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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">The daily computed weighted averaging basic reproduction number <inline-formula><mml:math id="M1" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for MERS-CoV in South Korea</title><author><name sortKey="Jeong, Darae" sort="Jeong, Darae" uniqKey="Jeong D" first="Darae" last="Jeong">Darae Jeong</name><affiliation><nlm:aff id="af000005">Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea</nlm:aff></affiliation></author><author><name sortKey="Lee, Chang Hyeong" sort="Lee, Chang Hyeong" uniqKey="Lee C" first="Chang Hyeong" last="Lee">Chang Hyeong Lee</name><affiliation><nlm:aff id="af000010">Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan 689-798, Republic of Korea</nlm:aff></affiliation></author><author><name sortKey="Choi, Yongho" sort="Choi, Yongho" uniqKey="Choi Y" first="Yongho" last="Choi">Yongho Choi</name><affiliation><nlm:aff id="af000005">Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea</nlm:aff></affiliation></author><author><name sortKey="Kim, Junseok" sort="Kim, Junseok" uniqKey="Kim J" first="Junseok" last="Kim">Junseok Kim</name><affiliation><nlm:aff id="af000005">Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">32288098</idno><idno type="pmc">7126530</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7126530</idno><idno type="RBID">PMC:7126530</idno><idno type="doi">10.1016/j.physa.2016.01.072</idno><date when="2016">2016</date><idno type="wicri:Area/Pmc/Corpus">001123</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">001123</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">The daily computed weighted averaging basic reproduction number <inline-formula><mml:math id="M1" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for MERS-CoV in South Korea</title><author><name sortKey="Jeong, Darae" sort="Jeong, Darae" uniqKey="Jeong D" first="Darae" last="Jeong">Darae Jeong</name><affiliation><nlm:aff id="af000005">Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea</nlm:aff></affiliation></author><author><name sortKey="Lee, Chang Hyeong" sort="Lee, Chang Hyeong" uniqKey="Lee C" first="Chang Hyeong" last="Lee">Chang Hyeong Lee</name><affiliation><nlm:aff id="af000010">Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan 689-798, Republic of Korea</nlm:aff></affiliation></author><author><name sortKey="Choi, Yongho" sort="Choi, Yongho" uniqKey="Choi Y" first="Yongho" last="Choi">Yongho Choi</name><affiliation><nlm:aff id="af000005">Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea</nlm:aff></affiliation></author><author><name sortKey="Kim, Junseok" sort="Kim, Junseok" uniqKey="Kim J" first="Junseok" last="Kim">Junseok Kim</name><affiliation><nlm:aff id="af000005">Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea</nlm:aff></affiliation></author></analytic><series><title level="j">Physica a</title><idno type="ISSN">0378-4371</idno><idno type="eISSN">0378-4371</idno><imprint><date when="2016">2016</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>In this paper, we propose the daily computed weighted averaging basic reproduction number <inline-formula><mml:math id="M2" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for Middle East respiratory syndrome coronavirus (MERS-CoV) outbreak in South Korea, May to July 2015. We use an SIR model with piecewise constant parameters <inline-formula><mml:math id="M3" altimg="si31.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi></mml:math></inline-formula> (contact rate) and <inline-formula><mml:math id="M4" altimg="si32.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi></mml:math></inline-formula> (removed rate). We use the explicit Euler’s method for the solution of the SIR model and a nonlinear least-square fitting procedure for finding the best parameters. In <inline-formula><mml:math id="M5" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, the parameters <inline-formula><mml:math id="M6" altimg="si23.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M7" altimg="si35.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math></inline-formula>, and <inline-formula><mml:math id="M8" altimg="si36.gif" display="inline" overflow="scroll"><mml:mi>w</mml:mi></mml:math></inline-formula> denote days from a reference date, the number of days in averaging, and a weighting factor, respectively. We perform a series of numerical experiments and compare the results with the real-world data. In particular, using the predicted reproduction number based on the previous two consecutive reproduction numbers, we can predict the future behavior of the reproduction number.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Zhang, X" uniqKey="Zhang X">X. Zhang</name></author><author><name sortKey="Ge, B" uniqKey="Ge B">B. Ge</name></author><author><name sortKey="Wang, Q" uniqKey="Wang Q">Q. Wang</name></author><author><name sortKey="Jiang, J" uniqKey="Jiang J">J. Jiang</name></author><author><name sortKey="You, H" uniqKey="You H">H. You</name></author><author><name sortKey="Chen, Y" uniqKey="Chen Y">Y. Chen</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Cowling, B J" uniqKey="Cowling B">B.J. Cowling</name></author><author><name sortKey="Park, M" uniqKey="Park M">M. Park</name></author><author><name sortKey="Fang, V J" uniqKey="Fang V">V.J. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Physica A</journal-id><journal-id journal-id-type="iso-abbrev">Physica A</journal-id><journal-title-group><journal-title>Physica a</journal-title></journal-title-group><issn pub-type="ppub">0378-4371</issn><issn pub-type="epub">0378-4371</issn><publisher><publisher-name>Elsevier B.V.</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">32288098</article-id><article-id pub-id-type="pmc">7126530</article-id><article-id pub-id-type="publisher-id">S0378-4371(16)00126-6</article-id><article-id pub-id-type="doi">10.1016/j.physa.2016.01.072</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>The daily computed weighted averaging basic reproduction number <inline-formula><mml:math id="M1" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for MERS-CoV in South Korea</article-title></title-group><contrib-group><contrib contrib-type="author" id="au000005"><name><surname>Jeong</surname><given-names>Darae</given-names></name><xref rid="af000005" ref-type="aff">a</xref></contrib><contrib contrib-type="author" id="au000010"><name><surname>Lee</surname><given-names>Chang Hyeong</given-names></name><xref rid="af000010" ref-type="aff">b</xref></contrib><contrib contrib-type="author" id="au000015"><name><surname>Choi</surname><given-names>Yongho</given-names></name><xref rid="af000005" ref-type="aff">a</xref></contrib><contrib contrib-type="author" id="au000020"><name><surname>Kim</surname><given-names>Junseok</given-names></name><email>cfdkim@korea.ac.kr</email><email>http://math.korea.ac.kr/~cfdkim/</email><xref rid="af000005" ref-type="aff">a</xref><xref rid="cor000005" ref-type="corresp">⁎</xref></contrib></contrib-group><aff id="af000005"><label>a</label>Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea</aff><aff id="af000010"><label>b</label>Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan 689-798, Republic of Korea</aff><author-notes><corresp id="cor000005"><label>⁎</label>Corresponding author. <email>cfdkim@korea.ac.kr</email><email>http://math.korea.ac.kr/~cfdkim/</email></corresp></author-notes><pub-date pub-type="pmc-release"><day>2</day><month>2</month><year>2016</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"><day>1</day><month>6</month><year>2016</year></pub-date><pub-date pub-type="epub"><day>2</day><month>2</month><year>2016</year></pub-date><volume>451</volume><fpage>190</fpage><lpage>197</lpage><history><date date-type="received"><day>8</day><month>12</month><year>2015</year></date><date date-type="rev-recd"><day>8</day><month>1</month><year>2016</year></date></history><permissions><copyright-statement>Copyright © 2016 Elsevier B.V. All rights reserved.</copyright-statement><copyright-year>2016</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="a000005"><p>In this paper, we propose the daily computed weighted averaging basic reproduction number <inline-formula><mml:math id="M2" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for Middle East respiratory syndrome coronavirus (MERS-CoV) outbreak in South Korea, May to July 2015. We use an SIR model with piecewise constant parameters <inline-formula><mml:math id="M3" altimg="si31.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi></mml:math></inline-formula> (contact rate) and <inline-formula><mml:math id="M4" altimg="si32.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi></mml:math></inline-formula> (removed rate). We use the explicit Euler’s method for the solution of the SIR model and a nonlinear least-square fitting procedure for finding the best parameters. In <inline-formula><mml:math id="M5" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, the parameters <inline-formula><mml:math id="M6" altimg="si23.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M7" altimg="si35.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math></inline-formula>, and <inline-formula><mml:math id="M8" altimg="si36.gif" display="inline" overflow="scroll"><mml:mi>w</mml:mi></mml:math></inline-formula> denote days from a reference date, the number of days in averaging, and a weighting factor, respectively. We perform a series of numerical experiments and compare the results with the real-world data. In particular, using the predicted reproduction number based on the previous two consecutive reproduction numbers, we can predict the future behavior of the reproduction number.</p></abstract><abstract abstract-type="author-highlights" id="a000010"><title>Highlights</title><p><list list-type="simple" id="l000005"><list-item id="li000005"><label>•</label><p id="p000005">We propose the daily basic reproduction number for MERS-CoV in South Korea.</p></list-item><list-item id="li000010"><label>•</label><p id="p000010">We use an SIR model with piecewise constant contact and removed rates.</p></list-item><list-item id="li000015"><label>•</label><p id="p000015">We apply the explicit Euler’s method for the solution of the SIR model.</p></list-item><list-item id="li000020"><label>•</label><p id="p000020">We use nonlinear least-square fitting procedure for finding the best parameters.</p></list-item></list></p></abstract><kwd-group id="ks000005"><title>Keywords</title><kwd>MERS-CoV</kwd><kwd>SIR model</kwd><kwd>Explicit Euler’s method</kwd><kwd>Optimal parameter fitting</kwd></kwd-group></article-meta></front><body><sec id="s000005"><label>1</label><title>Introduction</title><p id="p000025">Middle East respiratory syndrome coronavirus (MERS-CoV), first identified in Saudi Arabia in June 2012, is a viral respiratory disease caused by a novel coronavirus <xref rid="br000005" ref-type="bibr">[1]</xref>. South Korea experienced the largest outbreak of MERS-CoV infections outside the Arabian peninsula <xref rid="br000010" ref-type="bibr">[2]</xref>. There have been 186 confirmed infective cases in South Korea within two months after the first infective person who returned from a trip to the Arabian peninsula was diagnosed on 11 May, 2015, and 38 people died and more than 16,000 people were quarantined due to the spread of the disease. <xref rid="f000005" ref-type="fig">Fig. 1</xref>represents the epidemic curve of MERS-CoV, South Korea, 20 May–17 July 2015 <xref rid="br000015" ref-type="bibr">[3]</xref>, <xref rid="br000020" ref-type="bibr">[4]</xref>.<fig id="f000005"><label>Fig. 1</label><caption><p>Epidemic curve of MERS-CoV, South Korea, 20 May–17 July 2015.</p></caption><graphic xlink:href="gr1_lrg"></graphic></fig></p><p id="p000030">In order to implement appropriate surveillance and control measures, it is very important to predict the future trend of the epidemic. Therefore, it is the purpose of the present paper to show daily computed reproduction numbers for epidemics, in particular, MERS-CoV, so that we can predict the future behavior of the reproduction number day-by-day.</p><p id="p000035">In the mathematical modeling of the spread of an epidemic disease, it is crucial to estimate the parameters (e.g., contact rate and recovery rate in an SIR model), but it is difficult to do so due to the lack of data available for the estimation. In case of the MERS-CoV spread in South Korea, there are almost complete daily data of the spread of the disease, and we choose the MERS-CoV spread case in South Korea as an epidemic model to verify the validity and accuracy of the method proposed in this paper. To the authors’ knowledge, there have been no previous works that present computational methods for estimation of the values of the epidemic parameters based on actual daily epidemic data.</p><p id="p000040">The rest of the paper is organized as follows. In Section <xref rid="s000010" ref-type="sec">2</xref>, we describe the mathematical model. In Section <xref rid="s000015" ref-type="sec">3</xref>, we provide a numerical algorithm for the estimation of the parameters. We perform several numerical experiments in Section <xref rid="s000020" ref-type="sec">4</xref>. In Section <xref rid="s000040" ref-type="sec">5</xref>, we provide a summary and present our conclusions.</p></sec><sec id="s000010"><label>2</label><title>The mathematical model</title><p id="p000045">In this paper, we consider the SIR model that was introduced in 1927 by A.G. McKendrick and W.O. Kermack <xref rid="br000025" ref-type="bibr">[5]</xref>. The model is simple and has been widely used so far, for instance, in multigroup epidemic modeling <xref rid="br000030" ref-type="bibr">[6]</xref>, online social network dynamics <xref rid="br000035" ref-type="bibr">[7]</xref>, the model adopting the delay term <xref rid="br000040" ref-type="bibr">[8]</xref>, stochastic model <xref rid="br000030" ref-type="bibr">[6]</xref>, <xref rid="br000045" ref-type="bibr">[9]</xref>, <xref rid="br000050" ref-type="bibr">[10]</xref>, vaccination strategy <xref rid="br000055" ref-type="bibr">[11]</xref>, <xref rid="br000060" ref-type="bibr">[12]</xref>. In this model, the population is divided into susceptible <inline-formula><mml:math id="M9" altimg="si37.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, infected <inline-formula><mml:math id="M10" altimg="si38.gif" display="inline" overflow="scroll"><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, and removed <inline-formula><mml:math id="M11" altimg="si39.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> individuals at time <inline-formula><mml:math id="M12" altimg="si40.gif" display="inline" overflow="scroll"><mml:mi>t</mml:mi></mml:math></inline-formula>. The governing ordinary differential equations for the SIR model are as follows: <disp-formula-group id="fd000005"><disp-formula id="fd000010"><label>(1)</label><mml:math id="M13" altimg="si41.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><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:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000015"><label>(2)</label><mml:math id="M14" altimg="si42.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>γ</mml:mi><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000020"><label>(3)</label><mml:math id="M15" altimg="si43.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>γ</mml:mi><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></disp-formula-group> with initial condition <inline-formula><mml:math id="M16" altimg="si44.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M17" altimg="si45.gif" display="inline" overflow="scroll"><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M18" altimg="si46.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. Here, the parameters <inline-formula><mml:math id="M19" altimg="si31.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M20" altimg="si32.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi></mml:math></inline-formula> denote the contact (susceptibility to disease) and removed (either dead or recovered) rates from disease, respectively <xref rid="br000065" ref-type="bibr">[13]</xref>. We assume that a removed individual can never be infected again. Let <inline-formula><mml:math id="M21" altimg="si49.gif" display="inline" overflow="scroll"><mml:mi>N</mml:mi></mml:math></inline-formula> be the constant total population size. Therefore, it is sufficient to solve only two Eqs. <xref rid="fd000010" ref-type="disp-formula">(1)</xref>, <xref rid="fd000015" ref-type="disp-formula">(2)</xref>, i.e., <inline-formula><mml:math id="M22" altimg="si50.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>. By assuming that <inline-formula><mml:math id="M23" altimg="si31.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M24" altimg="si32.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi></mml:math></inline-formula> are time-dependent parameters, we generalize Eqs. <xref rid="fd000010" ref-type="disp-formula">(1)</xref>, <xref rid="fd000015" ref-type="disp-formula">(2)</xref>, <xref rid="fd000020" ref-type="disp-formula">(3)</xref> as follows: <disp-formula-group id="fd000025"><disp-formula id="fd000030"><label>(4)</label><mml:math id="M25" altimg="si53.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000035"><label>(5)</label><mml:math id="M26" altimg="si54.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>γ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000040"><label>(6)</label><mml:math id="M27" altimg="si55.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>γ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>.</mml:mtext></mml:math></disp-formula></disp-formula-group> For example, the transmission of the vector-borne diseases such as Dengue fever and Malaria has strong seasonal patterns and thus the parameters are estimated as the time-dependent functions <xref rid="br000070" ref-type="bibr">[14]</xref>, <xref rid="br000075" ref-type="bibr">[15]</xref>. Another example is a seasonal SIR model for the spread of the whooping cough in which the contact rate <inline-formula><mml:math id="M28" altimg="si31.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi></mml:math></inline-formula> is given as a time-dependent periodic function which accounts seasonal changes <xref rid="br000080" ref-type="bibr">[16]</xref>.</p></sec><sec id="s000015"><label>3</label><title>The numerical method</title><p id="p000050">Let us assume that we have daily statistics about <inline-formula><mml:math id="M29" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M30" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>I</mml:mi></mml:math></inline-formula>, and <inline-formula><mml:math id="M31" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi></mml:math></inline-formula>. Let <inline-formula><mml:math id="M32" altimg="si60.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>ref</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>n</mml:mi></mml:math></inline-formula> be time, where <inline-formula><mml:math id="M33" altimg="si61.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>ref</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:math></inline-formula> is a reference time. Let <inline-formula><mml:math id="M34" altimg="si62.gif" display="inline" overflow="scroll"><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M35" altimg="si63.gif" display="inline" overflow="scroll"><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M36" altimg="si64.gif" display="inline" overflow="scroll"><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> be observed susceptible, infected, and removed data at time <inline-formula><mml:math id="M37" altimg="si65.gif" display="inline" overflow="scroll"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, respectively. Let <inline-formula><mml:math id="M38" altimg="si66.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M39" altimg="si67.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M40" altimg="si68.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> be numerical approximations of <inline-formula><mml:math id="M41" altimg="si69.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M42" altimg="si70.gif" display="inline" overflow="scroll"><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math id="M43" altimg="si71.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, respectively. In this paper, we propose a computation of <inline-formula><mml:math id="M44" altimg="si72.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M45" altimg="si73.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> on a daily basis, which fit best to the real data. To be more specific, for given data <inline-formula><mml:math id="M46" altimg="si74.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M47" altimg="si75.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, we want to find piecewise constant parameters <inline-formula><mml:math id="M48" altimg="si76.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="M49" altimg="si77.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> which minimize the following <disp-formula id="fd000045"><label>(7)</label><mml:math id="M50" altimg="si78.gif" display="block" overflow="scroll"><mml:munder><mml:mrow><mml:mo>min</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:munder><mml:mrow><mml:mo>[</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>]</mml:mo></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula> where <inline-formula><mml:math id="M51" altimg="si79.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M52" altimg="si80.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M53" altimg="si81.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> are the numerical solutions of Eqs. <xref rid="fd000030" ref-type="disp-formula">(4)</xref>, <xref rid="fd000035" ref-type="disp-formula">(5)</xref>, <xref rid="fd000040" ref-type="disp-formula">(6)</xref> with initial condition <inline-formula><mml:math id="M54" altimg="si82.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>. We divide one day between <inline-formula><mml:math id="M55" altimg="si83.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math id="M56" altimg="si84.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> into <inline-formula><mml:math id="M57" altimg="si85.gif" display="inline" overflow="scroll"><mml:mi>m</mml:mi></mml:math></inline-formula> subintervals, then <inline-formula><mml:math id="M58" altimg="si86.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="normal"><mml:mi>Δ</mml:mi></mml:mstyle><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>m</mml:mi></mml:math></inline-formula> is the time step size. Let <inline-formula><mml:math id="M59" altimg="si87.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="M60" altimg="si88.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, then by applying the explicit Euler’s method to Eqs. <xref rid="fd000030" ref-type="disp-formula">(4)</xref>, <xref rid="fd000035" ref-type="disp-formula">(5)</xref>, we have <disp-formula-group id="fd000050"><disp-formula id="fd000055"><label>(8)</label><mml:math id="M61" altimg="si89.gif" display="block" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mstyle mathvariant="normal"><mml:mi>Δ</mml:mi></mml:mstyle><mml:mi>t</mml:mi><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000060"><label>(9)</label><mml:math id="M62" altimg="si90.gif" display="block" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>Δ</mml:mi></mml:mstyle><mml:mi>t</mml:mi><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000065"><label>(10)</label><mml:math id="M63" altimg="si91.gif" display="block" overflow="scroll"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mtext>,</mml:mtext><mml:mspace width="1em" class="quad"></mml:mspace><mml:mtext>for </mml:mtext><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mtext>,</mml:mtext></mml:math></disp-formula></disp-formula-group> where we have used the condition <inline-formula><mml:math id="M64" altimg="si50.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>. Unless otherwise stated, we use <inline-formula><mml:math id="M65" altimg="si93.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="normal"><mml:mi>Δ</mml:mi></mml:mstyle><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>100</mml:mn></mml:math></inline-formula> in all numerical tests. If the solutions of Eqs. <xref rid="fd000055" ref-type="disp-formula">(8)</xref>, <xref rid="fd000060" ref-type="disp-formula">(9)</xref>, <xref rid="fd000065" ref-type="disp-formula">(10)</xref> are obtained for all <inline-formula><mml:math id="M66" altimg="si35.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math></inline-formula>, then we define <inline-formula><mml:math id="M67" altimg="si95.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M68" altimg="si96.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M69" altimg="si97.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. Note that while we can use higher-order numerical methods such as Runge–Kutta schemes to solve Eqs. <xref rid="fd000030" ref-type="disp-formula">(4)</xref>, <xref rid="fd000035" ref-type="disp-formula">(5)</xref>, we use the explicit Euler’s method for the sake of simplicity. Next, to find the best <inline-formula><mml:math id="M70" altimg="si76.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="M71" altimg="si77.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, we use a MATLAB routine called <bold>lsqcurvefit</bold>, based on the least-squares sense, which finds coefficients <inline-formula><mml:math id="M72" altimg="si76.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="M73" altimg="si77.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> that solve the problem <xref rid="fd000045" ref-type="disp-formula">(7)</xref>. We use <inline-formula><mml:math id="M74" altimg="si102.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1.0</mml:mn><mml:mstyle mathvariant="normal"><mml:mi>e</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula> and <inline-formula><mml:math id="M75" altimg="si103.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1.0</mml:mn><mml:mstyle mathvariant="normal"><mml:mi>e</mml:mi></mml:mstyle><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> as an initial guess in all numerical simulations. The MATLAB code is given in<xref rid="s000045" ref-type="sec">Appendix</xref> for the interested readers.</p></sec><sec id="s000020"><label>4</label><title>Numerical simulations</title><p id="p000055">In all computations, the initial conditions are taken to be <inline-formula><mml:math id="M76" altimg="si4.gif" display="inline" overflow="scroll"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>500</mml:mn><mml:mtext>,</mml:mtext><mml:mn>000</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M77" altimg="si105.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M78" altimg="si106.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>, and <inline-formula><mml:math id="M79" altimg="si107.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> with <inline-formula><mml:math id="M80" altimg="si108.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>ref</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>20</mml:mn></mml:math></inline-formula> May 2015. <xref rid="f000010" ref-type="fig">Fig. 2</xref>shows the time series of observed MERS-CoV epidemiological data and numerical solutions <inline-formula><mml:math id="M81" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M82" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>I</mml:mi></mml:math></inline-formula>, and <inline-formula><mml:math id="M83" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi></mml:math></inline-formula>, respectively.<fig id="f000010"><label>Fig. 2</label><caption><p>Time series of observed MERS-CoV epidemiological data and numerical solutions of (a) <inline-formula><mml:math id="M84" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi></mml:math></inline-formula>, (b) <inline-formula><mml:math id="M85" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>I</mml:mi></mml:math></inline-formula>, and (c) <inline-formula><mml:math id="M86" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi></mml:math></inline-formula>, respectively. Here, <inline-formula><mml:math id="M87" altimg="si4.gif" display="inline" overflow="scroll"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>500</mml:mn><mml:mtext>,</mml:mtext><mml:mn>000</mml:mn></mml:math></inline-formula> is used.</p></caption><graphic xlink:href="gr2_lrg"></graphic></fig></p><p id="p000060">The basic reproduction number <inline-formula><mml:math id="M88" altimg="si112.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> is the average number of secondary infections caused by one infectious individual in a completely susceptible population. It is an important measurement in that one can determine the stability of the disease-free equilibrium by computing the value of <inline-formula><mml:math id="M89" altimg="si112.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>; <inline-formula><mml:math id="M90" altimg="si114.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> implies that the disease-free equilibrium is locally asymptotically stable <xref rid="br000085" ref-type="bibr">[17]</xref>, and if <inline-formula><mml:math id="M91" altimg="si115.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>, it is unstable <xref rid="br000090" ref-type="bibr">[18]</xref>, <xref rid="br000095" ref-type="bibr">[19]</xref>. Finding basic reduction number <inline-formula><mml:math id="M92" altimg="si112.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> can be done through the next generation matrix <xref rid="br000100" ref-type="bibr">[20]</xref>, <xref rid="br000105" ref-type="bibr">[21]</xref> whose spectral radius is defined as <inline-formula><mml:math id="M93" altimg="si112.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>. In the SIR model, the basic reproduction number is computed as <inline-formula><mml:math id="M94" altimg="si118.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mi>N</mml:mi><mml:mo>/</mml:mo><mml:mi>γ</mml:mi></mml:math></inline-formula> <xref rid="br000110" ref-type="bibr">[22]</xref>, <xref rid="br000115" ref-type="bibr">[23]</xref>, which will be used for our method in the next section.</p><sec id="s000025"><label>4.1</label><title>Weighted averaging basic reproduction number <inline-formula><mml:math id="M95" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula></title><p id="p000065">In this section, we propose the weighted average values for infected and removed rates, and then define weighted averaging basic reproduction number. Let <inline-formula><mml:math id="M96" altimg="si120.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> for <inline-formula><mml:math id="M97" altimg="si121.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:math></inline-formula>. We define <inline-formula><mml:math id="M98" altimg="si35.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math></inline-formula>-days (<inline-formula><mml:math id="M99" altimg="si123.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>) weighted average <inline-formula><mml:math id="M100" altimg="si5.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> as <disp-formula id="fd000070"><label>(11)</label><mml:math id="M101" altimg="si125.gif" display="block" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mtable class="cases"><mml:mtr><mml:mtd><mml:munderover accentunder="false" accent="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:munderover accentunder="false" accent="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mspace width="1em" class="quad"></mml:mspace><mml:mtext>if </mml:mtext><mml:mspace class="nbsp"></mml:mspace><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mi>k</mml:mi><mml:mtext>,</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:munderover accentunder="false" accent="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:munderover accentunder="false" accent="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mspace width="1em" class="quad"></mml:mspace><mml:mtext>if </mml:mtext><mml:mi>n</mml:mi><mml:mo><</mml:mo><mml:mi>k</mml:mi><mml:mtext>,</mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula> where <inline-formula><mml:math id="M102" altimg="si126.gif" display="inline" overflow="scroll"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>ω</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> and the superscript <inline-formula><mml:math id="M103" altimg="si127.gif" display="inline" overflow="scroll"><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> for <inline-formula><mml:math id="M104" altimg="si128.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> represents an exponent. Also, <inline-formula><mml:math id="M105" altimg="si6.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> is defined similarly as Eq. <xref rid="fd000070" ref-type="disp-formula">(11)</xref>. Finally, we define the daily computed weighted averaging basic reproduction number as <inline-formula><mml:math id="M106" altimg="si130.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mi>N</mml:mi><mml:mo>/</mml:mo><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>. <xref rid="f000015" ref-type="fig">Fig. 3</xref>(a)–(c) represent <inline-formula><mml:math id="M107" altimg="si5.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M108" altimg="si6.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, and <inline-formula><mml:math id="M109" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for <inline-formula><mml:math id="M110" altimg="si8.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math></inline-formula>, respectively, with <inline-formula><mml:math id="M111" altimg="si135.gif" display="inline" overflow="scroll"><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, 0.3, and 1. <xref rid="f000020" ref-type="fig">Fig. 4</xref>shows the results of the parameters for <inline-formula><mml:math id="M112" altimg="si13.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn></mml:math></inline-formula>. Note that when <inline-formula><mml:math id="M113" altimg="si9.gif" display="inline" overflow="scroll"><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, we have <inline-formula><mml:math id="M114" altimg="si138.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for all <inline-formula><mml:math id="M115" altimg="si35.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math></inline-formula>. If <inline-formula><mml:math id="M116" altimg="si35.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math></inline-formula> is large, then we have more smooth profile and the transition point (<inline-formula><mml:math id="M117" altimg="si141.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>) moves to the right. If <inline-formula><mml:math id="M118" altimg="si142.gif" display="inline" overflow="scroll"><mml:mi>ω</mml:mi></mml:math></inline-formula> is close to zero, then we have a similar profile to <inline-formula><mml:math id="M119" altimg="si15.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>. Note that we have extraordinarily large values of <inline-formula><mml:math id="M120" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> from <inline-formula><mml:math id="M121" altimg="si145.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> to approximately <inline-formula><mml:math id="M122" altimg="si146.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>20</mml:mn></mml:math></inline-formula> since the recovered individuals were almost zero (<inline-formula><mml:math id="M123" altimg="si147.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>) in early times and the infected ones increased rapidly.<fig id="f000015"><label>Fig. 3</label><caption><p>Weighted averaging <inline-formula><mml:math id="M124" altimg="si5.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M125" altimg="si6.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, and <inline-formula><mml:math id="M126" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for <inline-formula><mml:math id="M127" altimg="si8.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math></inline-formula>. In each case, we compare the values with <inline-formula><mml:math id="M128" altimg="si9.gif" display="inline" overflow="scroll"><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, 0.3, and 1.</p></caption><graphic xlink:href="gr3_lrg"></graphic></fig><fig id="f000020"><label>Fig. 4</label><caption><p>Weighted averaging <inline-formula><mml:math id="M129" altimg="si5.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, <inline-formula><mml:math id="M130" altimg="si6.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, and <inline-formula><mml:math id="M131" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for <inline-formula><mml:math id="M132" altimg="si13.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn></mml:math></inline-formula>. In each case, we compare the values with <inline-formula><mml:math id="M133" altimg="si9.gif" display="inline" overflow="scroll"><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, 0.3, and 1.</p></caption><graphic xlink:href="gr4_lrg"></graphic></fig></p></sec><sec id="s000030"><label>4.2</label><title>Predictability of a linear extrapolation, <inline-formula><mml:math id="M134" altimg="si16.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="italic"><mml:mi>linear</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula></title><p id="p000070">In this section, we investigate the predictability of a linear extrapolation of <inline-formula><mml:math id="M135" altimg="si15.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>. Let us define the linear extrapolation <disp-formula id="fd000075"><label>(12)</label><mml:math id="M136" altimg="si150.gif" display="block" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="italic"><mml:mi>linear</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</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>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mtext>,</mml:mtext><mml:mspace width="1em" class="quad"></mml:mspace><mml:mtext>for </mml:mtext><mml:mn>3</mml:mn><mml:mo>≤</mml:mo><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mi>m</mml:mi><mml:mtext>,</mml:mtext></mml:math></disp-formula> where <inline-formula><mml:math id="M137" altimg="si16.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="italic"><mml:mi>linear</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> means the predicted reproduction number based on the previous two consecutive reproduction numbers and the error is defined as <disp-formula id="fd000080"><label>(13)</label><mml:math id="M138" altimg="si152.gif" display="block" overflow="scroll"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="italic"><mml:mi>linear</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mtext>,</mml:mtext><mml:mspace width="1em" class="quad"></mml:mspace><mml:mtext>for </mml:mtext><mml:mn>3</mml:mn><mml:mo>≤</mml:mo><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mi>m</mml:mi><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p id="p000075"><xref rid="f000025" ref-type="fig">Fig. 5</xref>(a) shows <inline-formula><mml:math id="M139" altimg="si16.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="italic"><mml:mi>linear</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> and <inline-formula><mml:math id="M140" altimg="si15.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>. Up to about three weeks, there are large differences between two reproduction numbers. However, we can observe a good agreement interval from <inline-formula><mml:math id="M141" altimg="si155.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn></mml:math></inline-formula> to <inline-formula><mml:math id="M142" altimg="si156.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>59</mml:mn></mml:math></inline-formula>, see <xref rid="f000025" ref-type="fig">Fig. 5</xref>(b). <xref rid="f000025" ref-type="fig">Fig. 5</xref>(c) and (d) are corresponding errors <inline-formula><mml:math id="M143" altimg="si24.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> of (a) and (b). The errors <inline-formula><mml:math id="M144" altimg="si24.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> from <inline-formula><mml:math id="M145" altimg="si155.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn></mml:math></inline-formula> to <inline-formula><mml:math id="M146" altimg="si156.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>59</mml:mn></mml:math></inline-formula> are listed in <xref rid="t000005" ref-type="table">Table 1</xref>. Except some days, the absolute errors <inline-formula><mml:math id="M147" altimg="si161.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>|</mml:mo><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>|</mml:mo></mml:mrow></mml:math></inline-formula> are less than one. We note that we have a spike in <xref rid="f000025" ref-type="fig">Fig. 5</xref>(b) and (d). That is due to no population data changes for two days, i.e., 11 July and 12 July as shown in <xref rid="f000005" ref-type="fig">Fig. 1</xref>.<fig id="f000025"><label>Fig. 5</label><caption><p>(a) Comparison of the <inline-formula><mml:math id="M148" altimg="si15.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> with <inline-formula><mml:math id="M149" altimg="si16.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="italic"><mml:mi>linear</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> which are denoted by the symbols (<inline-formula><mml:math id="M150" altimg="si17.gif" display="inline" overflow="scroll"><mml:mo>⋅</mml:mo></mml:math></inline-formula>) and (<inline-formula><mml:math id="M151" altimg="si18.gif" display="inline" overflow="scroll"><mml:mo>∘</mml:mo></mml:math></inline-formula>), respectively. (b) A magnified view of a rectangular box in (a). (c) Error for <inline-formula><mml:math id="M152" altimg="si15.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> from 20 May 2015 to 17 July 2015. (d) A magnified view of a rectangular box in (c) from 18 June 2015 to 17 July 2015.</p></caption><graphic xlink:href="gr5_lrg"></graphic></fig><table-wrap position="float" id="t000005"><label>Table 1</label><caption><p>Error which is expressed by <inline-formula><mml:math id="M153" altimg="si21.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="italic"><mml:mi>linear</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> from 18 June, 2015 to 17 July, 2015, i.e., <inline-formula><mml:math id="M154" altimg="si22.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mn>59</mml:mn></mml:math></inline-formula>.</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left"><inline-formula><mml:math id="M155" altimg="si23.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math></inline-formula></th><th align="left"><inline-formula><mml:math id="M156" altimg="si24.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula></th><th align="left"><inline-formula><mml:math id="M157" altimg="si23.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math></inline-formula></th><th align="left"><inline-formula><mml:math id="M158" altimg="si24.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula></th><th align="left"><inline-formula><mml:math id="M159" altimg="si23.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math></inline-formula></th><th align="left"><inline-formula><mml:math id="M160" altimg="si24.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula></th></tr></thead><tbody><tr><td align="left">30</td><td align="left">−2.4532</td><td align="left">40</td><td align="left">−0.1110</td><td align="left">50</td><td align="left">−0.0002</td></tr><tr><td align="left">31</td><td align="left">+0.3215</td><td align="left">41</td><td align="left">+0.0000</td><td align="left">51</td><td align="left">+0.0001</td></tr><tr><td align="left">32</td><td align="left">−0.6608</td><td align="left">42</td><td align="left">−0.0001</td><td align="left">52</td><td align="left">−0.0001</td></tr><tr><td align="left">33</td><td align="left">+0.3751</td><td align="left">43</td><td align="left">−0.1999</td><td align="left">53</td><td align="left">−0.0002</td></tr><tr><td align="left">34</td><td align="left">−0.2251</td><td align="left">44</td><td align="left">+0.2571</td><td align="left">54</td><td align="left">−1.6766</td></tr><tr><td align="left">35</td><td align="left">+0.5175</td><td align="left">45</td><td align="left">−0.4144</td><td align="left">55</td><td align="left">+3.3534</td></tr><tr><td align="left">36</td><td align="left">−0.0958</td><td align="left">46</td><td align="left">+0.6574</td><td align="left">56</td><td align="left">−1.6766</td></tr><tr><td align="left">37</td><td align="left">−0.1967</td><td align="left">47</td><td align="left">−0.1004</td><td align="left">57</td><td align="left">−0.0000</td></tr><tr><td align="left">38</td><td align="left">−0.0000</td><td align="left">48</td><td align="left">−0.1997</td><td align="left">58</td><td align="left">+0.0009</td></tr><tr><td align="left">39</td><td align="left">+0.1110</td><td align="left">49</td><td align="left">+0.0002</td><td align="left">59</td><td align="left">−0.0000</td></tr></tbody></table></table-wrap></p></sec><sec id="s000035"><label>4.3</label><title>The SIR model with disease-related death</title><p id="p000080">In this section, we consider the modified SIR model with disease-related death. Let the total population <inline-formula><mml:math id="M161" altimg="si49.gif" display="inline" overflow="scroll"><mml:mi>N</mml:mi></mml:math></inline-formula> be divided into susceptible <inline-formula><mml:math id="M162" altimg="si37.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, infected <inline-formula><mml:math id="M163" altimg="si38.gif" display="inline" overflow="scroll"><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, recovered <inline-formula><mml:math id="M164" altimg="si39.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, and dead <inline-formula><mml:math id="M165" altimg="si166.gif" display="inline" overflow="scroll"><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> individuals at time <inline-formula><mml:math id="M166" altimg="si40.gif" display="inline" overflow="scroll"><mml:mi>t</mml:mi></mml:math></inline-formula>, then the SIR model with fatality rate is given by <disp-formula-group id="fd000085"><disp-formula id="fd000090"><label>(14)</label><mml:math id="M167" altimg="si53.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000095"><label>(15)</label><mml:math id="M168" altimg="si169.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>γ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000100"><label>(16)</label><mml:math id="M169" altimg="si170.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>γ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd000105"><label>(17)</label><mml:math id="M170" altimg="si171.gif" display="block" overflow="scroll"><mml:mfrac><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula></disp-formula-group> with initial condition <inline-formula><mml:math id="M171" altimg="si44.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M172" altimg="si45.gif" display="inline" overflow="scroll"><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M173" altimg="si46.gif" display="inline" overflow="scroll"><mml:mi>R</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M174" altimg="si175.gif" display="inline" overflow="scroll"><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. Here, the time-dependent parameters <inline-formula><mml:math id="M175" altimg="si72.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M176" altimg="si73.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math id="M177" altimg="si178.gif" display="inline" overflow="scroll"><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> denote the contact, recovered, and dead rates from disease, respectively <xref rid="br000110" ref-type="bibr">[22]</xref>. Now, by using the proposed method, we evaluate the numerical solution of the modified SIR model <xref rid="fd000090" ref-type="disp-formula">(14)</xref>, <xref rid="fd000095" ref-type="disp-formula">(15)</xref>, <xref rid="fd000100" ref-type="disp-formula">(16)</xref>, <xref rid="fd000105" ref-type="disp-formula">(17)</xref>. We take the same initial conditions: <inline-formula><mml:math id="M178" altimg="si4.gif" display="inline" overflow="scroll"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>500</mml:mn><mml:mtext>,</mml:mtext><mml:mn>000</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M179" altimg="si105.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M180" altimg="si106.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="M181" altimg="si107.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, and <inline-formula><mml:math id="M182" altimg="si183.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>. The initial guesses for finding coefficients <inline-formula><mml:math id="M183" altimg="si31.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M184" altimg="si32.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi></mml:math></inline-formula>, and <inline-formula><mml:math id="M185" altimg="si186.gif" display="inline" overflow="scroll"><mml:mi>d</mml:mi></mml:math></inline-formula> are 1.0e−6, 1.0e−1, and 1.0e−1, respectively. Using this model, we can predict the ratio of individuals who died by disease to the infected population at a given time. To investigate the ratio, we introduce the fatality percentage rate <inline-formula><mml:math id="M186" altimg="si20.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> which is defined as <disp-formula id="fd000110"><mml:math id="M187" altimg="si188.gif" display="block" overflow="scroll"><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mspace class="nbsp"></mml:mspace><mml:mrow><mml:mo>(</mml:mo><mml:mi>%</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p id="p000085">In <xref rid="f000030" ref-type="fig">Fig. 6</xref>, we show the fatality percentage rate <inline-formula><mml:math id="M188" altimg="si20.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> from the real observation data and the numerical approximations. Here, the fatality percentage rate <inline-formula><mml:math id="M189" altimg="si20.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> from the numerical approximation is obtained by the numerical solutions <inline-formula><mml:math id="M190" altimg="si66.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M191" altimg="si67.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M192" altimg="si68.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M193" altimg="si194.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>. The numerical solutions <inline-formula><mml:math id="M194" altimg="si66.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M195" altimg="si67.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M196" altimg="si68.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M197" altimg="si194.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> are evaluated by using the best-fitting parameters <inline-formula><mml:math id="M198" altimg="si199.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math id="M199" altimg="si200.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula><mml:math id="M200" altimg="si201.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> at time <inline-formula><mml:math id="M201" altimg="si202.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>. As shown in <xref rid="f000030" ref-type="fig">Fig. 6</xref>, the both results are in good agreement in later times.<fig id="f000030"><label>Fig. 6</label><caption><p>Fatality percentage rate <inline-formula><mml:math id="M202" altimg="si20.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> by the observation data (circle marker) and the numerical approximations (solid line).</p></caption><graphic xlink:href="gr6_lrg"></graphic></fig></p></sec></sec><sec id="s000040"><label>5</label><title>Conclusion</title><p id="p000090">We have proposed the daily computed weighted averaging basic reproduction number <inline-formula><mml:math id="M203" altimg="si7.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> for MERS-CoV outbreak in South Korea, May to July 2015. The linearly extrapolated reproduction number, <inline-formula><mml:math id="M204" altimg="si16.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="italic"><mml:mi>linear</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>, has important implications for the future prediction of the trend of the reproduction numbers. We showed that the results computed by our method match very well with the actual data of the MERS-CoV spread in South Korea. Using the method proposed in this paper, one can predict the progress of the spread of infections on the use of real-time epidemic data, so that proper control policies can be performed in an appropriate time for reducing the spread of the infections. 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The following command returns the optimal parameters.</p><p id="p000105"><fig id="f000035" position="anchor"><graphic xlink:href="fx1_lrg"></graphic></fig></p><p id="p000110">In this routine, <inline-formula><mml:math id="M206" altimg="si205.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="M207" altimg="si206.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> are an initial guess, <inline-formula><mml:math id="M208" altimg="si207.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>[</mml:mo><mml:mi>n</mml:mi><mml:mspace class="nbsp"></mml:mspace><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> is a time interval, and <inline-formula><mml:math id="M209" altimg="si208.gif" display="inline" overflow="scroll"><mml:mtext>SIRdata</mml:mtext><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mspace class="nbsp"></mml:mspace><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mspace class="nbsp"></mml:mspace><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>. Also, <inline-formula><mml:math id="M210" altimg="si209.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mspace class="nbsp"></mml:mspace><mml:mspace class="nbsp"></mml:mspace><mml:mn>0</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M211" altimg="si210.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mspace class="nbsp"></mml:mspace><mml:mspace class="nbsp"></mml:mspace><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> represent the lower and upper bounds for the fitting parameters <inline-formula><mml:math id="M212" altimg="si76.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="M213" altimg="si77.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. Here, the function SIRmodel is given as follows.</p><p id="p000115"><fig id="f000040" position="anchor"><graphic xlink:href="fx2_lrg"></graphic></fig></p></sec><ack id="ac000005"><title>Acknowledgments</title><p>The author (D. Jeong) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the <funding-source id="gs000005">Ministry of Education, Science and Technology</funding-source> (2014R1A6A3A01009812). C.H. Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the <funding-source id="gs000010">Ministry of Education</funding-source> (2014R1A1A2054976). The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the <funding-source id="gs000015">Korea government (MSIP)</funding-source> (NRF-2014R1A2A2A01003683). The authors would like to thank the reviewers for their comments that helped to improve the manuscript.</p></ack></back></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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