <italic>t</italic>−1 to the corresponding features of the individual at time <italic>t</italic>; else the state of the individual keeps unchanged.</p></list-item><list-item id="u0340"><p id="p0905"><bold>Step 4:</bold> Compute the IPI value of the individual by formula (13), if its IPI value is improved when comparing to that at time <italic>t</italic>−1, then the individual evolves to the new state <italic>s</italic> at time <italic>t</italic>; else the individual still stay at its old state at time <italic>t</italic>−1.</p></list-item></list></td></tr><tr><td><bold> END FOR</bold></td></tr><tr><td>Operator REINIT is activated at certain frequency;</td></tr><tr><td><bold>END FOR</bold></td></tr></tbody></table></table-wrap></p><p id="p0910">The psuedo-code of SEIQRA can be seen in <xref rid="s0155" ref-type="sec">Appendix A</xref>. The strategy of taking values of the parameters in SEIQRA is shown in <xref rid="t0015" ref-type="table">Table 3</xref>.</p></sec><sec id="s0095"><label>2.6</label><title>Time complexity of SEIQRA</title><p id="p0915">The time complexity of SEIQRA is evaluated as shown in
<xref rid="t0020" ref-type="table">Table 4</xref>, it has relations with the maximum evolution times <italic>G</italic>, the number of individuals <italic>N</italic>, the number of variables <italic>n</italic>, the time complexity of operators: S–S, S–E, S–R, E–E, E–I, E–R, I–I, I–Q, I–R, Q–Q, Q–R, R–R, R–S, Growth and REINIT, operations: <italic>AVG</italic>, <italic>DE</italic>, <italic>EPN</italic>, <italic>CHV</italic>, <italic>RFL</italic>, <italic>CRS</italic> and other auxiliary operations.<table-wrap position="float" id="t0020"><label>Table 4</label><caption><p>The time complexity of SEIQRA.</p></caption><alt-text id="at0120">Table 4</alt-text><table frame="hsides" rules="groups"><thead><tr><th><bold>Operation</bold></th><th><bold>Average time complexity</bold></th><th><bold>Maximum loop times</bold></th></tr></thead><tbody><tr><td>Computation of <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>E</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>Q</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>), <italic>SEIQR</italic><sub><italic>i</italic></sub>(<italic>t</italic>)</td><td><italic>O</italic>(12)</td><td>(<italic>G</italic>+<italic>N</italic>+9)<italic>N</italic></td></tr><tr><td>Operators: S–S,S–E,E–E,E–I,E–R,I–I,I–Q,I–R,Q–Q,Q–R,R–R,R–S</td><td><italic>O</italic>(3<italic>nE</italic><sub>0</sub>/13)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>S–R</td><td><italic>O</italic>(6<italic>nE</italic><sub>0</sub>/13)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Operation <italic>AVG</italic></td><td><italic>O</italic>((5<italic>N</italic>+5<italic>L</italic>+3)<italic>nE</italic><sub>0</sub>/6)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Operation <italic>DE</italic></td><td><italic>O</italic>((5<italic>N</italic>+5<italic>L</italic>+3)<italic>nE</italic><sub>0</sub>/6)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Operation <italic>EPN</italic></td><td><italic>O</italic>((5<italic>N</italic>+5<italic>L</italic>+3)<italic>nE</italic><sub>0</sub>/6)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Operation <italic>CHV</italic></td><td><italic>O</italic>((5<italic>N</italic>+5<italic>L</italic>+3)<italic>nE</italic><sub>0</sub>/6)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Operation <italic>RFL</italic></td><td><italic>O</italic>((5<italic>N</italic>+5<italic>L</italic>+3)<italic>nE</italic><sub>0</sub>/6)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Operation <italic>CRS</italic></td><td><italic>O</italic>((5<italic>N</italic>+5<italic>L</italic>+3)<italic>nE</italic><sub>0</sub>/6)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Maintenance of states</td><td><italic>O</italic>((1–7<italic>E</italic><sub>0</sub>/10)<italic>n</italic>)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Calculation of objective function</td><td><italic>O</italic>(<italic>n</italic>)–<italic>O</italic>(<italic>n</italic><sup>2</sup>)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Operator growth</td><td><italic>O</italic>(3<italic>n</italic>)</td><td>(<italic>G</italic>+<italic>N</italic>+9)(<italic>N</italic>+3)</td></tr><tr><td>Operator REINIT</td><td><italic>O</italic>(3<italic>n</italic>+7(<italic>n</italic>+1)<italic>N</italic>+<italic>n</italic><sup>2</sup><italic>N</italic>)</td><td><italic>G</italic>+<italic>N</italic>+9</td></tr><tr><td>Output of results</td><td><italic>O</italic>(<italic>n</italic>)</td><td>1</td></tr></tbody></table></table-wrap></p></sec></sec><sec id="s0100"><label>3</label><title>Performance study and parameter selection of SEIQRA</title><p id="p0920">The computer used to test is a Toshiba notebook computer, its CPU is Intel Core™ I5, M520 @ 2.40 GHz, its memory size is 4 GB, its OS is Windows 7.</p><sec id="s0105"><label>3.1</label><title>Performance study and parameter selection of SEIQRA</title><p id="p0925">Now we use the Bump function <italic>F</italic><sub>0</sub>(<italic><bold>X</bold></italic>) <xref rid="bib35" ref-type="bibr">[35]</xref> and Michalewicz function <italic>F</italic><sub>1</sub>(<italic><bold>X</bold></italic>) <xref rid="bib35" ref-type="bibr">[35]</xref> as an example to test the performance of SEIQRA, and determine the setting of some key parameters. The mathematical models of <italic>F</italic><sub>0</sub>(<italic><bold>X</bold></italic>) and <italic>F</italic><sub>1</sub>(<italic><bold>X</bold></italic>) are as follows:</p><p id="p0930">Bump function:<disp-formula id="eq0185"><mml:math id="M141" altimg="si0084.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="true">{</mml:mo><mml:mrow><mml:mtable><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>min</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="true">|</mml:mo><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:msup><mml:mrow><mml:mi>cos</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:msup><mml:mrow><mml:mi>cos</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi mathvariant="normal">s</mml:mi><mml:mi mathvariant="normal">.t</mml:mi><mml:mo>.</mml:mo><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>≥</mml:mo><mml:mn>0.75</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>≤</mml:mo><mml:mn>7.5</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"></mml:mspace><mml:mspace width="1em"></mml:mspace><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:math></disp-formula></p><p id="p0935">Michalewicz function:<disp-formula id="eq0190"><mml:math id="M142" altimg="si0085.gif" overflow="scroll"><mml:mrow><mml:mi>min</mml:mi><mml:mspace width=".25em"></mml:mspace><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:mi>sin</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:mi>sin</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mi mathvariant="normal">π</mml:mi></mml:mfrac></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>15</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula></p><p id="p0940">The two functions are carefully selected because they have such properties as shown in
<xref rid="t0025" ref-type="table">Table 5</xref>.<table-wrap position="float" id="t0025"><label>Table 5</label><caption><p>Properties of the Bump function <italic>F</italic><sub>0</sub>(<italic><bold>X</bold></italic>) and Michalewicz function <italic>F</italic><sub>1</sub>(<italic><bold>X</bold></italic>).</p></caption><alt-text id="at0125">Table 5</alt-text><table frame="hsides" rules="groups"><thead><tr><th><bold>Property</bold></th><th><bold>Bump function <italic>F</italic><sub>0</sub>(</bold><italic><bold>X</bold></italic><bold>)</bold></th><th><bold>Michalewicz function <italic>F</italic></bold><sub><bold>1</bold></sub><bold>(</bold><italic><bold>X</bold></italic><bold>)</bold></th></tr></thead><tbody><tr><td>Optimization problem with constraints?</td><td>Yes</td><td>No</td></tr><tr><td>Multimodal?</td><td>Yes</td><td>Yes</td></tr><tr><td>Rotated and Shifted?</td><td>Yes, the original point (0,0,…,0) is not the global optimum solution.</td><td>Yes, the original point (0,0,…,0) is not the global optimum solution.</td></tr><tr><td>Domain of <italic><bold>X</bold></italic></td><td><italic><bold>X</bold></italic><inline-formula><mml:math id="M143" altimg="si0011.gif" overflow="scroll"><mml:mo>∈</mml:mo></mml:math></inline-formula>[0, π]<sup><italic>n</italic></sup></td><td><italic><bold>X</bold></italic><inline-formula><mml:math id="M144" altimg="si0011.gif" overflow="scroll"><mml:mo>∈</mml:mo></mml:math></inline-formula>[0, 10]<sup><italic>n</italic></sup></td></tr><tr><td>Non-separable?</td><td>Yes</td><td>No</td></tr><tr><td>Global optimum is known?</td><td>No, the global optimum objective function value varies with dimensionality <italic>n</italic></td><td>No, the global optimum objective function value varies with dimensionality <italic>n</italic></td></tr><tr><td>The function contains a cyclic function?</td><td>Yes</td><td>Yes</td></tr><tr><td>It can be solved easily?</td><td>No</td><td>No</td></tr></tbody></table></table-wrap></p><p id="p0945">The reason we select the Bump and Michalewicz function to determine the setting of parameters in SEIQRA is that properties of the two functions have a typical representative in engineering applications, and hence the generalized configuration of parameters in SEIQRA can be approximately found by the two functions.</p><p id="p0950"><xref rid="f0025" ref-type="fig">Fig. 5</xref> is the graphs of the Bump and Michalewicz function when <italic>n</italic>=2. When <italic>n</italic> is greater, the Bump and Michalewicz function are very difficult to optimize.<fig id="f0025"><label>Fig. 5</label><caption><p>The graphs of the Bump and Michalewicz function when <italic>n</italic>=2. (a) The Bump function; (b) Michalewicz function.</p></caption><alt-text id="at0025">Fig. 5</alt-text><graphic xlink:href="gr5"></graphic></fig></p><p id="p0955">To make clear that the setting of parameters in SEIQRA gives influence on its evolving process when it solves the Bump and Michalewicz function, a deep investigation is made into the important parameters such as <italic>L</italic>, <italic>E</italic><sub>0</sub>, <italic>N</italic>, <italic>T</italic>, <italic>τ</italic> and <italic>λ</italic>.</p><p id="p0960"><xref rid="t0030" ref-type="table">Table 6</xref> describes the relationship among average best objective function value, parameter <italic>L</italic> and average CPU time;
<xref rid="f0030" ref-type="fig">Fig. 6</xref> shows that relation of objective function value with CPU time when <italic>L</italic> takes different values. <xref rid="t0030" ref-type="table">Table 6</xref> and <xref rid="f0030" ref-type="fig">Fig. 6</xref> show that when <italic>L</italic> is valued differently, for the Bump function there is little difference in CPU time, for the Michalewicz function there is great difference in CPU time for some <italic>L</italic>s; but when <italic>L</italic>>1, there is little difference in precision of average optimum objective function values. Therefore, for SEIQRA, <italic>L</italic>=2–10 is more appropriate, we suggest <italic>L</italic>=6 is the best setting based on comprehensive consideration of precision of objective function value and CPU time.<table-wrap position="float" id="t0030"><label>Table 6</label><caption><p>Effect of parameter <italic>L</italic> when <italic>n</italic>=50, <italic>E</italic><sub>0</sub>=0.06, <italic>N</italic>=200, <italic>N</italic><sub>0</sub>=2000, <italic>T</italic>=4, <italic>λ</italic>=1, <italic>τ</italic>=1, <italic>G</italic>=1.0E+5 and run times=20; REINIT is prohibited.</p></caption><alt-text id="at0130">Table 6</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2"><italic>L</italic></th><th colspan="2">The Bump function<hr></hr></th><th colspan="2">The Michalewicz function<hr></hr></th></tr><tr><th>Average best objective function value</th><th>Average CPU time/s</th><th>Average best objective function value</th><th>Average CPU time/s</th></tr></thead><tbody><tr><td>1</td><td align="char">−0.827953281950753</td><td align="char">1875</td><td align="char">−49.5957355933971</td><td align="char">502</td></tr><tr><td>2</td><td align="char">−0.831606013670455</td><td align="char">1922</td><td align="char">−49.6091287496027</td><td align="char">226</td></tr><tr><td>3</td><td align="char">−0.831606013670455</td><td align="char">1922</td><td align="char">−49.6154517305820</td><td align="char">1166</td></tr><tr><td>4</td><td align="char">−0.832688629582492</td><td align="char">1884</td><td align="char">−49.6069852705723</td><td align="char">307</td></tr><tr><td>5</td><td align="char">−0.832970269937945</td><td align="char">1981</td><td align="char">−49.6148346625030</td><td align="char">264</td></tr><tr><td><bold>6</bold></td><td align="char">−<bold>0.834826053656805</bold></td><td align="char"><bold>1922</bold></td><td align="char">−<bold>49.6161619228999</bold></td><td align="char"><bold>855</bold></td></tr><tr><td>7</td><td align="char">−0.833787056092329</td><td align="char">1908</td><td align="char">−49.6101059078490</td><td align="char">514</td></tr><tr><td>8</td><td align="char">−0.833787056092329</td><td align="char">1908</td><td align="char">−49.6227683161410</td><td align="char">1224</td></tr><tr><td>9</td><td align="char">−0.834897470880660</td><td align="char">1904</td><td align="char">−49.6226863498109</td><td align="char">262</td></tr><tr><td>10</td><td align="char">−0.834647833760122</td><td align="char">1917</td><td align="char">−49.6176302033132</td><td align="char">300</td></tr></tbody></table></table-wrap><fig id="f0030"><label>Fig. 6</label><caption><p>Relation of objective function value with CPU time when <italic>L</italic> takes different values. (a) The Bump function; (b) The Michalewicz function.</p></caption><alt-text id="at0030">Fig. 6</alt-text><graphic xlink:href="gr6"></graphic></fig></p><p id="p0965"><xref rid="t0035" ref-type="table">Table 7</xref> describes the relationship among parameter <italic>E</italic><sub>0</sub>, average best objective function value and average CPU time. It shows that for the Bump function, when <italic>E</italic><sub>0</sub>=0.008–0.2, the precision is relatively high, but CPU time increases moderately; when <italic>E</italic><sub>0</sub>>0.2, CPU time varies greatly, but the precision decreases greatly also; especially when <italic>E</italic><sub>0</sub>=1, it is unable to obtain the optimum solution; for the Michalewicz function, when <italic>E</italic><sub>0</sub>=0.001–0.1, the precision is relatively high, but CPU time varies greatly; when <italic>E</italic><sub>0</sub>>0.1, CPU time varies greatly, but the precision decreases greatly also; especially when <italic>E</italic><sub>0</sub>=1, it is unable to obtain the optimum solution. Therefore, when <italic>E</italic><sub>0</sub>=0.008–0.1, SEIQRA performs best. When <italic>E</italic><sub>0</sub>=0.008–0.1, it means that there are at most only 0.8%–10% variables to be involved in computation, but the CPU time consumed by SEIQRA is lower than that when <italic>E</italic><sub>0</sub> takes other values, SEIQRA can still obtain the optimum solution with relatively high precision.<table-wrap position="float" id="t0035"><label>Table 7</label><caption><p>Effect of parameter <italic>E</italic><sub>0</sub> when <italic>n</italic>=50, <italic>L</italic>=6, <italic>N</italic>=200, <italic>N</italic><sub>0</sub>=2000, <italic>G</italic>=1.0E+5, <italic>T</italic>=4, <italic>λ</italic>=1, <italic>τ</italic>=1, and run times=20; REINT is prohibited.</p></caption><alt-text id="at0135">Table 7</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2"><italic>E</italic><sub>0</sub></th><th colspan="2">The Bump function<hr></hr></th><th colspan="2">The Michalewicz function<hr></hr></th></tr><tr><th>Average best objective function value</th><th>Average CPU time/s</th><th>Average best objective function value</th><th>Average CPU time/s</th></tr></thead><tbody><tr><td>0.001</td><td align="char">−0.687779446604573</td><td>898</td><td align="char">−49.6155521313138</td><td>707</td></tr><tr><td>0.002</td><td align="char">−0.718737770254323</td><td>904</td><td align="char">−49.6228212529016</td><td>712</td></tr><tr><td>0.004</td><td align="char">−0.760642273836502</td><td>1091</td><td align="char">−49.6228212529016</td><td>428</td></tr><tr><td>0.006</td><td align="char">−0.790099822986605</td><td>1095</td><td align="char">−49.6228212529016</td><td>488</td></tr><tr><td>0.008</td><td align="char">−0.809000316375438</td><td>1161</td><td align="char">−49.6228212529016</td><td>840</td></tr><tr><td>0.01</td><td align="char">−0.815311674554719</td><td>1182</td><td align="char">−49.6228212529016</td><td>346</td></tr><tr><td>0.02</td><td align="char">−0.824890036082204</td><td>1349</td><td align="char">−49.6227922678471</td><td>289</td></tr><tr><td>0.04</td><td align="char">−0.830856211441936</td><td>1618</td><td align="char">−49.6228212529017</td><td>272</td></tr><tr><td><bold>0.06</bold></td><td align="char">−<bold>0.833803362869570</bold></td><td><bold>1908</bold></td><td align="char">−<bold>49.6228122188572</bold></td><td><bold>217</bold></td></tr><tr><td>0.08</td><td align="char">−0.831795692900186</td><td>2149</td><td align="char">−49.6132970801741</td><td>217</td></tr><tr><td>0.1</td><td align="char">−0.828502259790528</td><td>2332</td><td align="char">−49.589768503263</td><td>1719</td></tr><tr><td>0.2</td><td align="char">−0.818150482452084</td><td>3697</td><td align="char">−48.8793535376783</td><td>396</td></tr><tr><td>0.4</td><td align="char">−0.697848195144567</td><td>6495</td><td align="char">−47.7219912917489</td><td>1350</td></tr><tr><td>0.6</td><td align="char">−0.741233476291840</td><td>3921</td><td align="char">−47.583935279112</td><td>1412</td></tr><tr><td>0.8</td><td align="char">−0.687309180877045</td><td>1937</td><td align="char">−47.0371451853083</td><td>1234</td></tr><tr><td>1</td><td align="char">−0.666689009262077</td><td>1413</td><td align="char">−46.4085466749631</td><td>3421</td></tr></tbody></table></table-wrap></p><p id="p0970"><xref rid="f0035" ref-type="fig">Fig. 7</xref>, which is logarithmic scales on vertical and horizontal axis, shows the relation of IPI (objective function value can be transformed into IPI value with formula <xref rid="eq0070" ref-type="disp-formula">(13)</xref>) with CPU time when <italic>E</italic><sub>0</sub> takes different values. From <xref rid="f0035" ref-type="fig">Fig. 7</xref> we can see that when <italic>E</italic><sub>0</sub> increases from 0.0001 to 1, the ascending rate of IPI value, which corresponds to the descending rate of objective function value, increases greatly, but the precision of objective function value decreases moderately; from which we can find that the most suitable value for <italic>E</italic><sub>0</sub> is <italic>E</italic><sub>0</sub>=0.06.<fig id="f0035"><label>Fig. 7</label><caption><p>Relation of IPI value with CPU time when <italic>E</italic><sub>0</sub> takes different values. (a) The Bump function; (b) The Michalewicz function.</p></caption><alt-text id="at0035">Fig. 7</alt-text><graphic xlink:href="gr7"></graphic></fig></p><p id="p0975"><xref rid="t0040" ref-type="table">Table 8</xref> describes the relation among parameter <italic>N</italic>, average best objective function value and average CPU time consumed by SEIQRA;
<xref rid="f0040" ref-type="fig">Fig. 8</xref> shows the relation of objective function value with CPU time when <italic>N</italic> takes different values. From <xref rid="t0040" ref-type="table">Table 8</xref> and <xref rid="f0040" ref-type="fig">Fig. 8</xref>, we can see that for average objective function value, when <italic>N</italic> increases, the precision of the average objective function value increases a little; for average CPU time, when <italic>N</italic> increases, the average CPU time increases greatly. Therefore, <italic>N</italic>=150–350 is the most suitable setting, we suggest <italic>N</italic>=200 is the suitable setting based on comprehensive consideration of precision of objective function value and CPU time.<table-wrap position="float" id="t0040"><label>Table 8</label><caption><p>Relation of average best objective function value with average CPU time when <italic>N</italic> takes different values, and <italic>n</italic>=50, <italic>L</italic>=6, <italic>E</italic><sub>0</sub>=0.06, <italic>N</italic><sub>0</sub>=2000, <italic>G</italic>=1.0E+5, <italic>T</italic>=4, <italic>τ</italic>=1, <italic>λ</italic>=1 and run times=20; REINT is prohibited.</p></caption><alt-text id="at0140">Table 8</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2"><italic>N</italic></th><th colspan="2">The Bump function<hr></hr></th><th colspan="2">The Michalewicz function<hr></hr></th></tr><tr><th>Average best objective function value</th><th>Average CPU time/s</th><th>Average best objective function value</th><th>Average CPU time/s</th></tr></thead><tbody><tr><td>50</td><td align="char">−0.823492909251594</td><td>341</td><td align="char">−49.4590588403967</td><td>37</td></tr><tr><td>100</td><td align="char">−0.831597657955816</td><td>756</td><td align="char">−49.6085383998470</td><td>57</td></tr><tr><td>150</td><td align="char">−0.834587575202624</td><td>1287</td><td align="char">−49.6041680740384</td><td>128</td></tr><tr><td><bold>200</bold></td><td align="char">−<bold>0.834904200815997</bold></td><td><bold>1889</bold></td><td align="char">−<bold>49.6207649262707</bold></td><td><bold>115</bold></td></tr><tr><td>250</td><td align="char">−0.834368405804599</td><td>2464</td><td align="char">−49.6193576435355</td><td>186</td></tr><tr><td>300</td><td align="char">−0.834107295117368</td><td>4246</td><td align="char">−49.6174447715844</td><td>238</td></tr><tr><td>350</td><td align="char">−0.833079266813661</td><td>4228</td><td align="char">−49.6227683161410</td><td>260</td></tr><tr><td>400</td><td align="char">−0.831765847386432</td><td>5358</td><td align="char">−49.6221674761665</td><td>414</td></tr><tr><td>450</td><td align="char">−0.832726350044014</td><td>6453</td><td align="char">−49.6228212529016</td><td>608</td></tr><tr><td>500</td><td align="char">−0.831655895034941</td><td>7611</td><td align="char">−49.6215008404875</td><td>1113</td></tr></tbody></table></table-wrap><fig id="f0040"><label>Fig. 8</label><caption><p>Relation of objective function value with CPU time when <italic>N</italic> takes different values. (a) The Bump function; (b) The Michalewicz function.</p></caption><alt-text id="at0040">Fig. 8</alt-text><graphic xlink:href="gr8"></graphic></fig></p><p id="p0980"><xref rid="t0045" ref-type="table">Table 9</xref>, <xref rid="t0050" ref-type="table">Table 10</xref>, <xref rid="t0055" ref-type="table">Table 11</xref> show relation of average objective function value with average CPU time under different vaccination cycles, latency periods and immunity periods respectively;
<xref rid="f0045" ref-type="fig">Fig. 9</xref>, <xref rid="f0050" ref-type="fig">Fig. 10</xref>, <xref rid="f0055" ref-type="fig">Fig. 11</xref> show relation of objective function value with CPU time under different vaccination cycles, latency periods and immunity periods respectively. From <xref rid="t0045" ref-type="table">Table 9</xref>, <xref rid="t0050" ref-type="table">Table 10</xref>, <xref rid="t0055" ref-type="table">Table 11</xref> and <xref rid="f0045" ref-type="fig">Fig. 9</xref>, <xref rid="f0050" ref-type="fig">Fig. 10</xref>, <xref rid="f0055" ref-type="fig">Fig. 11</xref> we can see that for different vaccination cycles, latency periods and immunity periods, average objective function value and average CPU Time do not change greatly. The reason is that the 6 operations are executed equi-probably, the time consumption is almost the same for different vaccination cycles, latency periods and immunity periods. We suggest <italic>T</italic>=4, <italic>τ</italic>=1 and <italic>λ</italic>=1 are the best setting based on comprehensive consideration of precision of objective function value and CPU time.<table-wrap position="float" id="t0045"><label>Table 9</label><caption><p>Relation of average best objective function value with average CPU time under different vaccination cycles <italic>T</italic> when <italic>n</italic>=50, <italic>E</italic><sub>0</sub>=0.06, <italic>N</italic>=200, <italic>N</italic><sub>0</sub>=2000, <italic>L</italic>=6, <italic>τ</italic>=1, <italic>λ</italic>=1, <italic>G</italic>=80,000 and run times=20; REINT is prohibited.</p></caption><alt-text id="at0145">Table 9</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2"><italic>T</italic></th><th colspan="2">The Bump function<hr></hr></th><th colspan="2">The Michalewicz function<hr></hr></th></tr><tr><th>Average best objective function value</th><th>Average CPU time/s</th><th>Average best objective function value</th><th>Average CPU time/s</th></tr></thead><tbody><tr><td>2</td><td align="char">−0.832266925553587</td><td>375</td><td align="char">−49.6169127665783</td><td>188</td></tr><tr><td><bold>4</bold></td><td align="char">−<bold>0.829913555844357</bold></td><td><bold>372</bold></td><td align="char">−<bold>49.6220370470490</bold></td><td><bold>226</bold></td></tr><tr><td>6</td><td align="char">−0.832552777450192</td><td>373</td><td align="char">−49.6212884223517</td><td>176</td></tr><tr><td>8</td><td align="char">−0.831113846663504</td><td>370</td><td align="char">−49.6200359088558</td><td>286</td></tr><tr><td>10</td><td align="char">−0.831731827910368</td><td>370</td><td align="char">−49.6171262764610</td><td>159</td></tr><tr><td>12</td><td align="char">−0.831162154102009</td><td>372</td><td align="char">−49.6215181903295</td><td>180</td></tr><tr><td>14</td><td align="char">−0.831960998149279</td><td>296</td><td align="char">−49.6172562470701</td><td>268</td></tr><tr><td>16</td><td align="char">−0.830513944599602</td><td>372</td><td align="char">−49.6209997605222</td><td>204</td></tr><tr><td>18</td><td align="char">−0.831892355828711</td><td>371</td><td align="char">−49.6224605571573</td><td>306</td></tr><tr><td>20</td><td align="char">−0.829500740118085</td><td>375</td><td align="char">−49.6211005825003</td><td>306</td></tr></tbody></table></table-wrap><table-wrap position="float" id="t0050"><label>Table 10</label><caption><p>Relation of average best objective function value with average CPU time under different latency periods <italic>τ</italic> when <italic>n</italic>=50, <italic>E</italic><sub>0</sub>=0.06, <italic>N</italic>=200, <italic>N</italic><sub>0</sub>=2000, <italic>L</italic>=6, <italic>T</italic>=4, <italic>λ</italic>=1, <italic>G</italic>=80,000 and run times=20; REINT is prohibited.</p></caption><alt-text id="at0150">Table 10</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2"><italic>τ</italic></th><th colspan="2">The Bump function<hr></hr></th><th colspan="2">The Michalewicz function<hr></hr></th></tr><tr><th>Average best objective function value</th><th>Average CPU time/s</th><th>Average best objective function value</th><th>Average CPU time/s</th></tr></thead><tbody><tr><td><bold>1</bold></td><td align="char"><bold>−0.832264972912071</bold></td><td><bold>374</bold></td><td align="char"><bold>−49.6226411044064</bold></td><td><bold>208</bold></td></tr><tr><td>2</td><td align="char"><bold>−</bold>0.832089019848813</td><td>370</td><td align="char"><bold>−</bold>49.6195936998853</td><td>194</td></tr><tr><td>3</td><td align="char"><bold>−</bold>0.832106201210614</td><td>371</td><td align="char"><bold>−</bold>49.619302645638</td><td>196</td></tr><tr><td>4</td><td align="char"><bold>−</bold>0.830133500000392</td><td>369</td><td align="char"><bold>−</bold>49.6202190684598</td><td>200</td></tr><tr><td>5</td><td align="char"><bold>−</bold>0.831524847058460</td><td>367</td><td align="char"><bold>−</bold>49.6150659802303</td><td>317</td></tr><tr><td>6</td><td align="char"><bold>−</bold>0.829945399638853</td><td>371</td><td align="char"><bold>−</bold>49.6149880010193</td><td>135</td></tr><tr><td>7</td><td align="char"><bold>−</bold>0.832489306938715</td><td>363</td><td align="char"><bold>−</bold>49.621078127580</td><td>203</td></tr><tr><td>8</td><td align="char"><bold>−</bold>0.832031698894371</td><td>354</td><td align="char"><bold>−</bold>49.6162557853017</td><td>284</td></tr><tr><td>9</td><td align="char"><bold>−</bold>0.830629150090071</td><td>288</td><td align="char"><bold>−</bold>49.6164115369954</td><td>251</td></tr><tr><td>10</td><td align="char"><bold>−</bold>0.829828723701112</td><td>371</td><td align="char"><bold>−</bold>49.6125735983749</td><td>147</td></tr></tbody></table></table-wrap><table-wrap position="float" id="t0055"><label>Table 11</label><caption><p>Relation of average best objective function value with average CPU time under different immunity periods <italic>λ</italic> when <italic>n</italic>=50, <italic>E</italic><sub>0</sub>=0.06, <italic>N</italic>=200, <italic>N</italic><sub>0</sub>=2000, <italic>L</italic>=6, <italic>T</italic>=4, <italic>τ</italic>=1, <italic>G</italic>=80,000 and run times=20; REINT is prohibited.</p></caption><alt-text id="at0155">Table 11</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2"><italic>λ</italic></th><th colspan="2">The Bump function<hr></hr></th><th colspan="2">The Michalewicz function<hr></hr></th></tr><tr><th>Average best objective function value</th><th>Average CPU time/s</th><th>Average best objective function value</th><th>Average CPU time/s</th></tr></thead><tbody><tr><td><bold>1</bold></td><td align="char"><bold>−0.829709813549924</bold></td><td><bold>373</bold></td><td align="char"><bold>−49.616733008549</bold></td><td><bold>162</bold></td></tr><tr><td>2</td><td align="char"><bold>−</bold>0.832503794330900</td><td>370</td><td align="char"><bold>−</bold>49.6160884750357</td><td>237</td></tr><tr><td>3</td><td align="char"><bold>−</bold>0.827466380640695</td><td>369</td><td align="char"><bold>−</bold>49.6190711754889</td><td>133</td></tr><tr><td>4</td><td align="char"><bold>−</bold>0.830384696699006</td><td>369</td><td align="char"><bold>−</bold>49.6108273829626</td><td>238</td></tr><tr><td>5</td><td align="char"><bold>−</bold>0.82998943759237</td><td>367</td><td align="char"><bold>−</bold>49.6147830621168</td><td>212</td></tr><tr><td>6</td><td align="char"><bold>−</bold>0.829456526977736</td><td>367</td><td align="char"><bold>−</bold>49.6170709900992</td><td>174</td></tr><tr><td>7</td><td align="char"><bold>−</bold>0.829905775418647</td><td>505</td><td align="char"><bold>−</bold>49.6165828146807</td><td>317</td></tr><tr><td>8</td><td align="char"><bold>−</bold>0.830430239230583</td><td>438</td><td align="char"><bold>−</bold>49.6205720290063</td><td>262</td></tr><tr><td>9</td><td align="char"><bold>−</bold>0.830675022518777</td><td>499</td><td align="char"><bold>−</bold>49.6179419076148</td><td>189</td></tr><tr><td>10</td><td align="char"><bold>−</bold>0.828715260929138</td><td>457</td><td align="char"><bold>−</bold>49.6189395939285</td><td>232</td></tr></tbody></table></table-wrap><fig id="f0045"><label>Fig. 9</label><caption><p>Relation of objective function value with CPU time when <italic>T</italic> takes different values. (a) The Bump Function; (b) The Michalewicz function.</p></caption><alt-text id="at0045">Fig. 9</alt-text><graphic xlink:href="gr9"></graphic></fig><fig id="f0050"><label>Fig. 10</label><caption><p>Relation of objective function value with CPU time when <italic>τ</italic> takes different values. (a) The Bump Function; (b) The Michalewicz function.</p></caption><alt-text id="at0050">Fig. 10</alt-text><graphic xlink:href="gr10"></graphic></fig><fig id="f0055"><label>Fig. 11</label><caption><p>Relation of objective function value with CPU time when <italic>λ</italic> takes different values. (a) The Bump Function; (b) The Michalewicz function.</p></caption><alt-text id="at0055">Fig. 11</alt-text><graphic xlink:href="gr11"></graphic></fig></p><p id="p0985">Since in an ecosystem, <italic>β</italic><sup><italic>t</italic></sup>, <inline-formula><mml:math id="M145" altimg="si0032.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M146" altimg="si0033.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M147" altimg="si0034.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <italic>p</italic><sup><italic>t</italic></sup>, <italic>δ</italic><sup><italic>t</italic></sup> and <italic>ρ</italic><sup><italic>t</italic></sup> of population fluctuate with time, therefore, optimum objective function value obtained by SEIQRA always fluctuates in the vicinity of theoretical optimum objective function value. In order to increase precision of optimal objective function value, each setting of parameters should be made reasonably and allow SEIQRA to run many times.</p><p id="p0990"><xref rid="t0060" ref-type="table">Table 12</xref> shows that relation among <italic>n</italic> (the number of variables of optimization problem <xref rid="eq0005" ref-type="disp-formula">(1)</xref>), average best objective function value and average CPU time. From <xref rid="t0060" ref-type="table">Table 12</xref> we can see that when <italic>n</italic> increases, the consumed CPU time increases greatly, but the precision of objective function value decreases a little also.
<xref rid="f0060" ref-type="fig">Fig. 12</xref> describes the convergence graphs when <italic>n</italic> takes different values. Owing to the “each-step-is-not-bad” search strategy, the convergence process of SEIQRA is fairly smooth <xref rid="bib83" ref-type="bibr">[83]</xref>, <xref rid="bib86" ref-type="bibr">[86]</xref>, <xref rid="bib87" ref-type="bibr">[87]</xref>, <xref rid="bib88" ref-type="bibr">[88]</xref>.<table-wrap position="float" id="t0060"><label>Table 12</label><caption><p>Relation among <italic>n</italic>, <italic>N</italic>, average objective function value and average CPU time when <italic>E</italic><sub>0</sub>=0.06, <italic>L</italic>=6, <italic>N</italic><sub>0</sub>=2000, <italic>G</italic>=1.0E+5, <italic>T</italic>=4, <italic>τ</italic>=1, <italic>λ</italic>=1 and run times=20; REINT is invoked.</p></caption><alt-text id="at0160">Table 12</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2"><italic>n</italic></th><th colspan="2">The Bump Function<hr></hr></th><th colspan="2">The Michalewicz Function<hr></hr></th></tr><tr><th>Average Objective Function Value</th><th>Average CPU Time/s</th><th>Average Objective Function Value</th><th>Average CPU Time/s</th></tr></thead><tbody><tr><td>100</td><td align="char">−0.842074788323724</td><td>3639</td><td align="char">−99.5852089056127</td><td>4629</td></tr><tr><td>200</td><td align="char">−0.835425654584833</td><td>7026</td><td align="char">−197.590732022399</td><td>6764</td></tr><tr><td>300</td><td align="char">−0.812782304160569</td><td>16,557</td><td align="char">−296.228141716020</td><td>7647</td></tr><tr><td>400</td><td align="char">−0.823113677539460</td><td>14,066</td><td align="char">−382.365976205754</td><td>10,489</td></tr><tr><td>500</td><td align="char">−0.838622479339227</td><td>17,710</td><td align="char">−472.813854215453</td><td>16,892</td></tr><tr><td>600</td><td align="char">−0.823781525014319</td><td>20,485</td><td align="char">−573.012868088253</td><td>20,338</td></tr></tbody></table></table-wrap><fig id="f0060"><label>Fig. 12</label><caption><p>The convergence process of SEIQRA. (a) The Bump Function; (b) The Michalewicz Function.</p></caption><alt-text id="at0060">Fig. 12</alt-text><graphic xlink:href="gr12"></graphic></fig></p></sec><sec id="s0110"><label>3.2</label><title>Analysis of parameter setting</title><p id="p0995">The parameters in SEIQRA are classified into three classes, namely<list list-type="simple" id="li0115"><list-item id="o0345"><label>(1)</label><p id="p1000">Class A: the basic parameters a population-based intelligent optimization algorithms may possess of: <italic>G</italic> (or <italic>ε</italic>) and <italic>N</italic>;</p></list-item><list-item id="o0350"><label>(2)</label><p id="p1005">Class B: the parameters associated with the SEIQRA epidemic model: <italic>L</italic>, <italic>E</italic><sub>0</sub>, <italic>T</italic>, <italic>τ</italic> and <italic>λ</italic>;</p></list-item><list-item id="o0355"><label>(3)</label><p id="p1010">Class C: the parameters associated with operator REINIT: <italic>N</italic><sub>0</sub>, <italic>t</italic><sub>max</sub>, <italic>ρ</italic>, <italic>LU</italic><sub>0</sub>, <italic>r</italic>;</p></list-item></list></p><p id="p1015">The parameters in Class A is a basis; the parameters in Class B is very important for SEIQRA, but <italic>L</italic>, <italic>T</italic>, <italic>τ</italic> and <italic>λ</italic> produce little influence on precision of the best objective function value and CPU time, so <italic>L</italic>, <italic>T</italic>, <italic>τ</italic> and <italic>λ</italic> can take fixed values; the parameters in Class C is not important for SEIQRA. If REINIT is prohibited, then the parameters in Class C do not need to be set.</p><p id="p1020">The standard parameter configuration in SEIQRA is as follows:<list list-type="simple" id="li0120"><list-item id="o0360"><label>(1)</label><p id="p1025"><italic>G</italic>=1.0E+5, <italic>N</italic>=150–350, generally <italic>N</italic>=200, <italic>ε</italic> is defined by the engineering problem to be solved.</p></list-item><list-item id="o0365"><label>(2)</label><p id="p1030">Always set <italic>L</italic>=6, <italic>T</italic>=4, <italic>τ</italic>=1 and <italic>λ</italic>=1.</p></list-item><list-item id="o0370"><label>(3)</label><p id="p1035">Set <italic>E</italic><sub>0</sub>=0.06(for low dimensional problems) or <italic>E</italic><sub>0</sub>=0.001(for high dimensional problems). Conclusively, when SEIQRA solves a problem, only does 1/10–1/1000 of variables of the problem need to take part in computation in each iteration.</p></list-item><list-item id="o0375"><label>(4)</label><p id="p1040">Always set <italic>N</italic><sub>0</sub>=2000, <italic>t</italic><sub><italic>max</italic></sub>=30, <italic>ρ</italic>=0.8, <italic>LU</italic><sub>0</sub>=<italic>ε</italic>, <italic>r</italic>=0.8.</p></list-item></list></p><p id="p1045">If we want to enhance the probability of finding global optima, we only increase <italic>N</italic><sub>0</sub>, for example, set <italic>N</italic><sub>0</sub>=20,000, the probability of finding global optima will be increased greatly; if we want to enhance the precision of global optima, we can only decrease <italic>LU</italic><sub>0</sub> to much lower level than <italic>ε</italic>.</p></sec></sec><sec id="s0115"><label>4</label><title>Scalability of SEIQRA</title><p id="p1050">We use 28 benchmark functions listed in CEC’2013 <xref rid="bib85" ref-type="bibr">[85]</xref> to test the scalability of SEIQRA, these benchmarks are F1–F28, as shown in
<xref rid="t0065" ref-type="table">Table 13</xref>. The benchmark functions that we minimized are functions that are representative of those used in the literature of performance analysis of populations-based optimization algorithms. More information about these functions can be found in <xref rid="bib85" ref-type="bibr">[85]</xref>.<table-wrap position="float" id="t0065"><label>Table 13</label><caption><p>28 Benchmark function optimization problems <xref rid="bib85" ref-type="bibr">[85]</xref>.</p></caption><alt-text id="at0165">Table 13</alt-text><table frame="hsides" rules="groups"><thead><tr><th colspan="2" rowspan="2">Benchmark function</th><th rowspan="2">Range of each variable</th><th rowspan="2">Theoretical global optimum solution</th><th colspan="3">Theoretical global optimum objective function value<hr></hr></th></tr><tr><th><italic>D</italic>=10</th><th><italic>D</italic>=30</th><th><italic>D</italic>=50</th></tr></thead><tbody><tr><td>F1</td><td>Sphere function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−1400</td><td>−1400</td><td>−1400</td></tr><tr><td>F2</td><td>Rotated high conditioned elliptic function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−1300</td><td>−1300</td><td>−1300</td></tr><tr><td>F3</td><td>Rotated Bent cigar function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−1200</td><td>−1200</td><td>−1200</td></tr><tr><td>F4</td><td>Rotated Discus function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−1100</td><td>−1100</td><td>−1100</td></tr><tr><td>F5</td><td>Different Powers function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−1000</td><td>−1000</td><td>−1000</td></tr><tr><td>F6</td><td>Rotated Rosenbrock function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−900</td><td>−900</td><td>−900</td></tr><tr><td>F7</td><td>Rotated Schaffers F7 function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−800</td><td>−800</td><td>−800</td></tr><tr><td>F8</td><td>Rotated Ackley׳s function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−700</td><td>−700</td><td>−700</td></tr><tr><td>F9</td><td>Rotated Weierstrass function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−600</td><td>−600</td><td>−600</td></tr><tr><td>F10</td><td>Rotated Griewank׳s function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−500</td><td>−500</td><td>−500</td></tr><tr><td>F11</td><td>Rastrigin׳s function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−400</td><td>−400</td><td>−400</td></tr><tr><td>F12</td><td>Rotated Rastrigin׳s function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−300</td><td>−300</td><td>−300</td></tr><tr><td>F13</td><td>Non-continuous rotated Rastrigin׳s function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>−200</td><td>−200</td><td>−200</td></tr><tr><td>F14</td><td>Schwefel׳s function</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe −804.4948)</td><td>Unknown (maybe −2262.9856)</td><td>Unknown (maybe −3721.4764)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>F15</td><td>Rotated Schwefel׳s function</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe −604.4948)</td><td>Unknown (maybe −2062.9856)</td><td>Unknown (maybe −3521.4764)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>F16</td><td>Rotated Katsuura function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>200</td><td>200</td><td>200</td></tr><tr><td>F17</td><td>Lunacek bi-Ratrigin function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>300</td><td>300</td><td>300</td></tr><tr><td>F18</td><td>Rotated Lunacek bi-Ratrigin function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>400</td><td>400</td><td>400</td></tr><tr><td>F19</td><td>Rotated Expanded Griewank׳s plus Rosenbrock׳s function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>500</td><td>500</td><td>500</td></tr><tr><td>F20</td><td>Rotated Expanded Scaffer׳s F6 function</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>600</td><td>600</td><td>600</td></tr><tr><td>F21</td><td>Composition function 1</td><td>[−100,100]</td><td><italic><bold>O</bold></italic></td><td>900</td><td>900</td><td>900</td></tr><tr><td>F22</td><td>Composition function 2</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe 440)</td><td>Unknown (maybe −498.2879)</td><td>Unknown (maybe −1965.7202)</td></tr><tr><td>F23</td><td>Composition function 3</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe 390)</td><td>Unknown (maybe −1200.3297)</td><td>Unknown (maybe −2721.4763)</td></tr><tr><td>F24</td><td>Composition function 4</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe 800)</td><td>Unknown (maybe 1200)</td><td>Unknown (maybe 631.2470)</td></tr><tr><td>F25</td><td>Composition function 5</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe 1200)</td><td>Unknown (maybe 1200)</td><td>Unknown (maybe 1200.000)</td></tr><tr><td>F26</td><td>Composition function 6</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe 1000)</td><td>Unknown (maybe 650)</td><td>Unknown (maybe 294.1417)</td></tr><tr><td>F27</td><td>Composition function 7</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe 1600)</td><td>Unknown (maybe 1600)</td><td>Unknown (maybe 1600.000)</td></tr><tr><td>F28</td><td>Composition function 8</td><td>[−100,100]</td><td>Unknown</td><td>Unknown (maybe 1700)</td><td>Unknown (maybe 1500)</td><td>Unknown (maybe −4744.3417 )</td></tr></tbody></table></table-wrap></p><p id="p1055">In <xref rid="t0065" ref-type="table">Table 13</xref>, <italic>D</italic> is dimensions of an optimization problem, here <italic>D</italic>=<italic>n</italic>; <italic><bold>O</bold></italic> is an <italic>n</italic>-dimensional decision vector.
<xref rid="t0070" ref-type="table">Table 14</xref> shows biases of theoretical global optimum objective function values of F1–F28. We use SEIQRA to solve 28 benchmark functions listed in <xref rid="t0065" ref-type="table">Table 13</xref>, the results are shown in
<xref rid="t0075" ref-type="table">Table 15</xref>, the setting of parameters used in SEIQRA are <italic>D</italic>=<italic>n</italic>=10, 30, 50, <italic>ε</italic>=1.0E−12, <italic>E</italic><sub>0</sub>=0.06, <italic>L</italic>=6, <italic>T</italic>=4, <italic>τ</italic>=1, <italic>λ</italic>=1, <italic>N</italic>=200, <italic>N</italic><sub>0</sub>=2000, <italic>t</italic><sub>max</sub>=30, <italic>ρ</italic>=0.8, <italic>LU</italic><sub>0</sub>=<italic>ε</italic>, <italic>r</italic>=0.8, <italic>G</italic>=2.0E+5 and run times=20; parameter<italic><bold>M</bold></italic>1 and <italic><bold>M</bold></italic>2 in F1–F28 are set according to the method mentioned in article <xref rid="bib85" ref-type="bibr">[85]</xref>; For convenience of analysis, let <italic><bold>O</bold></italic> is randomly generated; REINIT is invoked for some benchmark functions.<table-wrap position="float" id="t0070"><label>Table 14</label><caption><p>Biases of theoretical global optimum objective function values of F1–F28.</p></caption><alt-text id="at0170">Table 14</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2">Benchmark function</th><th colspan="3">Bias of theoretical global optimum objective function value<hr></hr></th></tr><tr><th><italic>D</italic>=10</th><th><italic>D</italic>=30</th><th><italic>D</italic>=50</th></tr></thead><tbody><tr><td>F1</td><td align="char">1400</td><td align="char">1400</td><td align="char">1400</td></tr><tr><td>F2</td><td align="char">1300</td><td align="char">1300</td><td align="char">1300</td></tr><tr><td>F3</td><td align="char">1200</td><td align="char">1200</td><td align="char">1200</td></tr><tr><td>F4</td><td align="char">1100</td><td align="char">1100</td><td align="char">1100</td></tr><tr><td>F5</td><td align="char">1000</td><td align="char">1000</td><td align="char">1000</td></tr><tr><td>F6</td><td align="char">900</td><td align="char">900</td><td align="char">900</td></tr><tr><td>F7</td><td align="char">800</td><td align="char">800</td><td align="char">800</td></tr><tr><td>F8</td><td align="char">700</td><td align="char">700</td><td align="char">700</td></tr><tr><td>F9</td><td align="char">600.0022</td><td align="char">600.0022</td><td align="char">600.0022</td></tr><tr><td>F10</td><td align="char">500</td><td align="char">500</td><td align="char">500</td></tr><tr><td>F11</td><td align="char">400</td><td align="char">400</td><td align="char">400</td></tr><tr><td>F12</td><td align="char">300</td><td align="char">300</td><td align="char">300</td></tr><tr><td>F13</td><td align="char">200</td><td align="char">200</td><td align="char">200</td></tr><tr><td>F14</td><td align="char">804.4948</td><td align="char">2262.9856</td><td align="char">3721.4765</td></tr><tr><td>F15</td><td align="char">604.4948</td><td align="char">2062.9856</td><td align="char">3521.4764</td></tr><tr><td>F16</td><td align="char">−200</td><td align="char">−200</td><td align="char">−200</td></tr><tr><td>F17</td><td align="char">−300</td><td align="char">−300</td><td align="char">−300</td></tr><tr><td>F18</td><td align="char">−400</td><td align="char">−400</td><td align="char">−400</td></tr><tr><td>F19</td><td align="char">−500</td><td align="char">−500</td><td align="char">−500</td></tr><tr><td>F20</td><td align="char">−600</td><td align="char">−600</td><td align="char">−600</td></tr><tr><td>F21</td><td align="char">−900</td><td align="char">−900</td><td align="char">−900</td></tr><tr><td>F22</td><td align="char">−440</td><td align="char">498.287</td><td align="char">1965.7202</td></tr><tr><td>F23</td><td align="char">−390</td><td align="char">1200.3297</td><td align="char">2721.4763</td></tr><tr><td>F24</td><td align="char">−800</td><td align="char">−1200</td><td align="char">−631.2470</td></tr><tr><td>F25</td><td align="char">−1200</td><td align="char">−1200</td><td align="char">−1200.000</td></tr><tr><td>F26</td><td align="char">−1000</td><td align="char">−650</td><td align="char">−294.1417</td></tr><tr><td>F27</td><td align="char">−1600</td><td align="char">−1700</td><td align="char">−1600</td></tr><tr><td>F28</td><td align="char">−1700</td><td align="char">-1500</td><td align="char">4744.3417</td></tr></tbody></table></table-wrap><table-wrap position="float" id="t0075"><label>Table 15</label><caption><p>Result deviation of 28 benchmark functions solved by SEIQRA (<italic>D</italic>=<italic>n</italic>=10).</p></caption><alt-text id="at0175">Table 15</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2">Benchmark function</th><th colspan="4">Objective function value<hr></hr></th><th colspan="4">CPU time/s<hr></hr></th></tr><tr><th>Best</th><th>Worst</th><th>Mean</th><th>Standard deviation</th><th>Smallest</th><th>Biggest</th><th>Mean</th><th>Standard deviation</th></tr></thead><tbody><tr><td>F1</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>7</td><td>9</td><td>8</td><td>1.1920E−01</td></tr><tr><td>F2</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>37</td><td>45</td><td>41</td><td>4.7655E−01</td></tr><tr><td>F3</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>10</td><td>15</td><td>13</td><td>3.3065E−01</td></tr><tr><td>F4</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>607</td><td>635</td><td>621</td><td>1.7664E+00</td></tr><tr><td>F5</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>118</td><td>154</td><td>134</td><td>2.1151E+00</td></tr><tr><td>F6</td><td><bold>0.0000E+00</bold></td><td>5.2400E−02</td><td>2.8500E−02</td><td>3.6065E−03</td><td>443</td><td>487</td><td>467</td><td>3.0284E+00</td></tr><tr><td>F7</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>44</td><td>51</td><td>48</td><td>4.0527E−01</td></tr><tr><td>F8</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>27</td><td>34</td><td>30</td><td>5.0380E−01</td></tr><tr><td>F9</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>−1.2000E−03</td><td>1.4027E−04</td><td>127</td><td>146</td><td>136</td><td>1.2114E+00</td></tr><tr><td>F10</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>24</td><td>35</td><td>28</td><td>7.6681E−01</td></tr><tr><td>F11</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>14</td><td>18</td><td>16</td><td>3.0878E−01</td></tr><tr><td>F12</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>35</td><td>44</td><td>41</td><td>5.6196E−01</td></tr><tr><td>F13</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>10</td><td>17</td><td>14</td><td>4.0010E−01</td></tr><tr><td>F14</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>5.0842E−14</td><td>145</td><td>186</td><td>162</td><td>2.0160E+00</td></tr><tr><td>F15</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>2.5421E−14</td><td>39</td><td>47</td><td>42</td><td>3.2819E−01</td></tr><tr><td>F16</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>28</td><td>33</td><td>30</td><td>3.8753E−01</td></tr><tr><td>F17</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>0</td><td>0</td><td>0</td><td>0.0000E+00</td></tr><tr><td>F18/</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>40</td><td>48</td><td>44</td><td>3.7347E−01</td></tr><tr><td>F19</td><td><bold>2.5710E−01</bold></td><td>4.7585E+00</td><td>2.3441E+00</td><td>3.3849E−01</td><td>113</td><td>126</td><td>119</td><td>9.7755E−01</td></tr><tr><td>F20</td><td><bold>0.0000E+00</bold></td><td>1.0756E+00</td><td>5.4020E−01</td><td>6.1315E−02</td><td>226</td><td>270</td><td>248</td><td>2.5082E+00</td></tr><tr><td>F21</td><td><bold>2.0000E+02</bold></td><td>2.0000E+02</td><td>2.0000E+02</td><td>0.0000E+00</td><td>60</td><td>68</td><td>63</td><td>4.5128E−01</td></tr><tr><td>F22</td><td><bold>1.2760E−01</bold></td><td>3.7539E+00</td><td>2.0645E+00</td><td>1.9783E−01</td><td>1610</td><td>1890</td><td>1760</td><td>1.5276E+01</td></tr><tr><td>F23</td><td><bold>7.4390E−01</bold></td><td>1.1188E+01</td><td>6.1489E+00</td><td>6.1107E−01</td><td>1610</td><td>1788</td><td>1702</td><td>1.0415E+01</td></tr><tr><td>F24</td><td><bold>2.0185E+01</bold></td><td>2.8724E+01</td><td>2.4406E+01</td><td>5.4742E−01</td><td>2610</td><td>2978</td><td>2792</td><td>2.3593E+01</td></tr><tr><td>F25</td><td><bold>1.7800E−02</bold></td><td>2.3698E+01</td><td>1.2443E+01</td><td>1.5730E+00</td><td>2100</td><td>2300</td><td>2205</td><td>1.3285E+01</td></tr><tr><td>F26</td><td><bold>1.4170E−01</bold></td><td>2.3177E+01</td><td>1.0148E+01</td><td>1.5937E+00</td><td>510</td><td>600</td><td>549</td><td>6.2267E+00</td></tr><tr><td>F27</td><td><bold>0.0000E+00</bold></td><td>7.8120E−01</td><td>3.4080E−01</td><td>4.9952E−02</td><td>1900</td><td>2300</td><td>2075</td><td>2.5577E+01</td></tr><tr><td>F28</td><td><bold>6.5155E+01</bold></td><td>1.6577E+02</td><td>1.0063E+02</td><td>6.1548E+00</td><td>8000</td><td>9000</td><td>8353</td><td>6.1171E+01</td></tr></tbody></table></table-wrap></p><p id="p1060">From
<xref rid="t0075" ref-type="table">Table 15</xref>, <xref rid="t0080" ref-type="table">Table 16</xref>, <xref rid="t0085" ref-type="table">Table 17</xref> we can see that SEIQRA can solve F1–F28 when <italic>n</italic>=10, 30 and 50, it means the algorithm has good scalability.<table-wrap position="float" id="t0080"><label>Table 16</label><caption><p>Result deviation of 28 benchmark functions solved by SEIQRA (<italic>D</italic>=<italic>n</italic>=30).</p></caption><alt-text id="at0180">Table 16</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2">Benchmark function</th><th colspan="4">Objective function value<hr></hr></th><th colspan="4">CPU time/s<hr></hr></th></tr><tr><th>Best</th><th>Worst</th><th>Mean</th><th>Standard deviation</th><th>Smallest</th><th>Biggest</th><th>Mean</th><th>Standard deviation</th></tr></thead><tbody><tr><td>F1</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>20</td><td>25</td><td>23</td><td>2.9800E−01</td></tr><tr><td>F2</td><td><bold>0.0000E+00</bold></td><td>8.4800E−02</td><td>3.8500E−02</td><td>5.0514E−03</td><td>178</td><td>183</td><td>180</td><td>2.9784E−01</td></tr><tr><td>F3</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>78</td><td>82</td><td>80</td><td>2.6452E−01</td></tr><tr><td>F4</td><td><bold>0.0000E+00</bold></td><td>8.9600E−02</td><td>4.5800E−02</td><td>5.6526E−03</td><td>1114</td><td>1134</td><td>1124</td><td>1.2617E+00</td></tr><tr><td>F5</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>208</td><td>223</td><td>215</td><td>8.8131E−01</td></tr><tr><td>F6</td><td><bold>0.0000E+00</bold></td><td>7.1700E−02</td><td>3.9000E−02</td><td>4.9349E−03</td><td>281</td><td>293</td><td>288</td><td>8.2592E−01</td></tr><tr><td>F7</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>498</td><td>512</td><td>506</td><td>8.1054E−01</td></tr><tr><td>F8</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>170</td><td>183</td><td>176</td><td>9.3564E−01</td></tr><tr><td>F9</td><td><bold>0.0000E+00</bold></td><td>−2.2000E−03</td><td>−1.0000E−03</td><td>1.4027E−04</td><td>82</td><td>93</td><td>87</td><td>7.0136E−01</td></tr><tr><td>F10</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>71</td><td>80</td><td>75</td><td>6.2739E−01</td></tr><tr><td>F11</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>8</td><td>12</td><td>10</td><td>3.0878E−01</td></tr><tr><td>F12</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>177</td><td>193</td><td>188</td><td>9.9904E−01</td></tr><tr><td>F13</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>57</td><td>65</td><td>61</td><td>4.5725E−01</td></tr><tr><td>F14</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>176</td><td>185</td><td>180</td><td>4.4253E−01</td></tr><tr><td>F15</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>76</td><td>83</td><td>78</td><td>2.8717E−01</td></tr><tr><td>F16</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>269</td><td>283</td><td>275</td><td>1.0851E+00</td></tr><tr><td>F17</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>1</td><td>1</td><td>1</td><td>0.0000E+00</td></tr><tr><td>F18/</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>143</td><td>153</td><td>148</td><td>4.6683E−01</td></tr><tr><td>F19</td><td><bold>5.5170E</bold>−<bold>01</bold></td><td>2.4742E+01</td><td>1.1767E+01</td><td>1.8190E+00</td><td>598</td><td>613</td><td>605</td><td>1.1279E+00</td></tr><tr><td>F20</td><td><bold>1.2670E</bold>−<bold>01</bold></td><td>1.0756E+00</td><td>6.0330E−01</td><td>5.4092E−02</td><td>316</td><td>329</td><td>323</td><td>7.4107E−01</td></tr><tr><td>F21</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>309</td><td>321</td><td>314</td><td>6.7692E−01</td></tr><tr><td>F22</td><td>−<bold>9.0000E</bold>−<bold>04</bold></td><td>2.1607E+00</td><td>1.1536E+00</td><td>1.1793E−01</td><td>12,319</td><td>12,593</td><td>12,465</td><td>1.4948E+01</td></tr><tr><td>F23</td><td><bold>0.0000E+00</bold></td><td>−4.4897E+01</td><td>−2.3236E+01</td><td>2.6269E+00</td><td>3179</td><td>3356</td><td>3271</td><td>1.0356E+01</td></tr><tr><td>F24</td><td>−<bold>8.5270E</bold>−<bold>01</bold></td><td>2.3780E−01</td><td>−3.1360E−01</td><td>6.9914E−02</td><td>7320</td><td>7410</td><td>7364</td><td>5.7700E+00</td></tr><tr><td>F25</td><td><bold>1.7800E</bold>−<bold>02</bold></td><td>5.4738E+01</td><td>2.8730E+01</td><td>3.6349E+00</td><td>2619</td><td>2703</td><td>2663</td><td>5.5798E+00</td></tr><tr><td>F26</td><td><bold>1.4170E</bold>−<bold>01</bold></td><td>9.2781E+00</td><td>3.5569E+00</td><td>3.4264E+00</td><td>5688</td><td>5876</td><td>5770</td><td>1.3007E+01</td></tr><tr><td>F27</td><td><bold>1.0720E</bold>−<bold>01</bold></td><td>1.3459E+01</td><td>5.9322E+00</td><td>8.5373E−01</td><td>4801</td><td>4903</td><td>4846</td><td>6.5222E+00</td></tr><tr><td>F28</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>16,023</td><td>16,745</td><td>16,278</td><td>4.4165E+01</td></tr></tbody></table></table-wrap><table-wrap position="float" id="t0085"><label>Table 17</label><caption><p>Results of 28 benchmark functions solved by SEIQRA (<italic>D</italic>=<italic>n</italic>=50).</p></caption><alt-text id="at0185">Table 17</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2">Benchmark function</th><th colspan="4">Objective function value<hr></hr></th><th colspan="4">CPU time/s<hr></hr></th></tr><tr><th>Best</th><th>Worst</th><th>Mean</th><th>Standard deviation</th><th>Smallest</th><th>Biggest</th><th>Mean</th><th>Standard deviation</th></tr></thead><tbody><tr><td>F1</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>50</td><td>54</td><td>51</td><td>1.1048E+00</td></tr><tr><td>F2</td><td><bold>0.0000E+00</bold></td><td>8.4800E−02</td><td>6.5300E−02</td><td>2.3556E−02</td><td>11,604</td><td>11,677</td><td>11,628</td><td>2.1543E+01</td></tr><tr><td>F3</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>194</td><td>199</td><td>197</td><td>1.3809E+00</td></tr><tr><td>F4</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>7711</td><td>7723</td><td>7719</td><td>3.5904E+00</td></tr><tr><td>F5</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>1348</td><td>1353</td><td>1351</td><td>1.3809E+00</td></tr><tr><td>F6</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>2268</td><td>2278</td><td>2274</td><td>2.7619E+00</td></tr><tr><td>F7</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>350</td><td>355</td><td>353</td><td>1.3809E+00</td></tr><tr><td>F8</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>157</td><td>162</td><td>160</td><td>1.6571E+00</td></tr><tr><td>F9</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>485</td><td>491</td><td>489</td><td>1.4235E+00</td></tr><tr><td>F10</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>243</td><td>250</td><td>246</td><td>1.9333E+00</td></tr><tr><td>F11</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>58</td><td>61</td><td>60</td><td>8.2860E−01</td></tr><tr><td>F12</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>453</td><td>460</td><td>456</td><td>1.8392E+00</td></tr><tr><td>F13</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>322</td><td>331</td><td>326</td><td>2.4857E+00</td></tr><tr><td>F14</td><td><bold>1.0000E−05</bold></td><td>0.0000E+00</td><td>0.0000E−04</td><td>4.4714E−08</td><td>230</td><td>235</td><td>232</td><td>1.2812E+00</td></tr><tr><td>F15</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>5.4464E−08</td><td>185</td><td>190</td><td>188</td><td>1.1810E−01</td></tr><tr><td>F16</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>1239</td><td>1245</td><td>1242</td><td>1.6571E+00</td></tr><tr><td>F17</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>0</td><td>0</td><td>0</td><td>0.0000E+00</td></tr><tr><td>F18/</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>269</td><td>278</td><td>274</td><td>2.4857E+00</td></tr><tr><td>F19</td><td><bold>2.5710E−01</bold></td><td>4.7585E+00</td><td>1.1733E+00</td><td>2.9003E+00</td><td>15,543</td><td>15,712</td><td>15,665</td><td>4.6676E+01</td></tr><tr><td>F20</td><td><bold>3.2770E−01</bold></td><td>1.0756E+00</td><td>9.7850E−01</td><td>2.0660E−01</td><td>3847</td><td>3872</td><td>3860</td><td>6.9046E+00</td></tr><tr><td>F21</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td><td>703</td><td>714</td><td>711</td><td>3.0381E+00</td></tr><tr><td>F22</td><td><bold>−8.1980E−01</bold></td><td>1.0392E+01</td><td>0.0000E+00</td><td>1.1563E+01</td><td>4104</td><td>5274</td><td>4187</td><td>3.2314E+02</td></tr><tr><td>F23</td><td><bold>−9.8350E−01</bold></td><td>8.1736E+00</td><td>0.0000E+00</td><td>1.2205E+01</td><td>674</td><td>712</td><td>683</td><td>1.0495E+01</td></tr><tr><td>F24</td><td><bold>3.5650E+02</bold></td><td>3.5792E+02</td><td>3.5742E+02</td><td>1.1659E+00</td><td>11,046</td><td>11,287</td><td>11,152</td><td>6.6561E+01</td></tr><tr><td>F25</td><td><bold>5.9018E+01</bold></td><td>6.4738E+01</td><td>6.2533E+01</td><td>1.5797E+00</td><td>2231</td><td>2276</td><td>2249</td><td>1.2428E+01</td></tr><tr><td>F26</td><td><bold>0.0000E+00</bold></td><td>1.0351E+00</td><td>4.9730E−01</td><td>3.9630E−01</td><td>6101</td><td>6159</td><td>6137</td><td>1.6019E+01</td></tr><tr><td>F27</td><td><bold>4.0176E+00</bold></td><td>1.2781E+01</td><td>1.1446E+01</td><td>2.4204E+00</td><td>1759</td><td>1781</td><td>1765</td><td>6.0761E+00</td></tr><tr><td>F28</td><td><bold>−7.8500E−01</bold></td><td>1.6261E+00</td><td>0.0000E+00</td><td>4.4910E−01</td><td>110,212</td><td>110,786</td><td>110,475</td><td>1.5853E+02</td></tr></tbody></table></table-wrap></p></sec><sec id="s0120"><label>5</label><title>Comparison of SEIQRA with other population-based optimization algorithms</title><sec id="s0125"><label>5.1</label><title>Comparison study of SEIQRA</title><p id="p1065">We choose another 7 population-based optimization algorithms to make comparison to solve F1–F28; these algorithms include Real Code Genetic Algorithm (RC-GA) <xref rid="bib32" ref-type="bibr">[32]</xref>, Differential Ant-Stigmergy Algorithm (DASA) <xref rid="bib84" ref-type="bibr">[84]</xref>, Non-parametric Particle Swarm Optimization (NP-PSO) <xref rid="bib38" ref-type="bibr">[38]</xref>, Biogeography-based Optimization (BBO) <xref rid="bib42" ref-type="bibr">[42]</xref>, Differential Evolution (DE) <xref rid="bib89" ref-type="bibr">[89]</xref>, Self-adaptive Differential Evolution (SaDE) <xref rid="bib43" ref-type="bibr">[43]</xref>, and Artificial Bee Colony (ABC) <xref rid="bib47" ref-type="bibr">[47]</xref>.</p><p id="p1070">When calculating, the setting of parameters in the 7 population-based optimization algorithms is initialized according to
<xref rid="t0090" ref-type="table">Table 18</xref>.<table-wrap position="float" id="t0090"><label>Table 18</label><caption><p>The setting of parameters in the 7 population-based optimization algorithms.</p></caption><alt-text id="at0190">Table 18</alt-text><table frame="hsides" rules="groups"><thead><tr><th>Optimization algorithm</th><th>Parameters</th></tr></thead><tbody><tr><td>RC-GA <xref rid="bib32" ref-type="bibr">[32]</xref></td><td>The number of chromosomes <italic>N</italic>=100, mutation probability=0.01, the number of parents=0.5 <italic>N</italic>, <italic>G</italic>=100,000</td></tr><tr><td>DASA <xref rid="bib84" ref-type="bibr">[84]</xref></td><td>The number of ants <italic>m</italic>=30, discrete base <italic>b</italic>=10, pheromone decay rate <italic>ρ</italic>=0.2, global scale-increasing factor <italic>s</italic><sub>+</sub>=0.02, global scale-decreasing factor <italic>s</italic><sub>-</sub>=0.01, the maximum precision of variables <italic>ε</italic>=1.0e-15, <italic>G</italic>=300,000</td></tr><tr><td>NP-PSO <xref rid="bib38" ref-type="bibr">[38]</xref></td><td><italic>N</italic>=100, <italic>G</italic>=100,000</td></tr><tr><td>BBO <xref rid="bib42" ref-type="bibr">[42]</xref></td><td>Habitat modification probability=1, immigration probability bounds per gene=[0, 1], step size for numerical integration of probabilities=1, maximum immigration and migration rates for each island=1, and mutation probability=0.02, <italic>N</italic>=200, elitism=2, <italic>G</italic>=100,000</td></tr><tr><td>DE <xref rid="bib89" ref-type="bibr">[89]</xref></td><td>Weighting factor <italic>F</italic>=0.5, crossover constant <italic>CR</italic>=0.9, <italic>N</italic>=100, <italic>G</italic>=1.0E+5</td></tr><tr><td>SaDE <xref rid="bib43" ref-type="bibr">[43]</xref></td><td>Interval of weighting factor=[0.45,0.55]; crossover constant=[0.85,0.95], <italic>N</italic>=100, <italic>G</italic>=1.0E+5</td></tr><tr><td>ABC <xref rid="bib47" ref-type="bibr">[47]</xref></td><td>Employed bees or onlookers=300, trying times=300 <italic>n</italic>, <italic>G</italic>=8.0E+5</td></tr></tbody></table></table-wrap></p><p id="p1075">The algorithms are run independently 20 times for each benchmark function and average best objective function value, standard deviation (STD), median of the best objective function values, CPU time for finding the best objective function value, and rank of each algorithm are reported in
<xref rid="t0095" ref-type="table">Table 19</xref>. The algorithms are ranked based on Rank 1 and Rank 2 in <xref rid="t0095" ref-type="table">Table 19</xref>. Rank 1 is based on precision of average best objective function value; Rank 2 is based on precision of average best objective function value and CPU time. From <xref rid="t0095" ref-type="table">Table 19</xref> we can see that the ranks of SEIQRA, RC-GA, NP-PSO, BBO, DE, SaDE and ABC based on precision or precision and CPU time are as follows:<disp-formula id="eq0195"><mml:math id="M148" altimg="si0086.gif" overflow="scroll"><mml:mrow><mml:mtext>SEIQRA</mml:mtext><mml:mo>></mml:mo><mml:mtext>SaDE</mml:mtext><mml:mo>></mml:mo><mml:mtext>DE</mml:mtext><mml:mo>></mml:mo><mml:mtext>ABC</mml:mtext><mml:mo>></mml:mo><mml:mtext>NP</mml:mtext><mml:mo>-</mml:mo><mml:mtext>PSO</mml:mtext><mml:mo>></mml:mo><mml:mtext>DSDA</mml:mtext><mml:mo>></mml:mo><mml:mtext>RC</mml:mtext><mml:mo>-</mml:mo><mml:mtext>GA</mml:mtext><mml:mo>></mml:mo><mml:mtext>BBO</mml:mtext></mml:mrow></mml:math></disp-formula><table-wrap position="float" id="t0095"><label>Table 19</label><caption><p>The minimization results of benchmark functions of 8 population-based optimization algorithms.</p></caption><alt-text id="at0195">Table 19</alt-text><table frame="hsides" rules="groups"><thead><tr><th>Benchmark function</th><th>SEIQRA</th><th>RC-GA</th><th>DSDA</th><th>NP-PSO</th><th>BBO</th><th>DE</th><th>SaDE</th><th>ABC</th></tr></thead><tbody><tr><td colspan="9">F1</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>2.5279E+02</td><td>8.5400E−02</td><td>6.5880E+00</td><td>4.2350E−01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td>2.5122E+02</td><td>7.5700E−02</td><td>6.5823E+00</td><td>4.1110E−01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td></tr><tr><td>STD</td><td>0.0000E+00</td><td>3.2044E−01</td><td>4.1660E−03</td><td>5.2464E−03</td><td>3.8377E−05</td><td>0.0000E+00</td><td>0.0000E+00</td><td>0.0000E+00</td></tr><tr><td>Time/s</td><td>51</td><td>2538</td><td>147</td><td>6119</td><td>638</td><td>5</td><td>7</td><td>787</td></tr><tr><td>Rank1</td><td>1</td><td>8</td><td>5</td><td>7</td><td>6</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Rank2</td><td>3</td><td>8</td><td>5</td><td>7</td><td>6</td><td>1</td><td>2</td><td>4</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F2</td></tr><tr><td>Average</td><td><bold>6.5300E</bold>−<bold>02</bold></td><td>8.3237E+06</td><td>2.8254E+06</td><td>3.1024E+08</td><td>1.0862E+07</td><td>6.3938E+00</td><td>5.3910E−01</td><td>3.0186E+08</td></tr><tr><td>Median</td><td><bold>4.3600E</bold>−<bold>02</bold></td><td>8.3237E+06</td><td>2.8254E+06</td><td>3.1024E+08</td><td>1.0862E+07</td><td>6.0822E+00</td><td>4.7850E−01</td><td>3.0186E+08</td></tr><tr><td>STD</td><td>3.2652E−03</td><td>1.8185E+01</td><td>1.7340E+01</td><td>1.9922E+01</td><td>1.9121E+01</td><td>4.1929E−03</td><td>3.5835E−03</td><td>1.5725E+01</td></tr><tr><td>Time/s</td><td>11,628</td><td>12,141</td><td>810</td><td>2053</td><td>1669</td><td>9551</td><td>5755</td><td>51</td></tr><tr><td>Rank1</td><td>1</td><td>5</td><td>4</td><td>8</td><td>6</td><td>3</td><td>2</td><td>7</td></tr><tr><td>Rank2</td><td>1</td><td>5</td><td>4</td><td>8</td><td>6</td><td>3</td><td>2</td><td>7</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F3</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>2.3867E+01</td><td>5.6600E−01</td><td>1.6895E+07</td><td>3.5110E−01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>2.4823E+04</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td>2.3841E+01</td><td>4.7530E−01</td><td>1.6895E+07</td><td>2.8070E−01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>2.4817E+04</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>4.0653E−03</td><td>3.4497E−03</td><td>1.6168E+01</td><td>3.9527E−03</td><td>0.0000E+00</td><td>0.0000E+00</td><td>1.4357E+01</td></tr><tr><td>Time/s</td><td>197</td><td>20,789</td><td>645</td><td>65</td><td>1103</td><td>53</td><td>43</td><td>1121</td></tr><tr><td>Rank1</td><td>1</td><td>5</td><td>5</td><td>8</td><td>4</td><td>1</td><td>1</td><td>7</td></tr><tr><td>Rank2</td><td>3</td><td>6</td><td>5</td><td>8</td><td>4</td><td>2</td><td>1</td><td>7</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F4</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>9.5427E+04</td><td>7.9466E+02</td><td>4.1266E+03</td><td>2.9388E+03</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>4.9956E+04</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td>9.5429E+04</td><td>7.9456E+02</td><td>4.1270E+03</td><td>2.9386E+03</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>4.9956E+04</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>2.1040E+01</td><td>3.7581E−03</td><td>4.1422E+00</td><td>3.7088E+00</td><td>0.0000E+00</td><td>4.0371E−05</td><td>2.1602E+01</td></tr><tr><td>Time/s</td><td>7719</td><td>58</td><td>5113</td><td>3427</td><td>3141</td><td>515</td><td>1178</td><td>297</td></tr><tr><td>Rank1</td><td>1</td><td>8</td><td>4</td><td>6</td><td>5</td><td>1</td><td>1</td><td>7</td></tr><tr><td>Rank2</td><td>3</td><td>8</td><td>4</td><td>6</td><td>5</td><td>1</td><td>2</td><td>7</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F5</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>5.3927E+02</td><td>3.5660E−01</td><td>7.5855E+03</td><td>1.5110E−01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>8.9469E+03</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td>5.3902E+02</td><td>3.6630E−01</td><td>7.5848E+03</td><td>2.1880E−01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>8.9472E+03</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>3.8820E−03</td><td>3.2402E−02</td><td>3.8064E+00</td><td>3.3413E−03</td><td>3.6289E−05</td><td>3.6234E−05</td><td>3.1642E+00</td></tr><tr><td>Time/s</td><td>1351</td><td>6205</td><td>1887</td><td>3598</td><td>773</td><td>1069</td><td>1086</td><td>80</td></tr><tr><td>Rank1</td><td>1</td><td>6</td><td>5</td><td>7</td><td>4</td><td>1</td><td>1</td><td>8</td></tr><tr><td>Rank2</td><td>3</td><td>6</td><td>5</td><td>7</td><td>4</td><td>1</td><td>2</td><td>8</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F6</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>3.0151E+02</td><td>2.2495E+02</td><td>3.6243E+03</td><td>6.8390E+01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>3.8696E+02</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td>3.0134E+02</td><td>2.2436E+02</td><td>3.6251E+03</td><td>6.8276E+01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>3.8659E+02</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>3.5841E−02</td><td>4.5944E−02</td><td>3.9308E+00</td><td>3.9877E−02</td><td>3.5218E−05</td><td>3.2570E−05</td><td>3.7471E−02</td></tr><tr><td>Time/s</td><td>2274</td><td>9999</td><td>244</td><td>3892</td><td>3276</td><td>1372</td><td>1058</td><td>760</td></tr><tr><td>Rank1</td><td>1</td><td>6</td><td>5</td><td>8</td><td>4</td><td>1</td><td>1</td><td>7</td></tr><tr><td>Rank2</td><td>3</td><td>6</td><td>5</td><td>8</td><td>4</td><td>2</td><td>1</td><td>7</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F7</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>3.0632E+02</td><td>5.4000E−03</td><td>8.8506E+00</td><td>1.1000E−03</td><td>1.6400E−02</td><td>2.5230E−01</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>3.0632E+02</td><td>1.1400E−02</td><td>8.8327E+00</td><td>1.1000E−03</td><td>2.8000E−03</td><td>1.5470E−01</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>0.0000E+00</td><td>3.0426E−03</td><td>4.6054E−04</td><td>4.0900E−04</td><td>2.8791E−05</td><td>4.1191E−04</td><td>3.1404E−03</td></tr><tr><td>Time/s</td><td>353</td><td>7</td><td>2927</td><td>77</td><td>32</td><td>457</td><td>2086</td><td>5597</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>8</td><td>4</td><td>7</td><td>3</td><td>5</td><td>6</td></tr><tr><td>Rank2</td><td>2</td><td>1</td><td>8</td><td>4</td><td>7</td><td>3</td><td>5</td><td>6</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F8</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>4.7000E−03</td><td><bold>0.0000E+00</bold></td><td>4.4000E−03</td><td>1.0000E−04</td><td>1.3000E−03</td><td>2.3000E−03</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>2.5000E−03</td><td><bold>0.0000E+00</bold></td><td>2.8000E−03</td><td>1.0000E−04</td><td>3.0000E−04</td><td>1.5000E−03</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>0.0000E+00</td><td>3.6148E−04</td><td>0.0000E+00</td><td>3.3552E−04</td><td>3.3273E−05</td><td>4.0662E−05</td><td>4.4275E−05</td></tr><tr><td>Time/s</td><td>160</td><td>6</td><td>16</td><td>122</td><td>360</td><td>2115</td><td>491</td><td>328</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>8</td><td>1</td><td>7</td><td>4</td><td>5</td><td>6</td></tr><tr><td>Rank2</td><td>3</td><td>1</td><td>8</td><td>2</td><td>7</td><td>4</td><td>5</td><td>6</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F9</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>6.9000E−03</td><td>2.9100E−02</td><td>3.3000E−03</td><td>2.7000E−03</td><td>5.5000E−03</td><td>3.3000E−03</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>4.6000E−03</td><td>1.9100E−02</td><td>3.9000E−03</td><td>2.6000E−03</td><td>3.7000E−03</td><td>3.7000E−03</td></tr><tr><td>STD</td><td>3.0479E−05</td><td>4.6563E−05</td><td>3.2479E−04</td><td>3.9442E−04</td><td>4.3589E−04</td><td>3.8901E−05</td><td>3.4213E−04</td><td>3.8233E−05</td></tr><tr><td>Time/s</td><td>489</td><td>11</td><td>1171</td><td>48</td><td>99</td><td>1615</td><td>469</td><td>472</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>7</td><td>8</td><td>4</td><td>3</td><td>6</td><td>4</td></tr><tr><td>Rank2</td><td>2</td><td>1</td><td>7</td><td>8</td><td>4</td><td>3</td><td>6</td><td>5</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F10</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>5.0000E−04</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>4.0000E−04</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td></tr><tr><td>STD</td><td>0.0000E+00</td><td>0.0000E+00</td><td>3.5156E−05</td><td>0.0000E+00</td><td>2.9912E−05</td><td>3.4835E−05</td><td>0.0000E+00</td><td>2.7635E−05</td></tr><tr><td>Time/s</td><td>246</td><td>2</td><td>0</td><td>24</td><td>11</td><td>99</td><td>263</td><td>137</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Rank2</td><td>8</td><td>2</td><td>1</td><td>4</td><td>3</td><td>5</td><td>7</td><td>6</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F11</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>1.6300E</bold>−<bold>02</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>1.5500E</bold>−<bold>02</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td></tr><tr><td>STD</td><td>0.0000E+00</td><td>0.0000E+00</td><td>3.8533E−05</td><td>0.0000E+00</td><td>3.8176E−04</td><td>2.2409E−05</td><td>0.0000E+00</td><td>2.4235E−05</td></tr><tr><td>Time/s</td><td>60</td><td>2</td><td>544</td><td>736</td><td>14</td><td>383</td><td>251</td><td>247</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>8</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Rank2</td><td>2</td><td>1</td><td>6</td><td>7</td><td>8</td><td>5</td><td>4</td><td>3</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F12</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>1.0000E−04</td><td><bold>0.0000E+00</bold></td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>1.0000E−04</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>1.0000E−04</td><td><bold>0.0000E+00</bold></td></tr><tr><td>STD</td><td>0.0000E+00</td><td>0.0000E+00</td><td>4.3581E−05</td><td>4.0126E−05</td><td>3.2116E−05</td><td>3.5867E−05</td><td>3.5737E−05</td><td>4.2985E−05</td></tr><tr><td>Time/s</td><td>456</td><td>5</td><td>0</td><td>51</td><td>88</td><td>1408</td><td>347</td><td>140</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>8</td><td>1</td></tr><tr><td>Rank2</td><td>7</td><td>2</td><td>1</td><td>3</td><td>4</td><td>6</td><td>8</td><td>5</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F13</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td></tr><tr><td>STD</td><td>0.0000E+00</td><td>0.0000E+00</td><td>3.9454E−05</td><td>3.8178E−05</td><td>4.4735E−05</td><td>3.7556E−05</td><td>3.7292E−05</td><td>3.7416E−05</td></tr><tr><td>Time/s</td><td>326</td><td>3</td><td>1</td><td>1895</td><td>85</td><td>2604</td><td>482</td><td>199</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Rank2</td><td>5</td><td>2</td><td>1</td><td>7</td><td>3</td><td>8</td><td>6</td><td>4</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F14</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>1.1000E−03</td><td>1.5100E−02</td><td>2.8841E+03</td><td>9.7000E−03</td><td>7.7000E−03</td><td>0.0000E+00</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>1.1000E−03</td><td>1.5000E−02</td><td>2.8841E+03</td><td>9.6000E−03</td><td>7.7000E−03</td><td>0.0000E+00</td></tr><tr><td>STD</td><td>4.1160E−05</td><td>3.2139E−05</td><td>3.1498E−05</td><td>4.2935E−05</td><td>3.5650E−05</td><td>3.3448E−05</td><td>3.4866E−05</td><td>3.4194E−05</td></tr><tr><td>Time/s</td><td>232</td><td>8</td><td>1272</td><td>249</td><td>599</td><td>77</td><td>130</td><td>99</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>4</td><td>7</td><td>8</td><td>6</td><td>5</td><td>1</td></tr><tr><td>Rank2</td><td>3</td><td>1</td><td>4</td><td>7</td><td>8</td><td>6</td><td>5</td><td>2</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F15</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>1.0000E−04</td><td>5.0000E−04</td><td>1.4000E−03</td><td>3.0000E−04</td><td><bold>0.0000E+00</bold></td><td>5.0000E−04</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>1.0000E−04</td><td>5.0000E−04</td><td>1.3000E−03</td><td>1.0000E−04</td><td>−1.0000E−04</td><td>5.0000E−04</td></tr><tr><td>STD</td><td>2.8171E−05</td><td>3.3254E−05</td><td>4.1701E−05</td><td>3.5130E−05</td><td>3.5641E−05</td><td>3.4427E−05</td><td>4.9930E−05</td><td>4.0094E−05</td></tr><tr><td>Time/s</td><td>188</td><td>9</td><td>327</td><td>445</td><td>121</td><td>444</td><td>504</td><td>93</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>1</td><td>5</td><td>8</td><td>7</td><td>1</td><td>5</td></tr><tr><td>Rank2</td><td>2</td><td>1</td><td>3</td><td>6</td><td>8</td><td>7</td><td>4</td><td>5</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F16</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>1.0000E−04</td><td>1.0000E−04</td><td>3.0000E−04</td><td>1.0000E−04</td><td><bold>0.0000E+00</bold></td><td>1.0000E−04</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>1.0000E−04</td><td>1.0000E−04</td><td>3.0000E−04</td><td>1.0000E−04</td><td><bold>0.0000E+00</bold></td><td>0.0000E+00</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>0.0000E+00</td><td>3.7005E−05</td><td>2.6888E−05</td><td>4.4005E−05</td><td>3.1010E−05</td><td>0.0000E+00</td><td>2.6570E−05</td></tr><tr><td>Time/s</td><td>1242</td><td>6</td><td>1946</td><td>1844</td><td>141</td><td>394</td><td>552</td><td>479</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>1</td><td>5</td><td>8</td><td>5</td><td>1</td><td>5</td></tr><tr><td>Rank2</td><td>3</td><td>1</td><td>4</td><td>7</td><td>8</td><td>5</td><td>2</td><td>6</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F17</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>5.1000E−03</td><td>3.8000E−03</td><td>1.8000E−03</td><td>2.0247E+00</td><td>4.3200E−02</td><td>9.6000E−03</td><td><bold>0.0000E+00</bold></td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td>5.4000E−03</td><td>3.7000E−03</td><td>1.9000E−03</td><td>2.0246E+00</td><td>4.3300E−02</td><td>9.6000E−03</td><td><bold>0.0000E+00</bold></td></tr><tr><td>STD</td><td>0.0000E+00</td><td>3.7792E−05</td><td>4.0089E−05</td><td>4.4945E−05</td><td>3.1251E−05</td><td>4.0504E−05</td><td>4.1728E−05</td><td>0.0000E+00</td></tr><tr><td>Time/s</td><td>0</td><td>7</td><td>653</td><td>552</td><td>489</td><td>239</td><td>625</td><td>0</td></tr><tr><td>Rank1</td><td>1</td><td>5</td><td>4</td><td>3</td><td>8</td><td>7</td><td>6</td><td>1</td></tr><tr><td>Rank2</td><td>1</td><td>5</td><td>4</td><td>3</td><td>8</td><td>7</td><td>6</td><td>1</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F18</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>2.0000E−04</td><td>0.0000E+00</td><td>2.8252E+02</td><td>1.2140E−01</td><td><bold>0.0000E+00</bold></td><td>1.3000E−03</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>2.0000E−04</td><td>0.0000E+00</td><td>2.8252E+02</td><td>1.2560E−01</td><td><bold>0.0000E+00</bold></td><td>1.3000E−03</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>0.0000E+00</td><td>3.4344E−05</td><td>0.0000E+00</td><td>3.6611E−03</td><td>4.1844E−03</td><td>0.0000E+00</td><td>3.8839E−05</td></tr><tr><td>Time/s</td><td>274</td><td>3</td><td>1</td><td>570</td><td>188</td><td>18</td><td>647</td><td>26</td></tr><tr><td>Rank1</td><td>1</td><td>1</td><td>5</td><td>1</td><td>8</td><td>7</td><td>1</td><td>6</td></tr><tr><td>Rank2</td><td>2</td><td>1</td><td>5</td><td>3</td><td>8</td><td>7</td><td>4</td><td>6</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F19</td></tr><tr><td>Average</td><td><bold>1.1733E+00</bold></td><td>4.1841E+02</td><td>5.3082E+01</td><td>1.0809E+04</td><td>3.8540E+01</td><td>2.1548E+00</td><td>2.0825E+00</td><td>2.8233E+01</td></tr><tr><td>Median</td><td><bold>1.1159E+00</bold></td><td>4.1841E+02</td><td>5.3087E+01</td><td>1.0810E+04</td><td>3.8548E+01</td><td>2.1445E+00</td><td>2.0447E+00</td><td>2.8226E+01</td></tr><tr><td>STD</td><td>2.9033 E+00</td><td>4.4896E−04</td><td>4.1200E−04</td><td>2.3979E+01</td><td>4.1444E−03</td><td>2.7829E−03</td><td>4.1410E−03</td><td>3.6863E−03</td></tr><tr><td>Time/s</td><td>15,665</td><td>10,150</td><td>3123</td><td>10,328</td><td>2351</td><td>199</td><td>1296</td><td>720</td></tr><tr><td>Rank1</td><td>1</td><td>7</td><td>6</td><td>8</td><td>5</td><td>3</td><td>2</td><td>4</td></tr><tr><td>Rank2</td><td>1</td><td>7</td><td>6</td><td>8</td><td>5</td><td>3</td><td>2</td><td>4</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F20</td></tr><tr><td>Average</td><td><bold>9.7850E−01</bold></td><td>8.5818E+00</td><td>3.6954E+00</td><td>1.9024E+00</td><td>5.1151E+00</td><td>1.3028E+01</td><td>1.3953E+01</td><td>1.1739E+01</td></tr><tr><td>Median</td><td><bold>5.1430E−01</bold></td><td>8.5827E+00</td><td>3.6938E+00</td><td>1.9012E+00</td><td>5.1134E+00</td><td>1.3021E+01</td><td>1.3952E+01</td><td>1.1738E+01</td></tr><tr><td>STD</td><td>2.0661E−01</td><td>4.1805E−03</td><td>4.0235E−03</td><td>2.9896E−03</td><td>3.5409E−03</td><td>4.3044E−053</td><td>3.2601E−03</td><td>5.0311E−03</td></tr><tr><td>Time/s</td><td>3860</td><td>3701</td><td>716</td><td>499</td><td>4546</td><td>400</td><td>99</td><td>553</td></tr><tr><td>Rank1</td><td>1</td><td>5</td><td>3</td><td>2</td><td>4</td><td>7</td><td>8</td><td>6</td></tr><tr><td>Rank2</td><td>1</td><td>5</td><td>3</td><td>2</td><td>4</td><td>7</td><td>8</td><td>6</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F21</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>5.2485E+03</td><td>1.4045E+01</td><td>7.7911E+00</td><td>2.1558E+01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>4.7579E+00</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td>5.2483E+03</td><td>1.4044E+01</td><td>7.7902E+00</td><td>2.1551E+01</td><td><bold>0.0000E+00</bold></td><td><bold>0.0000E+00</bold></td><td>4.7513E+00</td></tr><tr><td>STD</td><td>0.0000E+00</td><td>9.6462E−01</td><td>4.0378E−03</td><td>2.9258E−03</td><td>3.6489E−03</td><td>0.0000E+00</td><td>0.0000E+00</td><td>3.4583E−03</td></tr><tr><td>Time/s</td><td>711</td><td>4543</td><td>2581</td><td>12,506</td><td>475</td><td>197</td><td>225</td><td>5379</td></tr><tr><td>Rank1</td><td>1</td><td>8</td><td>6</td><td>5</td><td>7</td><td>1</td><td>1</td><td>4</td></tr><tr><td>Rank2</td><td>3</td><td>8</td><td>6</td><td>5</td><td>7</td><td>1</td><td>2</td><td>4</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F22</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>2.9864E+05</td><td>−1.1441E+02</td><td>1.3629E+03</td><td>1.8143E+03</td><td>5.9107E+03</td><td>1.5730E+02</td><td>5.3596E+02</td></tr><tr><td>Median</td><td><bold>1.1284E+00</bold></td><td>2.9865E+05</td><td>−1.1441E+02</td><td>1.3629E+03</td><td>1.8143E+03</td><td>5.9105E+03</td><td>1.5730E+02</td><td>5.3596E+02</td></tr><tr><td>STD</td><td>1.1563E+01</td><td>2.0103E+01</td><td>3.7340E−05</td><td>3.6630E−05</td><td>5.4300E−05</td><td>4.1423E+00</td><td>5.1860E−05</td><td>3.9080E−05</td></tr><tr><td>Time/s</td><td>4187</td><td>249</td><td>1570</td><td>1931</td><td>2611</td><td>331</td><td>3998</td><td>1577</td></tr><tr><td>Rank1</td><td>2</td><td>8</td><td>1</td><td>5</td><td>6</td><td>7</td><td>3</td><td>4</td></tr><tr><td>Rank2</td><td>2</td><td>8</td><td>1</td><td>5</td><td>6</td><td>7</td><td>3</td><td>4</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F23</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>1.9792E+03</td><td>2.0031E+02</td><td>1.4306E+01</td><td>2.0492E+02</td><td>3.6572E+02</td><td>2.1345E+02</td><td>6.3643E+01</td></tr><tr><td>Median</td><td><bold>8.3770E−01</bold></td><td>1.9792E+03</td><td>2.0030E+02</td><td>1.4298E+01</td><td>2.0492E+02</td><td>3.6572E+02</td><td>2.1344E+02</td><td>6.3639E+01</td></tr><tr><td>STD</td><td>1.2205E+01</td><td>4.2570E−04</td><td>3.4260E−03</td><td>3.8950E−03</td><td>3.4220E−03</td><td>4.6370E−03</td><td>3.9050E−03</td><td>3.5760E−03</td></tr><tr><td>Time/s</td><td>683</td><td>2001</td><td>2660</td><td>1149</td><td>6804</td><td>654</td><td>2210</td><td>2720</td></tr><tr><td>Rank1</td><td>1</td><td>8</td><td>4</td><td>2</td><td>5</td><td>7</td><td>6</td><td>3</td></tr><tr><td>Rank2</td><td>1</td><td>8</td><td>4</td><td>2</td><td>5</td><td>7</td><td>6</td><td>3</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F24</td></tr><tr><td>Average</td><td>3.5742E+02</td><td>5.7192E+02</td><td>5.6956E+02</td><td>4.3088E+02</td><td>5.6900E+02</td><td>4.8506E+02</td><td><bold>0.0000E+00</bold></td><td>7.9204E+01</td></tr><tr><td>Median</td><td>3.5804E+02</td><td>5.7184E+02</td><td>5.6938E+02</td><td>4.3090E+02</td><td>5.6914E+02</td><td>4.8518E+02</td><td><bold>0.0000E+00</bold></td><td>7.9205E+01</td></tr><tr><td>STD</td><td>1.1659E+00</td><td>1.4154E+00</td><td>1.2980E+00</td><td>1.0883E+00</td><td>1.2643E+00</td><td>1.2272E+00</td><td>3.8259E−05</td><td>3.8754E−03</td></tr><tr><td>Time/s</td><td>11,152</td><td>946</td><td>1599</td><td>1164</td><td>1170</td><td>433</td><td>2567</td><td>1772</td></tr><tr><td>Rank1</td><td>3</td><td>8</td><td>7</td><td>4</td><td>6</td><td>5</td><td>1</td><td>2</td></tr><tr><td>Rank2</td><td>3</td><td>8</td><td>7</td><td>4</td><td>6</td><td>5</td><td>1</td><td>2</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F25</td></tr><tr><td>Average</td><td><bold>6.2533E+01</bold></td><td>7.9420E+01</td><td>8.9847E+01</td><td>6.7286E+01</td><td>9.7890E+01</td><td>6.2932E+01</td><td>8.7304E+01</td><td>7.3093E+01</td></tr><tr><td>Median</td><td><bold>6.1276E+01</bold></td><td>7.9719E+01</td><td>9.0006E+01</td><td>6.7203E+01</td><td>9.7872E+01</td><td>6.2600E+01</td><td>8.7701E+01</td><td>7.3341E+01</td></tr><tr><td>STD</td><td>1.5797E+00</td><td>1.1098E+00</td><td>1.1801E+00</td><td>1.0510E+00</td><td>1.2081E+00</td><td>1.6040E+00</td><td>9.3316E−01</td><td>1.2855E+00</td></tr><tr><td>Time/s</td><td>2249</td><td>581</td><td>7</td><td>2623</td><td>888</td><td>1136</td><td>588</td><td>1805</td></tr><tr><td>Rank1</td><td>1</td><td>5</td><td>7</td><td>3</td><td>8</td><td>2</td><td>6</td><td>4</td></tr><tr><td>Rank2</td><td>1</td><td>5</td><td>7</td><td>3</td><td>8</td><td>2</td><td>6</td><td>4</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F26</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>8.0700E−01</td><td>1.2060E+03</td><td>4.9700E−01</td><td>8.3540E+01</td><td>8.6240E−01</td><td>1.3381E+00</td><td>4.9690E−01</td></tr><tr><td>Median</td><td><bold>0.0000E+00</bold></td><td>8.0790E−01</td><td>1.2063E+03</td><td>4.9810E−01</td><td>8.3539E+01</td><td>8.6240E−01</td><td>1.3327E+00</td><td>4.9710E−01</td></tr><tr><td>STD</td><td>3.9631E−01</td><td>3.4304E−03</td><td>1.1094E+00</td><td>4.7387E−04</td><td>2.7967E−03</td><td>4.4850E−03</td><td>3.9554E−03</td><td>3.4717E−03</td></tr><tr><td>Time</td><td>6137</td><td>555</td><td>319</td><td>5649</td><td>575</td><td>3321</td><td>737</td><td>1272</td></tr><tr><td>Rank1</td><td>1</td><td>4</td><td>8</td><td>3</td><td>7</td><td>5</td><td>6</td><td>2</td></tr><tr><td>Rank2</td><td>1</td><td>4</td><td>8</td><td>3</td><td>7</td><td>5</td><td>6</td><td>2</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F27</td></tr><tr><td>Average</td><td>1.1446E+01</td><td>2.1338E+01</td><td><bold>1.5291E+00</bold></td><td>6.4663E+00</td><td>8.7212E+00</td><td>9.7071E+00</td><td>3.8324E+01</td><td>3.5825E+00</td></tr><tr><td>Median</td><td>1.2770E+01</td><td>2.0967E+01</td><td><bold>9.7730E−01</bold></td><td>6.4517E+00</td><td>8.6188E+00</td><td>1.0310E+01</td><td>3.8167E+01</td><td>3.6585E+00</td></tr><tr><td>STD</td><td>2.4204E+00</td><td>1.3671E+00</td><td>1.5052E+00</td><td>1.4331E+00</td><td>1.0570E+00</td><td>1.5268E+00</td><td>9.4364E−01</td><td>1.1675E+00</td></tr><tr><td>Time</td><td>1765</td><td>4155</td><td>3520</td><td>4285</td><td>120</td><td>3553</td><td>1162</td><td>3063</td></tr><tr><td>Rank1</td><td>6</td><td>7</td><td>1</td><td>3</td><td>4</td><td>5</td><td>8</td><td>2</td></tr><tr><td>Rank2</td><td>6</td><td>7</td><td>1</td><td>3</td><td>4</td><td>5</td><td>8</td><td>2</td></tr><tr><td colspan="9">
</td></tr><tr><td colspan="9">F28</td></tr><tr><td>Average</td><td><bold>0.0000E+00</bold></td><td>1.2589E+04</td><td>8.7236E+03</td><td>2.2516E+03</td><td>8.7726E+03</td><td>7.8117E+03</td><td>7.7779E+03</td><td>8.8673E+03</td></tr><tr><td>Median</td><td><bold>1.8380E−01</bold></td><td>1.2589E+04</td><td>8.7232E+03</td><td>2.2516E+03</td><td>8.7728E+03</td><td>7.8116E+03</td><td>7.7780E+03</td><td>8.8671E+03</td></tr><tr><td>STD</td><td>4.4912E−01</td><td>1.2685E+00</td><td>1.4660E+00</td><td>4.1750E−06</td><td>1.2277E+00</td><td>1.3148E+00</td><td>1.3627E+00</td><td>1.3491E+00</td></tr><tr><td>Time/s</td><td>110,475</td><td>5910</td><td>2565</td><td>15,121</td><td>2083</td><td>3167</td><td>4760</td><td>1438</td></tr><tr><td>Rank 1</td><td>1</td><td>8</td><td>5</td><td>2</td><td>6</td><td>4</td><td>3</td><td>7</td></tr><tr><td>Rank 2</td><td>1</td><td>8</td><td>5</td><td>2</td><td>6</td><td>4</td><td>3</td><td>7</td></tr><tr><td>Sum Rank 1</td><td>36</td><td>122</td><td>118</td><td>119</td><td>156</td><td>100</td><td>92</td><td>113</td></tr><tr><td>Sum Rank 2</td><td>76</td><td>126</td><td>128</td><td>142</td><td>163</td><td>122</td><td>117</td><td>133</td></tr><tr><td>Final Rank 1</td><td>1</td><td>7</td><td>6</td><td>5</td><td>8</td><td>3</td><td>2</td><td>4</td></tr><tr><td>Final Rank 2</td><td>1</td><td>7</td><td>6</td><td>5</td><td>8</td><td>3</td><td>2</td><td>4</td></tr></tbody></table></table-wrap></p><p id="p1080">Also, the non-parametric Wilcoxon rank sum test <xref rid="bib38" ref-type="bibr">[38]</xref>, <xref rid="bib89" ref-type="bibr">[89]</xref> is conducted between the SEIQRA’s result and the best results achieved by the other 7 algorithms listed in <xref rid="t0090" ref-type="table">Table 18</xref> for each benchmark function to determine whether the results generated by SEIQRA are statistically different from the results obtained by the other algorithms. With the help of the famous statistical software package SPSS (version 19.0) and Excel (version 2003) which is used to transfer <italic>z</italic>-value into <italic>p</italic>-value, the Wilcoxon rank sum test is achieved, the results are shown in
<xref rid="t0100" ref-type="table">Table 20</xref>, where <italic>h</italic>-value=1 indicates that the performances of SEIQRA is statistically different with 99% certainty, <italic>h</italic>-value=−1 represents that the compared algorithm are significantly better than SEIQRA, and <italic>h</italic>-value=0 denotes that the results of the two considered algorithms are not significantly different. In <xref rid="t0100" ref-type="table">Table 20</xref>, rows 1(Better), 0 (Same), and −1 (Worse) give the number of benchmark functions that SEIQRA performs significantly better than, almost the same as, and significantly worse than the compared algorithm, respectively.<table-wrap position="float" id="t0100"><label>Table 20</label><caption><p>Comparative results of SEIQRA for the benchmark functions using Wilcoxon’s rank sum test (<italic>α</italic>=0.01).</p></caption><alt-text id="at0200">Table 20</alt-text><table frame="hsides" rules="groups"><thead><tr><th rowspan="2">Benchmark function</th><th rowspan="2">Wilcoxon’s rank sum test</th><th colspan="7">SEIQRA vs.<hr></hr></th></tr><tr><th>RC-GA</th><th>DSDA</th><th>NP-PSO</th><th>BBO</th><th>DE</th><th>SaDE</th><th>ABC</th></tr></thead><tbody><tr><td>F1</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>0.000</td><td>0.000</td><td>0.000</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F2</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8182E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.921</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F3</td><td><italic>p</italic>-value</td><td>8.8182E−05</td><td>8.8182E−05</td><td>8.8549E−05</td><td>8.8182E−05</td><td>1</td><td>1</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>0</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.921</td><td>−3.921</td><td>−3.920</td><td>−3.921</td><td>0.000</td><td>0.000</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F4</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>1</td><td>2.3959E−01</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>0</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>0.000</td><td>−1.176</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F5</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>1.2349E−02</td><td>1.4539E−01</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>0</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−2.502</td><td>−1.456</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F6</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>9.9951E−03</td><td>1.6728E−01</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−2.576</td><td>−1.381</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F7</td><td><italic>p</italic>-value</td><td>1</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8182E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>0.000</td><td>−3.920</td><td>−3.920</td><td>−3.921</td><td>−3.920</td><td>−3.920</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F8</td><td><italic>p</italic>-value</td><td>1</td><td>8.8549E−05</td><td>1</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8182E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>1</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>0.000</td><td>−3.920</td><td>0.000</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.921</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F9</td><td><italic>p</italic>-value</td><td>3.3155E−01</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−0.971</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F10</td><td><italic>p</italic>-value</td><td>1</td><td>1.6894E−02</td><td>1</td><td>8.2276E−01</td><td>8.8549E−05</td><td>1</td><td>2.9602E−01</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td></tr><tr><td></td><td><italic>z</italic>val</td><td>0.000</td><td>−2.389</td><td>0.000</td><td>−0.224</td><td>−3.920</td><td>0.000</td><td>−1.045</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F11</td><td><italic>p</italic>-value</td><td>1</td><td>8.8155E−01</td><td>1</td><td>8.8549E−05</td><td>1.6728E−01</td><td>1</td><td>8.2564E−02</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td><italic>z</italic>val</td><td>0.000</td><td>−0.149</td><td>0.000</td><td>−3.920</td><td>−1.381</td><td>0.000</td><td>−1.736</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F12</td><td><italic>p</italic>-value</td><td>1</td><td>1.0162E−03</td><td>3.9034E−01</td><td>6.8107E−01</td><td>8.5166E−01</td><td>8.8549E−05</td><td>8.5979E−02</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td></tr><tr><td></td><td><italic>z</italic>val</td><td>0.000</td><td>−3.286</td><td>−0.859</td><td>−0.411</td><td>−0.187</td><td>−3.920</td><td>−1.717</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F13</td><td><italic>p</italic>-value</td><td>1</td><td>1.6728E−01</td><td>5.2258E−02</td><td>3.7025E−01</td><td>8.8155E−01</td><td>9.7049E−01</td><td>2.7881E−01</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td><italic>z</italic>val</td><td>0.000</td><td>−1.381</td><td>−1.941</td><td>−0.896</td><td>−0.149</td><td>−0.037</td><td>−1.083</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F14</td><td><italic>p</italic>-value</td><td>6.1902E−02</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8182E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−1.867</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.921</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F15</td><td><italic>p</italic>-value</td><td>3.7025E−01</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>7.7948E−01</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−.896</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−0.280</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F16</td><td><italic>p</italic>-value</td><td>1</td><td>1.4010E−04</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>1</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>0.000</td><td>−3.808</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>0</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F17</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>1</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>-3.920</td><td>0.000</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F18</td><td><italic>p</italic>-value</td><td>1</td><td>8.8549E−05</td><td>1</td><td>8.8549E−05</td><td>8.8549E−05</td><td>1</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>0</td><td>1</td><td>0</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>0</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>0</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F19</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F20</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F21</td><td><italic>p</italic>-value</td><td>8.6370E−05</td><td>8.6370E−05</td><td>8.6370E−05</td><td>8.6370E−05</td><td>1</td><td>1</td><td>8.6370E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>0</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>0.000</td><td>0.000</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F22</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8182E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.7817E−05</td><td>8.8182E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>−1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.921</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.922</td><td>−3.921</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F23</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.7453E−05</td><td>8.7453E−05</td><td>8.6370E−05</td><td>8.6730E−05</td><td>8.7091E−05</td><td>8.7091E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.923</td><td>−3.923</td><td>−3.926</td><td>−3.925</td><td>−3.924</td><td>−3.924</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F24</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>−1</td><td>−1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F25</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F26</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>−1</td><td>1</td><td>1</td><td>1</td><td>−1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td><td>−3.920</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F27</td><td><italic>p</italic>-value</td><td>1.7127E−03</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>−1</td><td>−1</td><td>−1</td><td>−1</td><td>1</td><td>−1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.136</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>F28</td><td><italic>p</italic>-value</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td><td>8.8549E−05</td></tr><tr><td></td><td><italic>h</italic>-value</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td><italic>z</italic>val</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td><td>−3.92</td></tr><tr><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td><td>
</td></tr><tr><td>1 (Better)</td><td></td><td>17</td><td>23</td><td>20</td><td>24</td><td>19</td><td>15</td><td>19</td></tr><tr><td>0 (Same)</td><td></td><td>11</td><td>3</td><td>6</td><td>3</td><td>8</td><td>12</td><td>6</td></tr><tr><td>−1 (Worse)</td><td></td><td>0</td><td>2</td><td>2</td><td>1</td><td>1</td><td>1</td><td>3</td></tr></tbody></table></table-wrap></p><p id="p1085">From <xref rid="t0100" ref-type="table">Table 20</xref> we can know that SEIQRA performs significantly better than the other 7 algorithms.</p><p id="p1090"><xref rid="s0165" ref-type="sec">Appendix B</xref> illustrates the sample convergence curves of SEIQRA, RC-GA, NP-PSO, BBO, DE, SaDE and ABC when they solve benchmark F1–F28. In order to highlight the change of these sample convergence curves, the horizontal and vertical axis are illustrated with logarithmic scale.</p></sec><sec id="s0130"><label>5.2</label><title>Discussion</title><sec id="s0135"><label>5.2.1</label><title>Dynamic behaviors of operators in SEIQRA</title><p id="p1095">Analysis of dynamic behaviors of operators in SEIQRA is as follows:<list list-type="simple" id="li0125"><list-item id="o0380"><label>(1)</label><p id="p1100">Dynamic behaviors of the ecological operators: S–S, S–E, S–R, E–E, E–I, E–R, I–I, I–Q, I–R, Q–Q, Q–R, R–R and R–S.
<xref rid="f0065" ref-type="fig">Fig. 13</xref> describes that the relationship between times of state transition S→S, S→E, S→R, E→E, E→I, E→R, I→I, I→Q, I→R, Q→Q, Q→R, R→R and R→S being triggered and CPU time when SEIQRA solves F1. It shows that triggered times of these state transitions vary stochastically with time, but the average triggered times are stable for each state transition because the average triggered times approximate to level for all state transitions. Because each state transition corresponds to an operator, these ecological operators are triggered evenly.<fig id="f0065"><label>Fig. 13</label><caption><p>Times of state transition S→S, S→E, S→R, E→E, E→I, E→R, I→I, I→Q, I→R, Q→Q, Q→R, R→R and R→S being triggered during evolution of individuals when SEIQRA solves F1.</p></caption><alt-text id="at0065">Fig. 13</alt-text><graphic xlink:href="gr13"></graphic></fig></p></list-item></list><list list-type="simple" id="li0130"><list-item id="o0385"><label>(2)</label><p id="p1105">The heart rhythm of SEIQRA. SEIQRA possesses of 5 states which divide a population into 5 classes dynamically and automatically. The number of individuals in each class varies with time. The dynamic change of the number of individuals in 5 classes can be thought as the heart beating of SEIQRA, or the heart rhythm of SEIQRA.
<xref rid="f0070" ref-type="fig">Fig. 14</xref> describes that the relationship of the number of individuals staying at state S, E, I, Q and R with CPU time when SEIQRA solves F1. It shows that the number of individuals staying at state S, E, I, Q and R vary stochastically with time, but the average number of individuals staying at each state is also stable because the average number of individuals staying at each state approximates to level.<fig id="f0070"><label>Fig. 14</label><caption><p>The number of individuals staying at state S, E, I, Q and R during evolution of individuals when SEIQRA solves F1.</p></caption><alt-text id="at0070">Fig. 14</alt-text><graphic xlink:href="gr14"></graphic></fig></p></list-item></list></p><p id="p1110">Conclusively, the stability of the heart rhythm of SEIQRA can result in equi-probably triggering of the ecological operators S–S, S–E, S–R, E–E, E–I, E–R, I–I, I–Q, I–R, Q–Q, Q–R, R–R and R–S, because each state transition is equivalent to a triggering of the corresponding operator.</p><p id="p1115">The stability of the heart rhythm of SEIQRA is controlled by parameters <italic>β</italic><sub>0</sub>, <italic>γ</italic><sub>1</sub>, <italic>γ</italic><sub>2</sub>, <italic>γ</italic><sub>3</sub>, <italic>p</italic> and <italic>δ</italic>; when these parameters are set according to <xref rid="t0015" ref-type="table">Table 3</xref>, the stability of the heart rhythm of SEIQRA is guaranteed.</p></sec><sec id="s0140"><label>5.2.2</label><title>Dynamic behaviors of operations in SEIQRA</title><p id="p1120">SEIQRA possesses of 6 operations <italic>AVG</italic>, <italic>DE</italic>, <italic>EPN</italic>, <italic>CHV</italic>, <italic>RFL</italic> and <italic>CRS</italic>. The times of each operation being executed varies with time.
<xref rid="f0075" ref-type="fig">Fig. 15</xref> describes that the relationship of the times of each operation being executed with the CPU time when SEIQRA solves F1. It shows that the times of each operation being executed vary stochastically with time, but the average times of each operation being executed is also stable because the average times of each operation being executed approximates to level.<fig id="f0075"><label>Fig. 15</label><caption><p>Relationship of the times of each operation being executed with the CPU time when SEIQRA solves F1.</p></caption><alt-text id="at0075">Fig. 15</alt-text><graphic xlink:href="gr15"></graphic></fig></p></sec><sec id="s0145"><label>5.2.3</label><title>Dynamic behavior of operator REINIT in SEIQRA</title><p id="p1125">The dynamic behavior of REINIT is controlled by <italic>N</italic><sub>0</sub>, <italic>t</italic><sub><italic>max</italic></sub>, <italic>ρ</italic>, <italic>LU</italic><sub>0</sub> and <italic>r</italic>. REINIT is used to enhance the exploration and exploitation ability of SEIQRA. REINIT includes two operations: reinitialization and redistribution, as mentioned in <xref rid="s0080" ref-type="sec">Section 2.4.3.3</xref>.</p><p id="p1130">At stage of operation reinitialization, a large number of individuals are used to reinitialize a search space, After reinitialization, if a new global optimum solution is found, it means that the activation of REINIT is effective, we call it the effective triggering point, the current global optimum solution is updated into the newly-found global optimum solution; else the current global optimum solution keeps unchanged. The property of REINIT is called as the exploitation ability of REINIT.</p><p id="p1135">At stage of operation redistribution, a small number of individuals are used to reinitialize the search space determined by the current global optimum solution. After redistribution, a normal search is made by SEIQRA. The positions of the work individuals are always updated no matter whether the current global optimum solution is updated or not. Therefore sticky states of search may be broken. The property of REINIT is called as the exploration ability of REINIT.</p><p id="p1140">During searching process of SEIQRA, the identification of effectiveness of REINIT is illustrated in
<xref rid="f0080" ref-type="fig">Fig. 16</xref>, where ALM is the IPI convergence curve when SEIQRA solves an optimization problem; the series of points marked by circle is the assigned triggering points, namely at that time REINIT is invoked; point K is the starting time that REINIT is allowed to invoke; KN is a line constructed by use of the effective triggering points; point B is the first assigned triggering point, point C is the second assigned triggering point, …, point J is the last assigned triggering point; the series of the effective triggering points is marked by numbers 1, 2, 3, 4 and 5. Number 1 means the assigned triggering point C takes effect because the current global optimum solution is updated. Similarly, Numbers 2, 3, 4 and 5 mean that points D, E, G and I take effect respectively. Therefore, C, D, E, G and I are effective triggering points, while points B, F, H, J are ineffective triggering points.<fig id="f0080"><label>Fig. 16</label><caption><p>Identification of working effectiveness of REINIT.</p></caption><alt-text id="at0080">Fig. 16</alt-text><graphic xlink:href="gr16"></graphic></fig></p><p id="p1145">It is noteworthy that all assigned triggering points can always change positions of work individuals. Therefore it can help a search to overcome sticky states.</p><p id="p1150"><xref rid="f0085" ref-type="fig">Fig. 17</xref> illustrates a general case described in <xref rid="f0080" ref-type="fig">Fig. 16</xref> when SEIQRA solves benchmark function F14 and F6. When SEIQRA solves benchmark function F14, from <xref rid="f0085" ref-type="fig">Fig. 17</xref>(a) we can see that at 3 s, REINIT is invoked; at 95 s, the first assigned triggering point is produced; at 99 s, the first effective triggering point is produced. When search is approximate to the global optimum solution of F14, three effective triggering points are produced; when SEIQRA solves benchmark function F6, from <xref rid="f0085" ref-type="fig">Fig. 17</xref>(b) we can see that at 2 s, REINIT is invoked; at 592 s, the first assigned triggering point is produced; at 1009 s, the first effective triggering point is produced. When search is approximate to the global optimum solution of F6, a larger number of effective triggering points are produced; it means that REINIT continues to enhance the precision of the global optimum solution.<fig id="f0085"><label>Fig. 17</label><caption><p>Identification of working effectiveness of REINIT when SEIQRA solves benchmark functions F14 and F6, (a) F14 and (b) F6.</p></caption><alt-text id="at0085">Fig. 17</alt-text><graphic xlink:href="gr17"></graphic></fig></p><p id="p1155"><xref rid="f0090" ref-type="fig">Fig. 18</xref> illustrates a special case when SEIQRA solves basic benchmark function F18. From <xref rid="f0090" ref-type="fig">Fig. 18</xref> we can see that at 3 s, REINIT is invoked; at 57 s, the first assigned triggering point is produced. But no effective triggering points are generated during the whole process of search, it means that REINIT is only used to break sticky states of work individuals, namely operation reinitialization does not take effect, while operation redistribution takes effect.<fig id="f0090"><label>Fig. 18</label><caption><p>Identification of working effectiveness of REINIT when SEIQRA solves basic benchmark function F18.</p></caption><alt-text id="at0090">Fig. 18</alt-text><graphic xlink:href="gr18"></graphic></fig></p><p id="p1160">When an optimization problem can be solved without the help of REINIT, if the operator is invoked forcibly, then the convergence rate of solving the problem may be decreased because continuous triggering of REINIT can consume much CPU time.
<xref rid="f0095" ref-type="fig">Fig. 19</xref> illustrates a comparison of the convergence rate between “REINIT is not invoked” and “REINIT is invoked” when SEIQRA solves basic benchmark function F3. From <xref rid="f0095" ref-type="fig">Fig. 19</xref> we can see that the convergence rate when REINIT is not invoked is faster than that when REINIT is invoked; as time lapses, triggering of REINIT become more and more frequent, much time is consumed for triggering REINIT.<fig id="f0095"><label>Fig. 19</label><caption><p>A comparison of the convergence rate between “REINIT is not invoked” and “REINIT is invoked” when SEIQRA solves basic benchmark function F3.</p></caption><alt-text id="at0095">Fig. 19</alt-text><graphic xlink:href="gr19"></graphic></fig></p><p id="p1165">Conclusively, REINIT has two effects of the pros and cons, it can enhance the exploration and exploitation ability of SEIQRA, but it may consume much CPU time. Therefore, we suggest that if an optimization problem can be solved without REINIT, REINIT should be prohibited; if sticky states always occur when SEIQRA solves an optimization problem, REINIT should be invoked.</p></sec></sec></sec><sec id="s0150"><label>6</label><title>Conclusions</title><p id="p1170">In SEIQRA, Individuals come from one population; operators S–S, S–E, S–R, E–E, E–I, E–R, I–I, I–Q, I–R, Q–Q, Q–R, R–R and R–S are constructed based on the SEIQR epidemic model, they are not related to any actual optimization problems to be solved. Because the SEIQR epidemic model does not require the support of pathological knowledge, SEIQRA does not require the support of pathological knowledge also, this characteristic is beneficial to the research and improvement of SEIQRA.</p><p id="p1175">In conclusion, SEIQRA has the following properties:<list list-type="simple" id="li0135"><list-item id="o0390"><label>1.</label><p id="p1180">The stable rhythm of heart: the stable rhythm of heart in SEIQRA can ensure the ecological operators to be activated evenly.</p></list-item><list-item id="o0395"><label>2.</label><p id="p1185">Exploration and exploitation ability: it is represented by operator REINIT, which is activated at certain frequency to improve exploration and exploitation ability.</p></list-item><list-item id="o0400"><label>3.</label><p id="p1190">Wide adaptability: if an operator can solve one type of optimization problems, then synergy of many operators can solve many types of optimization problems. Because SEIQRA has 13 operators, it has wider adaptability to solve different types of optimization problems.</p></list-item><list-item id="o0405"><label>4.</label><p id="p1195">A small part of variables being involved in each iteration: Since a virus attacks only a small part of individual features every time, when individuals in different states exchange their feature information, there is just a small portion of features used in computation, even so, the individual’s IPI can still be well improved. Owing to the substantial reduction of features in calculation, when solving complicated optimization problems, especially high-dimensional optimization problems, SEIQRA can converge significantly and rapidly.</p></list-item><list-item id="o0410"><label>5.</label><p id="p1200">The stable setting of parameters: Although SEIQRA uses more parameters to control its running, the setting of its parameters is stable.</p></list-item><list-item id="o0415"><label>6.</label><p id="p1205">Multiple operations: Because SEIQRA possesses of 13 operators, it provides opportunities to contain many operations simultaneously.</p></list-item><list-item id="o0420"><label>7.</label><p id="p1210">Convergence: When evolving, the individuals with strong physique can continue to grow, while the individuals with weak physique will stop growing. This kind of search strategy can ensure global optima to be found at higher probability.</p></list-item><list-item id="o0425"><label>8.</label><p id="p1215">Visualization: The story-based design of SEIQRA makes it easy to visualize its process of evolution when solving an optimization problem.</p><p id="p1220">SEIQRA has characteristics of high adaptability, fast solving speed and convergence for complicated function optimization problems. Besides these, precision of global optimal solutions can also be improved by operator REINIT. These characteristics of SEIQRA may facilitate practical applications. Further study in the future includes:<list list-type="simple" id="li0140"><list-item id="o0430"><label>(1)</label><p id="p1225">How to use the mechanisms of immunity to construct immunity-based operators;</p></list-item><list-item id="o0435"><label>(2)</label><p id="p1230">How to use the mechanisms of virus attacking cells to construct virus transmission operators;</p></list-item><list-item id="o0440"><label>(3)</label><p id="p1235">How to use the mechanisms of infectious disease treatment to construct infectious disease treatment operators;</p></list-item><list-item id="o0445"><label>(4)</label><p id="p1240">How to implant independent virus description factor into features of an individual;</p></list-item><list-item id="o0450"><label>(5)</label><p id="p1245">How to embed various kinds of viruses in SEIQRA;</p></list-item><list-item id="o0455"><label>(6)</label><p id="p1250">How to bring more different types of individuals into SEIQRA;</p></list-item><list-item id="o0460"><label>(7)</label><p id="p1255">How to reflect time-delay differences of outbreak of different viruses in SEIQRA.</p></list-item></list></p></list-item></list></p><p id="p1260">In nature, there are millions of infectious diseases with different transmission mechanisms, infection mechanisms, immunity mechanisms and medical treatment methods, each infectious disease can be transformed into a population-based optimization algorithm with different performance. Besides these, the epidemic dynamics and the virus dynamics, which are built on mathematical models, provide solid mathematical foundations for developing artificial infectious disease optimization algorithms. 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Comput.</source><volume>1</volume><issue>1</issue><year>2011</year><fpage>3</fpage><lpage>18</lpage></element-citation></ref></ref-list><sec id="s0155"><title>Appendix A</title><sec id="s0160"><title>Psuedo-code of SEIQRA</title><p id="p1270"><list list-type="simple" id="li0145"><list-item id="o0465"><label>(1)</label><p id="p1275">Initialize the following parameters: (a) Let <italic>t</italic>=0, initialize the parameters SEIQRA involves as specified in <xref rid="t0015" ref-type="table">Table 3</xref>; (b) Take <italic>N</italic><sub>0</sub> initialization individuals, <italic>N</italic> work individuals as input parameters, use operator REINIT to produce <italic>N</italic> individuals <inline-formula><mml:math id="M149" altimg="si0087.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>; (c) Let <italic>LatentPeriod</italic>(<italic>i</italic>)=<italic>∞</italic>, <italic>ImmunityPeriod</italic>(<italic>i</italic>)=∞, <italic>i</italic>=1, 2, …, <italic>N</italic>; // <italic>LatentPeriod</italic>(<italic>i</italic>) is the starting time of latency of individual <italic>i</italic>; <italic>ImmunityPeriod</italic>(<italic>i</italic>) is the starting time of vaccination of individual <italic>i</italic>.</p></list-item><list-item id="o0470"><label>(2)</label><p id="p1280">Produce 5 random numbers: <inline-formula><mml:math id="M150" altimg="si0088.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>=<italic>Rnd</italic>(0,1), <inline-formula><mml:math id="M151" altimg="si0089.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>=<italic>Rnd</italic>(0,1), <inline-formula><mml:math id="M152" altimg="si0090.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>=<italic>Rnd</italic>(0,1), <inline-formula><mml:math id="M153" altimg="si0091.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>=<italic>Rnd</italic>(0,1), <inline-formula><mml:math id="M154" altimg="si0092.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>=<italic>Rnd(</italic>0,1); compute <inline-formula><mml:math id="M155" altimg="si0093.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>5</mml:mn></mml:munderover><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M156" altimg="si0094.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>5</mml:mn></mml:munderover><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M157" altimg="si0095.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>5</mml:mn></mml:munderover><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M158" altimg="si0096.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>5</mml:mn></mml:munderover><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M159" altimg="si0097.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, <italic>i</italic>=1, 2, …, <italic>N</italic>;</p></list-item><list-item id="o0475"><label>(3)</label><p id="p1285">Calculate the state of S, E, I, Q and R of individual <italic>i</italic>, <italic>SEIQR</italic><sub><italic>i</italic></sub>(0)=<italic>GetSEIQR</italic>{<italic>S</italic><sub><italic>i</italic></sub>(0), <italic>E</italic><sub><italic>i</italic></sub>(0), <italic>I</italic><sub><italic>i</italic></sub>(0), <italic>Q</italic><sub><italic>i</italic></sub>(0), <italic>R</italic><sub><italic>i</italic></sub>(0)}, <italic>i</italic>=1, 2, …, <italic>N</italic>; /* function <italic>GetSEIQR</italic>() is used to determine which state individual <italic>i</italic> will stay at.*/</p></list-item><list-item id="o0480"><label>(4)</label><p id="p1290">Execute the following operations:</p></list-item></list></p><p id="p1295"><inline-graphic xlink:href="fx1.gif"><alt-text id="at0225">Image 1</alt-text></inline-graphic></p><p id="p1300"><inline-graphic xlink:href="fx2.gif"><alt-text id="at0230">Image 2</alt-text></inline-graphic></p><p id="p1305"><inline-graphic xlink:href="fx3.gif"><alt-text id="at0235">Image 3</alt-text></inline-graphic></p></sec></sec><sec id="s0165"><title>Appendix B</title><p id="p1315">See
<xref rid="f0100" ref-type="fig">Fig. B1</xref><fig id="f0100"><label>Fig. B1</label><caption><p>Convergence graphs of F1–F28.</p></caption><alt-text id="at0100">Fig. B1</alt-text><graphic xlink:href="gr20a"></graphic><graphic xlink:href="gr20b"></graphic><graphic xlink:href="gr20c"></graphic></fig></p></sec><ack id="ack0005"><title>Acknowledgment</title><p>The article is supported by the <funding-source id="gs1">Foundation Research Project of Natural Science of Shaanxi Province-Key Project</funding-source> (2015JZ010) and <funding-source id="gs2">Society Science foundation of Shaanxi Province</funding-source> (2014P07).</p></ack></back></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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