Congruent epidemic models for unstructured and structured populations: Analytical reconstruction of a 2003 SARS outbreak
Identifieur interne : 001268 ( Pmc/Corpus ); précédent : 001267; suivant : 001269Congruent epidemic models for unstructured and structured populations: Analytical reconstruction of a 2003 SARS outbreak
Auteurs : John N. BombardtSource :
- Mathematical Biosciences [ 0025-5564 ] ; 2006.
Abstract
Both the threat of bioterrorism and the natural emergence of contagious diseases underscore the importance of quantitatively understanding disease transmission in structured human populations. Over the last few years, researchers have advanced the mathematical theory of scale-free networks and used such theoretical advancements in pilot epidemic models. Scale-free contact networks are particularly interesting in the realm of mathematical epidemiology, primarily because these networks may allow meaningfully structured populations to be incorporated in epidemic models at moderate or intermediate levels of complexity. Moreover, a scale-free contact network with node degree correlation is in accord with the well-known preferred mixing concept. The present author describes a semi-empirical and deterministic epidemic modeling approach that (a) focuses on time-varying rates of disease transmission in both unstructured and structured populations and (b) employs probability density functions to characterize disease progression and outbreak controls. Given an epidemic curve for a historical outbreak, this modeling approach calls for Monte Carlo calculations (that define the average new infection rate) and solutions to integro-differential equations (that describe outbreak dynamics in an aggregate population or across all network connectivity classes). Numerical results are obtained for the 2003 SARS outbreak in Taiwan and the dynamical implications of time-varying transmission rates and scale-free contact networks are discussed in some detail.
Url:
DOI: 10.1016/j.mbs.2006.05.004
PubMed: 16904134
PubMed Central: 7094332
Links to Exploration step
PMC:7094332Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Congruent epidemic models for unstructured and structured populations: Analytical reconstruction of a 2003 SARS outbreak</title><author><name sortKey="Bombardt, John N" sort="Bombardt, John N" uniqKey="Bombardt J" first="John N." last="Bombardt">John N. Bombardt</name></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">16904134</idno><idno type="pmc">7094332</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7094332</idno><idno type="RBID">PMC:7094332</idno><idno type="doi">10.1016/j.mbs.2006.05.004</idno><date when="2006">2006</date><idno type="wicri:Area/Pmc/Corpus">001268</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">001268</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Congruent epidemic models for unstructured and structured populations: Analytical reconstruction of a 2003 SARS outbreak</title><author><name sortKey="Bombardt, John N" sort="Bombardt, John N" uniqKey="Bombardt J" first="John N." last="Bombardt">John N. Bombardt</name></author></analytic><series><title level="j">Mathematical Biosciences</title><idno type="ISSN">0025-5564</idno><idno type="eISSN">1879-3134</idno><imprint><date when="2006">2006</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>Both the threat of bioterrorism and the natural emergence of contagious diseases underscore the importance of quantitatively understanding disease transmission in structured human populations. Over the last few years, researchers have advanced the mathematical theory of scale-free networks and used such theoretical advancements in pilot epidemic models. Scale-free contact networks are particularly interesting in the realm of mathematical epidemiology, primarily because these networks may allow meaningfully structured populations to be incorporated in epidemic models at moderate or intermediate levels of complexity. Moreover, a scale-free contact network with node degree correlation is in accord with the well-known preferred mixing concept. The present author describes a semi-empirical and deterministic epidemic modeling approach that (a) focuses on time-varying rates of disease transmission in both unstructured and structured populations and (b) employs probability density functions to characterize disease progression and outbreak controls. Given an epidemic curve for a historical outbreak, this modeling approach calls for Monte Carlo calculations (that define the average new infection rate) and solutions to integro-differential equations (that describe outbreak dynamics in an aggregate population or across all network connectivity classes). Numerical results are obtained for the 2003 SARS outbreak in Taiwan and the dynamical implications of time-varying transmission rates and scale-free contact networks are discussed in some detail.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Lloyd Smith, J O" uniqKey="Lloyd Smith J">J.O. Lloyd-Smith</name></author><author><name sortKey="Galvani, A P" uniqKey="Galvani A">A.P. Galvani</name></author><author><name sortKey="Getz, W M" uniqKey="Getz W">W.M. Getz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Riley, S" uniqKey="Riley S">S. Riley</name></author><author><name sortKey="Fraser, C" uniqKey="Fraser C">C. Fraser</name></author><author><name sortKey="Donnelly, C A" uniqKey="Donnelly C">C.A. Donnelly</name></author><author><name sortKey="Ghani, A C" uniqKey="Ghani A">A.C. Ghani</name></author><author><name sortKey="Abu Raddad, L J" uniqKey="Abu Raddad L">L.J. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Math Biosci</journal-id><journal-id journal-id-type="iso-abbrev">Math Biosci</journal-id><journal-title-group><journal-title>Mathematical Biosciences</journal-title></journal-title-group><issn pub-type="ppub">0025-5564</issn><issn pub-type="epub">1879-3134</issn><publisher><publisher-name>Elsevier Inc.</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">16904134</article-id><article-id pub-id-type="pmc">7094332</article-id><article-id pub-id-type="publisher-id">S0025-5564(06)00084-8</article-id><article-id pub-id-type="doi">10.1016/j.mbs.2006.05.004</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>Congruent epidemic models for unstructured and structured populations: Analytical reconstruction of a 2003 SARS outbreak</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Bombardt</surname><given-names>John N.</given-names></name><email>jbombard@ida.org</email><xref rid="cor1" ref-type="corresp">⁎</xref></contrib></contrib-group><aff>Institute for Defense Analyses, 4850 Mark Center Drive, Alexandria, VA 22311-1882, United States</aff><author-notes><corresp id="cor1"><label>⁎</label>Tel.: +1 703 845 2204; fax: +1 703 845 2255. <email>jbombard@ida.org</email></corresp></author-notes><pub-date pub-type="pmc-release"><day>9</day><month>6</month><year>2006</year></pub-date><pmc-comment> PMC Release delay is 0 months and 0 days and was based on .</pmc-comment>
<pub-date pub-type="ppub"><month>10</month><year>2006</year></pub-date><pub-date pub-type="epub"><day>9</day><month>6</month><year>2006</year></pub-date><volume>203</volume><issue>2</issue><fpage>171</fpage><lpage>203</lpage><history><date date-type="received"><day>16</day><month>8</month><year>2005</year></date><date date-type="rev-recd"><day>5</day><month>5</month><year>2006</year></date><date date-type="accepted"><day>9</day><month>5</month><year>2006</year></date></history><permissions><copyright-statement>Copyright © 2006 Elsevier Inc. All rights reserved.</copyright-statement><copyright-year>2006</copyright-year><copyright-holder>Elsevier Inc.</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><p>Both the threat of bioterrorism and the natural emergence of contagious diseases underscore the importance of quantitatively understanding disease transmission in structured human populations. Over the last few years, researchers have advanced the mathematical theory of scale-free networks and used such theoretical advancements in pilot epidemic models. Scale-free contact networks are particularly interesting in the realm of mathematical epidemiology, primarily because these networks may allow meaningfully structured populations to be incorporated in epidemic models at moderate or intermediate levels of complexity. Moreover, a scale-free contact network with node degree correlation is in accord with the well-known preferred mixing concept. The present author describes a semi-empirical and deterministic epidemic modeling approach that (a) focuses on time-varying rates of disease transmission in both unstructured and structured populations and (b) employs probability density functions to characterize disease progression and outbreak controls. Given an epidemic curve for a historical outbreak, this modeling approach calls for Monte Carlo calculations (that define the average new infection rate) and solutions to integro-differential equations (that describe outbreak dynamics in an aggregate population or across all network connectivity classes). Numerical results are obtained for the 2003 SARS outbreak in Taiwan and the dynamical implications of time-varying transmission rates and scale-free contact networks are discussed in some detail.</p></abstract><kwd-group><title>Keywords</title><kwd>Deterministic epidemic model</kwd><kwd>Scale-free network</kwd><kwd>Preferred mixing</kwd><kwd>Time-dependent transmission rate</kwd><kwd>SARS</kwd></kwd-group></article-meta></front><body><sec><label>1</label><title>Introduction</title><p>The presence and likely impact of contagion are critical issues that surround a bioterrorist attack, the natural emergence of a new disease, or the accidental release of a dangerous pathogen into the natural environment. Even if effective countermeasures are available, a failure to completely contain a contagious disease may give rise to a far-reaching epidemic with considerable social disruption and economic impact. Both civilian and military homeland biodefense planners are thus faced with a broad spectrum of contagious disease threats and a sound conceptual framework is necessary to shape and plan threat responses. Arguably, well-designed epidemic modeling studies can contribute significantly to such a conceptual framework.</p><p>Severe acute respiratory syndrome (SARS) is the first new contagious disease to emerge in the 21st century. Initial SARS cases occurred in southern China during November of 2002 and international air travel quickly spread the disease through Asia, North America and Europe. On the 5th of July 2003, the World Health Organization announced the global containment of SARS cases. This international outbreak was responsible for about 8100 cases and 775 deaths. Moreover, Asian economies (for example) suffered losses in the neighborhood of 15 billion dollars (plus additional billions of dollars for economic stimulus packages) <xref rid="bib1" ref-type="bibr">[1]</xref>.</p><p>The SARS outbreaks of 2002 and 2003 are instructive for several reasons. First, modern pharmaceuticals could neither prevent SARS infections nor abort the natural course of SARS cases. Each affected country controlled the spread of SARS in traditional ways: (a) case identification and contact tracing, (b) case-patient isolation (hospital infection control), and (c) exposure management (quarantine). Second, disease surveillance shortcomings and public health capacity limitations constrained SARS responses of Asian governments. Asian public health systems were clearly under stress and international assistance may have been critical in avoiding much larger outbreaks. Third and last, a great deal of SARS epidemiological information has already been published and more SARS data is likely to appear in the scientific literature.</p><p>The available SARS outbreak data has prompted researchers to reexamine the question of how best to quantitatively describe outbreak dynamics when traditional outbreak controls are the only viable countermeasures <xref rid="bib2" ref-type="bibr">[2]</xref>, <xref rid="bib3" ref-type="bibr">[3]</xref>, <xref rid="bib4" ref-type="bibr">[4]</xref>, <xref rid="bib5" ref-type="bibr">[5]</xref>. Good answers to this question have been (and will be) essential to the development of sound biodefense requirements and plans.</p><p>Epidemic models have long been valuable tools for studying the dynamics of contagious diseases in human populations. Assuming an unstructured population and the standard incidence, disease transmission occurs by means of homogeneous mixing, where each contagious individual is free to contact and infect any susceptible individual. But if the population is structured according to cultural, socio-economic, demographic or geographic factors, there is a mixing matrix that constrains opportunities for disease-causing contacts. Proportionate and preferred mixing matrices have enhanced the explanatory and predictive powers of epidemic models since the late 1980s <xref rid="bib6" ref-type="bibr">[6]</xref>. It is also true that sufficient data is seldom accessible to quantitatively define key mixing matrix elements for a large and highly structured population.</p><p>Viewing a structured population as a contact network, nodes or vertices in this network represent individuals and links or edges indicate potential disease-causing contacts. Mounting evidence shows that various real-world networks are scale-free; i.e., the probability distribution for the number of links per node (or the node ‘degree’) follows a power law <xref rid="bib7" ref-type="bibr">[7]</xref>, <xref rid="bib8" ref-type="bibr">[8]</xref>. This power law is associable with network evolution or growth. More specifically, in contrast to classical random graphs, a scale-free network is a growing open system that accommodates a new node by attaching it to certain preexisting network nodes. The growth process for a scale-free network gives rise to another distinguishing feature, correlated degrees of connected nodes <xref rid="bib9" ref-type="bibr">[9]</xref>. In the absence of node degree correlation, network-based epidemic models often entail proportionate (random) mixing assumptions.</p><p>May and Lloyd seem to have been the first researchers to incorporate scale-free network characteristics in a basic SIR (susceptible-infectious-removed) model for structured populations <xref rid="bib10" ref-type="bibr">[10]</xref>. These investigators relied upon a random mixing matrix and treated the discrete node degree as a continuous variable. They derived mathematical expressions and obtained numerical results describing (a) the infected fraction of the entire population as a function of a basic reproductive parameter and (b) the infected fraction by node degree for selected parameter values. These researchers focused on epidemic threshold issues and they did not discuss how a scale-free network could affect outbreak dynamics.</p><p>Barthelemy, Barrat, Boguna, Pastor-Satorras and Vespignani have published a series of interesting and informative papers that explore network effects on outbreak dynamics <xref rid="bib11" ref-type="bibr">[11]</xref>, <xref rid="bib12" ref-type="bibr">[12]</xref>, <xref rid="bib13" ref-type="bibr">[13]</xref>, <xref rid="bib14" ref-type="bibr">[14]</xref>, <xref rid="bib15" ref-type="bibr">[15]</xref>, <xref rid="bib16" ref-type="bibr">[16]</xref>. They have mathematically defined important properties of scale-free networks and examined epidemiological implications of these properties using susceptible-infectious (SI), susceptible-infectious-susceptible (SIS) and SIR models. In their network-based SI, SIS and SIR analyses, Barthelemy and coauthors routinely assumed an unbounded total population and employed a uniform initial condition (forcing all node degree or connectivity classes to be initially infected at the same level); additionally, they concentrated on an epidemic’s exponential growth phase. The principal finding of these analyses is that, once high connectivity classes become infected, a cascade of infection occurs and affects lower and lower connectivity classes over time. Lastly, with regard to a non-uniform initial condition, the investigators found that confining all initially infected individuals to a single high-connectivity class would lead to an outbreak involving a faster rise and a faster invasion of the network.</p><p>Two published studies describe stochastic SARS outbreak simulations based upon lattices or grids and ‘small world’ linkages. Masuda, Konno and Aihara <xref rid="bib17" ref-type="bibr">[17]</xref> argue that their simulations of the 2003 SARS outbreak in Singapore (as well as the general nature of SARS epidemics) are inconsistent with clustering properties and epidemic thresholds of scale-free networks. On the other hand, findings of the present author indicate that a finite scale-free network with node degree correlation is consistent with the 2003 SARS outbreak in Taiwan. Small and Tse <xref rid="bib18" ref-type="bibr">[18]</xref> discuss their simulation of the 2003 SARS outbreak in Hong Kong and they make the following observations: (a) a small world network with long distance links can exhibit scale-free behavior and (b) further work must be done to ascertain whether their simulation results reveal that type of behavior. Although clear evidence of scale-free contact networks is at hand for sexually transmitted diseases <xref rid="bib19" ref-type="bibr">[19]</xref>, similar published evidence of SARS-related contact networks is currently unavailable.</p><p>The main objectives of the present study are to: (a) develop congruent epidemic models for unstructured and structured populations that can better exploit currently available information on SARS progression and outbreak controls; (b) demonstrate the utility of these models by carefully reconstructing the dynamics of a historical SARS outbreak; and (c) employ historical time-varying rates of SARS transmission to assess effects of control measures in possible future outbreaks. Zoonotic infections and ordinary human activities in Asia may again be responsible for major SARS outbreaks, but it is also conceivable that a bioterrorist attack (for example, human vectors and intentional person-to-person SARS transmission) would cause even more human suffering and deaths. Epidemic models are clearly needed to quantitatively understand both natural and unnatural outbreaks of contagious diseases and to help improve or delimit outbreak response capabilities of health care systems.</p><p>This paper puts forward two complementary epidemic models for unstructured and structured populations. The SEIR-like compartmental framework
<xref rid="fn1" ref-type="fn">1</xref> for both models is deterministic and encompasses the standard incidence. An ‘X’ compartment (with <italic>X</italic><sub>S</sub>, <italic>X</italic><sub>E</sub> and <italic>X</italic><sub>I</sub> sub-compartments) is also a feature of this framework and it facilitates quantitative characterizations of outbreak control measures (quarantine, contact tracing and patient isolation). Most importantly, in order to enhance epidemiological realism, the two models accommodate a time-varying transmission rate and probability density functions (PDFs) for several important time intervals: namely, latent (incubational), contagious, onset-to-diagnosis, admission-to-discharge (from a hospital) and onset-to-death.</p><p>When exponential PDFs govern the latent and contagious periods, only mean values of these random variables enter a typical SEIR model. But alternative SEIR formulations make the assumption of exponentially distributed time intervals unnecessary. This is to say, by replacing differential equations (DEs) with integro-differential equations (IDEs), previous investigators have utilized non-exponential PDFs in fully characterizing latent and contagious periods <xref rid="bib20" ref-type="bibr">[20]</xref>, <xref rid="bib21" ref-type="bibr">[21]</xref>. The present author adopts a similar approach and obtains IDEs containing general PDFs for all key time intervals. Although numerical algorithms for these IDEs can require substantially more computer time than those for exponential PDFs and associated DEs, the higher level of epidemiological realism helps to confidently resolve critical dynamical aspects of network-based models.</p><p>The finite and discrete scale-free network of primary interest is the product of a rigorous mathematical analysis and the derived node degree PDF is in good agreement with results of numerical simulations <xref rid="bib22" ref-type="bibr">[22]</xref>. Network characteristics enter our epidemic model for a structured population through (a) the node degree PDF and (b) the Pearson correlation coefficient for degrees of connected nodes. The size of the total population and the node degree PDF determine sizes of all connectivity classes, which are the selected sub-populations for the epidemic model. Similarly, the aforementioned correlation coefficient can be related to the preferred mixing parameter for our structured population. This correlation coefficient has been evaluated for some scale-free social networks <xref rid="bib23" ref-type="bibr">[23]</xref>, but it is essentially an unknown quantity in the realm of SARS outbreaks and contact networks. Numerical results in the main body of this paper thus cover a range of correlation coefficient values as well as different locations (connectivity classes) of the index case.</p><p>Utilizing a network-based model to analytically reconstruct a historical outbreak indicates the model’s explanatory value and also ties its dynamical behavior to real epidemiological circumstances. For instance, if the analytical reconstruction of a 2003 SARS outbreak demonstrates a cascade of infection (from higher to lower connectivity classes) within the selected contact network, this resultant chain of disease transmission is at least connected with (a) the observed progression of one specific disease, (b) specific social (including health care) settings and (c) modern implementations of traditional control measures.</p><p>Since it has received relatively little attention from the epidemic modeling community at large, the 2003 SARS outbreak in Taiwan is analyzed herein. Ying-Hen Hsieh at the National Chung Hsing University (along with colleagues at other universities and government organizations in Taiwan) co-authored some interesting descriptive and analytical papers dealing with the Taiwan SARS outbreak. Hsieh and colleagues used SARS hospitalization and fatality data spanning one month (May 5 through June 4, 2003) to obtain parameter values for an epidemic model (i.e., a system of ordinary DEs) without quarantine or hospital isolation <xref rid="bib24" ref-type="bibr">[24]</xref>. Subsequently, these researchers included a two-level quarantine and hospital isolation in another system of ordinary DEs and they performed a stability analysis that identified necessary quarantine rates for outbreak containment <xref rid="bib25" ref-type="bibr">[25]</xref>.</p><p>A number of investigators have developed epidemic models (systems of ordinary DEs) that match certain temporal data from SARS outbreaks. In their study of SARS transmission and control in China, Zhou, Ma and Brauer assumed an inexhaustible supply of susceptible individuals, formulated a system of linear difference equations, and followed a process of trial and error in obtaining a time-varying transmission rate that enabled the model to closely match data on diagnosed cases per day <xref rid="bib26" ref-type="bibr">[26]</xref>. A more complex model of the SARS outbreak in China was later put forward by Zhang, Lou, Ma and Wu <xref rid="bib27" ref-type="bibr">[27]</xref>. These authors first quantified their ‘basic adequate contact rate’ using about three weeks of data on diagnosed SARS cases and then they obtained model results for the daily number of SARS patients (in good agreement with epidemiological data). Lastly, Gumel and co-authors studied a system of ordinary DEs with constant parameters and the chosen values of the two free parameters (transmission rates for contacts involving contagious individuals within and without hospital isolation) provided the ‘best’ agreement between calculated and reported cumulative SARS fatalities over time. After obtaining values of the two free parameters for four SARS outbreaks (Toronto, Hong Kong, Singapore and Beijing), calculations of cumulative probable cases over time generally conformed to reported data for each outbreak <xref rid="bib28" ref-type="bibr">[28]</xref>.</p><p>The three previously-summarized studies are just a few examples of how simpler SARS outbreak models (systems of ordinary DEs) have been structured and parameterized to produce results that fully or partially agree with selected epidemiological data. Perhaps it goes without saying that any epidemic modeler must work with the data that is accessible. In other words, even though an epidemic model is capable of emulating a particular temporal feature of an outbreak, this does not mean that the underlying modeling approach is necessarily a systematic or suitable vehicle for analyzing outbreak dynamics on the whole. To examine the relative value of the proposed epidemic models of intermediate complexity, the present author also formulates and numerically evaluates a parallel or companion system of ordinary DEs.</p></sec><sec><label>2</label><title>Basic epidemic model for an unstructured population</title><p>The dashed curve in the lower graph of <xref rid="fig1" ref-type="fig">Fig. 1</xref>identifies daily onsets of illness during the Taiwan outbreak <xref rid="bib29" ref-type="bibr">[29]</xref>, but this curve excludes the index case to draw attention to the <italic>new</italic> infections.
<xref rid="fn2" ref-type="fn">2</xref> These 670 onsets of illness (without the index case) were originally classified as <italic>probable</italic> new SARS cases. But the Center for Disease Control in Taiwan later published separate epidemic curves for both probable and <italic>laboratory-confirmed</italic> SARS cases <xref rid="bib30" ref-type="bibr">[30]</xref>. Only 346 probable SARS cases were confirmed through laboratory testing. Arguably, the true number of SARS cases in Taiwan could have been larger (even substantially larger) than 346. As a consequence, an epidemic curve representing a total of 671 probable onsets in Taiwan is deemed to be adequate for our modeling purposes.<fig id="fig1"><label>Fig. 1</label><caption><p>Average new infections per day (upper graph), probable daily onsets of illness (dashed curve in lower graph) and derived daily onsets (solid curve in lower graph) for the 2003 SARS outbreak in Taiwan.</p></caption><graphic xlink:href="gr1"></graphic></fig></p><p>One way of inferring new infections from symptom onsets is to uniformly shift these onsets backward in time according to the mean latent period. On the other hand, the Monte Carlo method is a more rigorous way to obtain average new infections per day. A useful Monte Carlo algorithm is straightforward and a single trial involves a tractable number of computations. First, obtain random latent periods for all onsets of illness (excluding the index case). Second, backtrack in time to determine when all infections began. And third, compile the total score for each time step. Averaging scores per day for a large number of Monte Carlo trials yields the desired infections per day.</p><p>The solid curve (vice points) in the upper graph of <xref rid="fig1" ref-type="fig">Fig. 1</xref> comes from 50 000 Monte Carlo trials. In these trials, the assumed PDF for the latent period was a Gamma distribution, <italic>G</italic><sub>1</sub>(<italic>x</italic>). The mean value (<italic>μ</italic><sub>1</sub>) and standard deviation (<italic>σ</italic><sub>1</sub>) of the latent period are in <xref rid="tbl1" ref-type="table">Table 1</xref>with a supporting note and reference. This table likewise displays mean values and standard deviations of several additional random time intervals that, by assumption, follow Gamma distributions <xref rid="bib31" ref-type="bibr">[31]</xref>, <xref rid="bib32" ref-type="bibr">[32]</xref>, <xref rid="bib33" ref-type="bibr">[33]</xref>, <xref rid="bib34" ref-type="bibr">[34]</xref>, <xref rid="bib35" ref-type="bibr">[35]</xref>, <xref rid="bib36" ref-type="bibr">[36]</xref>.<table-wrap position="float" id="tbl1"><label>Table 1</label><caption><p>Mean values and standard deviations of critical random time intervals</p></caption><table frame="hsides" rules="groups"><thead><tr><th>Time interval</th><th>Subscript (<italic>i</italic>)</th><th><italic>μ</italic><sub><italic>i</italic></sub></th><th><italic>σ</italic><sub><italic>i</italic></sub></th><th>Notes and references</th></tr></thead><tbody><tr><td>Latent</td><td>1</td><td align="char">4.49</td><td align="char">2.63</td><td>Guangdong and Beijing epidemic data <xref rid="bib31" ref-type="bibr">[31]</xref></td></tr><tr><td>Contagious</td><td>2</td><td align="char">12.5</td><td align="char">5.60</td><td>Nasopharyngeal aspirate data from Hong Kong Epidemic <xref rid="bib32" ref-type="bibr">[32]</xref></td></tr><tr><td>Onset-to-diagnosis</td><td>3</td><td align="char">2.89</td><td align="char">2.10</td><td><italic>μ</italic><sub>3</sub> from Taiwan epidemic data & <italic>σ</italic><sub>3</sub> Inferred from secondary source <xref rid="bib33" ref-type="bibr">[33]</xref>, <xref rid="bib34" ref-type="bibr">[34]</xref></td></tr><tr><td>Admission-to-discharge</td><td>4</td><td align="char">23.1</td><td align="char">7.80</td><td>Hong Kong epidemic data <xref rid="bib35" ref-type="bibr">[35]</xref></td></tr><tr><td>Onset-to-death</td><td>5</td><td align="char">23.5</td><td align="char">13.2</td><td>Hong Kong epidemic data <xref rid="bib36" ref-type="bibr">[36]</xref></td></tr></tbody></table></table-wrap></p><p>Solid curves in the upper and lower graphs of <xref rid="fig1" ref-type="fig">Fig. 1</xref> are related by means of a mathematical convolution. Treating the time <italic>t</italic> as a continuous independent variable, the desired convolution integral can be expressed in the following manner:<disp-formula id="fd1"><label>(1)</label><mml:math id="M1" altimg="si1.gif" overflow="scroll"><mml:mi>p</mml:mi><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mtext>.</mml:mtext></mml:math></disp-formula>The function <italic>p</italic>(<italic>u</italic>) is the new infection rate at time <italic>u</italic>, whereas <italic>G</italic><sub>1</sub>(<italic>t</italic> −
<italic>u</italic>) d<italic>u</italic> is essentially the probability of developing SARS at time <italic>t</italic> after becoming infected at time <italic>u</italic>. Thusly, Eq. <xref rid="fd1" ref-type="disp-formula">(1)</xref> defines the expected onsets of illness per unit time <xref rid="bib37" ref-type="bibr">[37]</xref>. Regarding the numerical evaluation of this convolution integral, series of discrete <italic>p</italic> and <italic>G</italic><sub>1</sub> values and fast Fourier transforms (FFTs) led to the calculated onsets (solid curve) in the lower graph of <xref rid="fig1" ref-type="fig">Fig. 1</xref>.</p><p>‘Nested’ convolution integrals appear in many of the equations that follow and the notation of Eq. <xref rid="fd1" ref-type="disp-formula">(1)</xref> can be extended to cover integrals of this type. Consider four continuous functions of time: <italic>f</italic><sub>A</sub>(<italic>t</italic>), <italic>f</italic><sub>B</sub>(<italic>t</italic>), <italic>f</italic><sub>C</sub>(<italic>t</italic>) and <italic>f</italic><sub>D</sub>(<italic>t</italic>). Two examples using these functions demonstrate the extended notation for nested convolution integrals, namely,<disp-formula><mml:math id="M2" altimg="si2.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">A</mml:mi></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">C</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">A</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">C</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi></mml:math></disp-formula>and<disp-formula><mml:math id="M3" altimg="si3.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">A</mml:mi></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">C</mml:mi></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">A</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mfenced open="{" close="}"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">v</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">C</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo>-</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>w</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p>The new infection rate, previously defined PDFs, the size of the fixed total population, <italic>N</italic><sub>0</sub>, and the number of initial or primary infections, <italic>E</italic>(0), are key elements of the basic epidemic model for an unstructured population. A general mathematical description of this model begins with a set of IDEs:<disp-formula id="fd11"><label>(2)</label><mml:math id="M4" altimg="si4.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd12"><label>(3)</label><mml:math id="M5" altimg="si5.gif" overflow="scroll"><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd13"><label>(4)</label><mml:math id="M6" altimg="si6.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd3"><label>(5)</label><mml:math id="M7" altimg="si7.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd4"><label>(6)</label><mml:math id="M8" altimg="si8.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>and<disp-formula id="fd5"><label>(7)</label><mml:math id="M9" altimg="si9.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>Remaining initial conditions become<disp-formula id="fd14"><label>(8)</label><mml:math id="M10" altimg="si10.gif" overflow="scroll"><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></disp-formula>with<disp-formula id="fd15"><label>(9)</label><mml:math id="M11" altimg="si11.gif" overflow="scroll"><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p>Laplace transforms simplify the integration of Eqs. <xref rid="fd3" ref-type="disp-formula">(5)</xref>, <xref rid="fd4" ref-type="disp-formula">(6)</xref>, <xref rid="fd5" ref-type="disp-formula">(7)</xref> and the results can be stated in terms of the time-dependent cumulative number of follow-on (new) infections, <italic>y</italic>(<italic>t</italic>):<disp-formula id="fd16"><label>(10)</label><mml:math id="M12" altimg="si12.gif" overflow="scroll"><mml:mi>y</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>Let <italic>z</italic> be the independent variable in the Laplace domain, let an underlined letter (e.g., <underline><italic>E</italic></underline>) denote a function in the Laplace domain and let <italic>L</italic><sup>−1</sup> signify an inverse Laplace transformation. Eq. <xref rid="fd3" ref-type="disp-formula">(5)</xref> in the Laplace domain becomes<disp-formula><mml:math id="M13" altimg="si13.gif" overflow="scroll"><mml:mi>z</mml:mi><mml:munder accentunder="true"><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>z</mml:mi><mml:msub><mml:mrow><mml:munder accentunder="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula>or<disp-formula><mml:math id="M14" altimg="si14.gif" overflow="scroll"><mml:munder accentunder="true"><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>1.0</mml:mn><mml:mo>-</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo>/</mml:mo><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>z</mml:mi><mml:mo>-</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">]</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:munder accentunder="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>Several inverse Laplace transforms are requisites for writing the last equation in the time domain: specifically,<disp-formula><mml:math id="M15" altimg="si15.gif" overflow="scroll"><mml:mtable columnspacing="0em"><mml:mtr><mml:mtd columnalign="right"></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>1.0</mml:mn><mml:mo>/</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1.0</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">for</mml:mi><mml:mspace width="0.35em"></mml:mspace><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mtext>,</mml:mtext></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>and<disp-formula><mml:math id="M16" altimg="si16.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">]</mml:mo><mml:munder accentunder="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo>_</mml:mo></mml:mrow></mml:munder><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p>Allocations of individuals among the <italic>E</italic>, <italic>I</italic> and <italic>R</italic> compartments are now defined by<disp-formula id="fd7"><label>(11)</label><mml:math id="M17" altimg="si17.gif" overflow="scroll"><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd8"><label>(12)</label><mml:math id="M18" altimg="si18.gif" overflow="scroll"><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>and<disp-formula id="fd9"><label>(13)</label><mml:math id="M19" altimg="si19.gif" overflow="scroll"><mml:mi>R</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>v</mml:mi><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>Integrals in Eqs. <xref rid="fd7" ref-type="disp-formula">(11)</xref>, <xref rid="fd8" ref-type="disp-formula">(12)</xref>, <xref rid="fd9" ref-type="disp-formula">(13)</xref> are readily interpretable and the parallel progression of initial and follow-on infections is apparent. For instance, the first term on the right-hand side of Eq. <xref rid="fd7" ref-type="disp-formula">(11)</xref> is the number of initially infected individuals who still remain in the <italic>E</italic> compartment at time <italic>t</italic>, while the difference between the second and third terms is the number of individuals whose follow-on infections began before time <italic>t</italic> and who have not yet moved into the <italic>I</italic> compartment.</p><p>Eqs. <xref rid="fd11" ref-type="disp-formula">(2)</xref>, <xref rid="fd12" ref-type="disp-formula">(3)</xref>, <xref rid="fd13" ref-type="disp-formula">(4)</xref> and <xref rid="fd14" ref-type="disp-formula">(8)</xref>, <xref rid="fd15" ref-type="disp-formula">(9)</xref>, <xref rid="fd16" ref-type="disp-formula">(10)</xref>, <xref rid="fd7" ref-type="disp-formula">(11)</xref>, <xref rid="fd8" ref-type="disp-formula">(12)</xref>, <xref rid="fd9" ref-type="disp-formula">(13)</xref> establish the general form of the basic epidemic model. Although the <italic>X</italic><sub>E</sub> sub-compartment is not part of our analytical reconstruction of the Taiwan SARS outbreak, <italic>X</italic><sub>S</sub> and <italic>X</italic><sub>I</sub> sub-compartmental dynamics are indeed required to emulate, respectively, the quarantine of susceptible individuals and the isolation of SARS patients in hospitals. And because <italic>R</italic>(<italic>t</italic>) is just the time-dependent cumulative number of individuals who are no longer contagious, extensions of the basic epidemic model are necessary to evaluate time-dependent cumulative deaths and recoveries (or recovered SARS case-patients who are discharged from a hospital). At the end of an outbreak, the total number of individuals in the <italic>R</italic> compartment must be the same as the total number of SARS-related deaths plus the total number of SARS recoveries.</p><p>Six parameters characterize the sub-compartment for quarantined susceptible individuals, <italic>X</italic><sub>S</sub>. The parameters <italic>ϕ</italic><sub>1</sub> and <italic>ϕ</italic><sub>2</sub> denote the respective proportions of initial and follow-on infections that result in hospital isolation upon symptom onset.
<xref rid="fn3" ref-type="fn">3</xref> Half of the quarantine parameters are time constants: (a) delay time, <italic>t</italic><sub>D</sub>, extending from day number 0 up to the start of contact tracing, (b) average time, <italic>t</italic><sub>T</sub>, to trace and quarantine the contacts of a case patient entering hospital isolation and (c) quarantine period, <italic>t</italic><sub>Q</sub>. Also, the average number of quarantined susceptible people per case-patient is <italic>λ</italic>. Values of all quarantine parameters for the SARS outbreak in Taiwan and supporting references are in <xref rid="tbl2" ref-type="table">Table 2</xref>.<table-wrap position="float" id="tbl2"><label>Table 2</label><caption><p>Parameter values for the basic and network-based epidemic models</p></caption><table frame="hsides" rules="groups"><thead><tr><th>Parameter</th><th>Parameter value(s)</th><th>Notes and references</th></tr></thead><tbody><tr><td><italic>N</italic><sub>0</sub></td><td>1 000 000</td><td>Computational resource limitations (network-based model)</td></tr><tr><td><italic>E</italic>(0)</td><td>1</td><td>Single index case in the basic model</td></tr><tr><td><italic>ϕ</italic><sub>1</sub></td><td>0</td><td>No early isolation of index case-patient</td></tr><tr><td><italic>ϕ</italic><sub>2</sub></td><td>0.0672</td><td>45/670 <xref rid="bib29" ref-type="bibr">[29]</xref></td></tr><tr><td><italic>t</italic><sub>D</sub> (days)</td><td>27</td><td>Quarantine in Taiwan began on March 18, 2003 <xref rid="bib29" ref-type="bibr">[29]</xref></td></tr><tr><td><italic>t</italic><sub>T</sub> (days)</td><td>3</td><td>By assumption</td></tr><tr><td><italic>t</italic><sub>Q</sub> (days)</td><td>12</td><td>Duration of Taiwan quarantine varied from 10–14 days <xref rid="bib33" ref-type="bibr">[33]</xref></td></tr><tr><td><italic>λ</italic> (day<sup>−1</sup>)</td><td>242</td><td>151,460/(671-45) <xref rid="bib29" ref-type="bibr">[29]</xref>, <xref rid="bib33" ref-type="bibr">[33]</xref></td></tr><tr><td><italic>θ</italic><sub>1</sub></td><td>0</td><td>Index case-patient survived</td></tr><tr><td><italic>θ</italic><sub>2</sub></td><td>0.109</td><td>73/670 <xref rid="bib29" ref-type="bibr">[29]</xref>, <xref rid="bib33" ref-type="bibr">[33]</xref></td></tr><tr><td><italic>E</italic><sub><italic>j</italic></sub>(0)</td><td><italic>δ</italic><sub><italic>jm</italic></sub></td><td>Location of the single index case in the network-based model: <italic>m</italic> = 3, 10, 30 or 100</td></tr><tr><td><italic>ε</italic></td><td><italic>r</italic> + 0.047</td><td>Combinative mixing parameter in the network-based model where <italic>r</italic> = 0.1, 0.2, 0.3 or 0.4</td></tr></tbody></table></table-wrap></p><p>If the diagnosis-to-isolation time is (on average) small, the time derivative of <italic>X</italic><sub>S</sub> basically depends on diagnoses per unit time. Letting <italic>q</italic><sub>1</sub> and <italic>q</italic><sub>2</sub> be time-dependent numbers of diagnosed initial and follow-on cases per unit time, respectively, the delay differential equation for <italic>X</italic><sub>S</sub> is<disp-formula id="fd21"><label>(14)</label><mml:math id="M20" altimg="si20.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula>where<disp-formula id="fd22"><label>(15)</label><mml:math id="M21" altimg="si21.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></disp-formula>and<disp-formula id="fd23"><label>(16)</label><mml:math id="M22" altimg="si22.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>p</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>Eqs. <xref rid="fd14" ref-type="disp-formula">(8)</xref>, <xref rid="fd21" ref-type="disp-formula">(14)</xref>, <xref rid="fd22" ref-type="disp-formula">(15)</xref>, <xref rid="fd23" ref-type="disp-formula">(16)</xref>, along with the average new infection rate in <xref rid="fig1" ref-type="fig">Fig. 1</xref> and parameter values in <xref rid="tbl1" ref-type="table">Table 1</xref>, <xref rid="tbl2" ref-type="table">Table 2</xref>, are sufficient to evaluate the quarantine rate, which is input for Eq. <xref rid="fd13" ref-type="disp-formula">(4)</xref>.</p><p>An early detection of symptoms or a later clinical diagnosis can trigger hospital isolation in the basic epidemic model. In other words, the hospital isolation sub-compartment, <italic>X</italic><sub>I</sub>, accommodates initial and follow-on case-patients who are identified by either tracing activities or clinicians. Generally, a case-patient stays in hospital isolation until he or she is no longer contagious; the patient then continues to receive medical care for some period of time (ending in hospital discharge or death). During the 2003 SARS outbreaks, efforts to isolate contagious people were not always successful. Nevertheless, in the model, everyone with a case of SARS is eventually admitted to a hospital and each case-patient within the <italic>X</italic><sub>I</sub> sub-compartment is assumed to be adequately isolated during the contagious period. The model <italic>does</italic> allow people with undiagnosed cases of SARS (who are outside the <italic>X</italic><sub>I</sub> sub-compartment) to infect other individuals (who may be in hospitals or elsewhere).</p><p>Since a SARS diagnosis and the associated hospital isolation are assumed to take place almost simultaneously, case-patients in the <italic>X</italic><sub>I</sub> sub-compartment fall into four distinct categories: (a) initial infection and diagnosis upon symptom onset, (b) initial infection and diagnosis after symptom onset, (c) follow-on infection and diagnosis upon symptom onset and (d) follow-on infection and diagnosis after symptom onset. As previously indicated, when case-patients in these categories cease to be contagious, they leave the <italic>X</italic><sub>I</sub> sub-compartment. The time-dependent number of case-patients in the <italic>X</italic><sub>I</sub> sub-compartment is below and it encompasses all four of the previously mentioned categories:<disp-formula id="fd37"><label>(17)</label><mml:math id="M23" altimg="si23.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mfenced open="{" close="}"><mml:mrow><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p>Cumulative time-dependent numbers of case-patient recoveries, <italic>H</italic><sub>R</sub>(<italic>t</italic>), and deaths, <italic>H</italic><sub>F</sub>(<italic>t</italic>), are the final elements of the basic epidemic model. To mathematically define these cumulative quantities, proportions of initial and follow-on cases resulting in death are useful parameters: respectively, <italic>θ</italic><sub>1</sub> and <italic>θ</italic><sub>2</sub>. (See <xref rid="tbl2" ref-type="table">Table 2</xref> for values of <italic>θ</italic><sub>1</sub> and <italic>θ</italic><sub>2</sub>.) Cumulative case fatalities are easily defined, i.e.,<disp-formula id="fd38"><label>(18)</label><mml:math id="M24" altimg="si24.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">F</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>v</mml:mi><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>Since the discharge of a case-patient from a hospital signifies a recovery, <italic>H</italic><sub>R</sub> depends on the above four case-patient categories and the desired mathematical definition is<disp-formula id="fd39"><label>(19)</label><mml:math id="M25" altimg="si25.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">R</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mfenced open="{" close="}"><mml:mrow><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>v</mml:mi><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">v</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo>-</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>w</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>y</mml:mi><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mi>∗</mml:mi><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p>The complete model for an unstructured population spans Eqs. <xref rid="fd11" ref-type="disp-formula">(2)</xref>, <xref rid="fd12" ref-type="disp-formula">(3)</xref>, <xref rid="fd13" ref-type="disp-formula">(4)</xref> and <xref rid="fd14" ref-type="disp-formula">(8)</xref>, <xref rid="fd15" ref-type="disp-formula">(9)</xref>, <xref rid="fd16" ref-type="disp-formula">(10)</xref>, <xref rid="fd7" ref-type="disp-formula">(11)</xref>, <xref rid="fd8" ref-type="disp-formula">(12)</xref>, <xref rid="fd9" ref-type="disp-formula">(13)</xref>, <xref rid="fd21" ref-type="disp-formula">(14)</xref>, <xref rid="fd22" ref-type="disp-formula">(15)</xref>, <xref rid="fd23" ref-type="disp-formula">(16)</xref>, <xref rid="fd37" ref-type="disp-formula">(17)</xref>, <xref rid="fd38" ref-type="disp-formula">(18)</xref>, <xref rid="fd39" ref-type="disp-formula">(19)</xref>. A known new infection rate and finite-difference replacements for Eqs. <xref rid="fd13" ref-type="disp-formula">(4)</xref>, <xref rid="fd21" ref-type="disp-formula">(14)</xref> enable numerical evaluations of <italic>X</italic><sub>S</sub> and <italic>S</italic> at successive (full-day) time steps. FFTs make it possible to efficiently and accurately convolve two or more Gamma PDFs, but convolution integrals with <italic>p</italic> (or its definite integral, <italic>y</italic>) in the integrands call for a standard integration algorithm like the trapezoidal rule. Numerical results for compartments and sub-compartments in the basic epidemic model are discussed in a subsequent section of this paper.</p><p>Because average new infections per unit time are derivable from a given epidemic curve, the quantitative analysis of populated compartments and sub-compartments in the basic epidemic model does not invoke the standard incidence. But once results of this analysis are in hand, assuming the standard incidence allows us to characterize a historical outbreak by means of a time-varying rate of disease transmission, <italic>β</italic>(<italic>t</italic>), where<disp-formula id="fd96"><label>(20)</label><mml:math id="M26" altimg="si26.gif" overflow="scroll"><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mtext>.</mml:mtext></mml:math></disp-formula>Under certain circumstances, historically derived time-varying rates of disease transmission may have predictive value <xref rid="bib38" ref-type="bibr">[38]</xref>.</p><p>The rate of disease transmission is essentially the product of (a) the probability of infection, given an ‘adequate’ contact, and (b) the rate of adequate contacts. That is, an adequate contact is a necessary (but not sufficient) condition for disease transmission. The conditional probability of infection could vary during an epidemic or outbreak if, for example, the disease-causing microorganism mutates and becomes more or less able to overcome the host’s defensive mechanisms. Even in the absence of mutations, changes in environmental conditions might alter a microorganism’s infectivity. Similarly, an actual rate of adequate contacts is likely to fluctuate with day-to-day human activities, imposed restrictions on these activities (e.g., quarantine) and a growing public awareness of an ongoing outbreak. An epidemic curve and an appropriately-derived time-varying rate of disease transmission may both be stepping-stones in resolving the fine structure of outbreak dynamics.</p><p>If a safe and effective SARS vaccine were to become widely available, public health officials might well rely on targeted vaccination programs (instead of quarantines) in controlling future SARS outbreaks. Minor modifications of Eq. <xref rid="fd21" ref-type="disp-formula">(14)</xref> and the supporting parameters yield the following mathematical description of a populated sub-compartment that retains successfully vaccinated people:<disp-formula><mml:math id="M27" altimg="si27.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">V</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">{</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">VPD</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">V</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">VPD</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">V</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">V</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">VPD</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">V</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">VPD</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">V</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">}</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula>where Eqs. <xref rid="fd22" ref-type="disp-formula">(15)</xref>, <xref rid="fd23" ref-type="disp-formula">(16)</xref> continue to specify the functions <italic>q</italic><sub>1</sub> and <italic>q</italic><sub>2</sub>. The two dimensionless parameters in the foregoing equation are <italic>λ</italic><sub>V</sub>, the average number of vaccinated people per case-patient, and <italic>ε</italic><sub>V</sub> or the vaccination efficacy (encompassing both the administration and in vivo performance of the vaccine). Also, the new time intervals are the elapsed time before the beginning of a targeted vaccination program, <italic>t</italic><sub>VPD</sub>, the average time to identify and vaccinate susceptible people who are at risk of becoming infected, <italic>t</italic><sub>V</sub>, and the average post-vaccination period of observation, <italic>t</italic><sub>O</sub>.</p><p>A SARS vaccine may be effective in preventing SARS infections yet ineffective in mitigating SARS cases. For this reason, new therapeutic drugs for SARS are as desirable as a SARS vaccine. Specific potential advantages of SARS chemotherapy include shorter contagious periods, reduced recovery times and fewer SARS-related deaths. In principle, the basic epidemic model can accommodate these attributes of chemotherapy through alterations of the relevant parameter values in <xref rid="tbl1" ref-type="table">Table 1</xref>, <xref rid="tbl2" ref-type="table">Table 2</xref>; namely, <italic>μ</italic><sub>2</sub>, <italic>σ</italic><sub>2</sub>, <italic>μ</italic><sub>4</sub>, <italic>σ</italic><sub>4</sub>, <italic>μ</italic><sub>5</sub>, <italic>σ</italic><sub>5</sub>, <italic>θ</italic><sub>1</sub> and <italic>θ</italic><sub>2</sub>. Interestingly, <italic>μ</italic><sub>2</sub> and <italic>σ</italic><sub>2</sub> are the only chemotherapy-related parameters affecting disease transmission in the epidemic models at hand and the efficacy of chemotherapy need not be specified explicitly. But until there is accessible data on a specific therapeutic drug, establishing values of chemotherapy-related epidemic modeling parameters is largely a matter of guesswork.</p></sec><sec><label>3</label><title>Network-based epidemic model for a structured population</title><p>Embedding a scale-free contact network in our basic epidemic model is a reasonably efficient way to explore disease transmission patterns in a structured population. That is, major functions of the desired network-based epidemic model include reconstructing historical outbreak dynamics with some fidelity and capturing disease transmission patterns at the sub-population level. These network-based modeling capabilities ought to facilitate the development of effective contact tracing, quarantine and/or vaccination plans.</p><p>Krapivsky and Redner <xref rid="bib39" ref-type="bibr">[39]</xref>, <xref rid="bib40" ref-type="bibr">[40]</xref> have carefully analyzed a fundamental model of network growth. In that model, the growth process for a network includes (a) an initial condition, (b) the addition of one node or vertex at each time step and (c) the attachment of a new node to a single pre-existing (or ancestral) network node. The initial condition of interest is a ‘dimer,’ involving a primordial node that is linked to the first network node.
<xref rid="fn4" ref-type="fn">4</xref> Most importantly, when the probability of a new node attaching to a pre-existing network node is proportional to the degree of the latter node, a scale-free network emerges in the manner of Barabasi and Albert <xref rid="bib41" ref-type="bibr">[41]</xref>.</p><p>As the number of nodes in a scale-free network becomes infinite, the discrete PDF for nodes of degree <italic>j</italic> is<disp-formula id="fd45"><label>(21)</label><mml:math id="M28" altimg="si28.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mo>/</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula>wherein <italic>j</italic> runs from 1 to infinity. With regard to correlating the degrees of linked nodes in an infinite scale-free network, researchers have obtained the following relative number of nodes of degree <italic>j</italic> that are linked (attached) to ancestral nodes of degree <italic>i</italic>:<disp-formula id="fd48"><label>(22)</label><mml:math id="M29" altimg="si29.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">ji</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo>+</mml:mo><mml:mn>12</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:math></disp-formula>with values of <italic>j</italic> as before and <italic>i</italic> running from 2 to infinity. The relative number of nodes of degree <italic>j</italic> with links to ancestral nodes of degree 1 is undefined.</p><p>Of particular interest in this paper is a finite scale-free network with <italic>N</italic><sub>0</sub> nodes and <italic>N</italic><sub>0</sub> links. The Krapivsky–Redner PDF for a finite scale-free network deviates from Eq. <xref rid="fd45" ref-type="disp-formula">(21)</xref>, especially at higher node degrees:<disp-formula id="fd46"><label>(23)</label><mml:math id="M30" altimg="si30.gif" overflow="scroll"><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mfenced open="{" close="}"><mml:mrow><mml:mtext>erfc</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mn>4</mml:mn><mml:mi mathvariant="normal">π</mml:mi></mml:mrow></mml:msqrt><mml:mo stretchy="false">]</mml:mo><mml:mi mathvariant="normal">exp</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>/</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:math></disp-formula>where erfc stands for the complementary error function and<disp-formula id="fd47"><label>(24)</label><mml:math id="M31" altimg="si31.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>j</mml:mi><mml:mo>/</mml:mo><mml:msubsup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mtext>.</mml:mtext></mml:math></disp-formula>If <italic>N</italic><sub>0</sub> = 10<sup>6</sup> nodes, Eqs. <xref rid="fd46" ref-type="disp-formula">(23)</xref>, <xref rid="fd47" ref-type="disp-formula">(24)</xref> tell us that virtually no nodes are in degree classes above <italic>j</italic> = 1482.
<xref rid="fn5" ref-type="fn">5</xref> In passing, this author is unaware of a published counterpart to Eq. <xref rid="fd48" ref-type="disp-formula">(22)</xref> that delineates node degree correlation in a finite scale-free network.</p><p>By and large, the basic epidemic model in the previous section is easy to disaggregate. Eqs. <xref rid="fd11" ref-type="disp-formula">(2)</xref>, <xref rid="fd12" ref-type="disp-formula">(3)</xref>, <xref rid="fd13" ref-type="disp-formula">(4)</xref> and <xref rid="fd14" ref-type="disp-formula">(8)</xref>, <xref rid="fd15" ref-type="disp-formula">(9)</xref>, <xref rid="fd16" ref-type="disp-formula">(10)</xref> can be restated for nodes (or individuals) in the <italic>j</italic>th node degree class (or <italic>j</italic>th connectivity class): viz.,<disp-formula id="fd75"><label>(25)</label><mml:math id="M32" altimg="si32.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd76"><label>(26)</label><mml:math id="M33" altimg="si33.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mtext>,</mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">E</mml:mi><mml:mtext>,</mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">I</mml:mi><mml:mtext>,</mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd77"><label>(27)</label><mml:math id="M34" altimg="si34.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mtext>,</mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd78"><label>(28)</label><mml:math id="M35" altimg="si35.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd79"><label>(29)</label><mml:math id="M36" altimg="si36.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>and<disp-formula id="fd80"><label>(30)</label><mml:math id="M37" altimg="si37.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mspace width="0.12em"></mml:mspace><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mtext>,</mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p>Invoking the standard incidence and employing the preferred mixing concept, the new infection rate for the <italic>j</italic>th connectivity class, <italic>p</italic><sub><italic>j</italic></sub>(<italic>t</italic>), and corresponding transmission matrix elements, <italic>β</italic><sub><italic>jk</italic></sub>(<italic>t</italic>)/<italic>N</italic><sub><italic>k</italic></sub>, can be written as<disp-formula id="fd55"><label>(31)</label><mml:math id="M38" altimg="si38.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">jk</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></disp-formula>and<disp-formula id="fd56"><label>(32)</label><mml:math id="M39" altimg="si39.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">jk</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>ε</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mfenced open="/"><mml:mrow><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:mfenced></mml:mrow><mml:mo>+</mml:mo><mml:mi>ε</mml:mi><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">jk</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mtext>.</mml:mtext></mml:math></disp-formula>Summation indices in Eqs. <xref rid="fd55" ref-type="disp-formula">(31)</xref>, <xref rid="fd56" ref-type="disp-formula">(32)</xref> range from 1 up to 1482, <italic>ε</italic> is the combinative mixing parameter, <italic>a</italic><sub><italic>j</italic></sub>(<italic>t</italic>) is the average number of disease-causing contacts per unit time per individual in the <italic>j</italic>th connectivity class, and <italic>δ</italic><sub><italic>jk</italic></sub> is the Kronecker delta <xref rid="bib42" ref-type="bibr">[42]</xref>. When <italic>ε</italic> = 0, members of different connectivity classes mix proportionately (randomly) and, when <italic>ε</italic> = 1, there is no mixing between members of different connectivity classes. Furthermore, since each individual in the <italic>j</italic>th connectivity class is linked to <italic>j</italic> other individuals, <italic>a</italic><sub><italic>j</italic></sub>(<italic>t</italic>) becomes<disp-formula id="fd58"><label>(33)</label><mml:math id="M40" altimg="si40.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>j</mml:mi><mml:mtext>,</mml:mtext></mml:math></disp-formula>where <italic>ω</italic>(<italic>t</italic>) is the average time-varying rate of disease transmission.</p><p>The transmission matrix elements can now be restated as<disp-formula id="fd136"><label>(34)</label><mml:math id="M41" altimg="si41.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">jk</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>j</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>ε</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="italic">kP</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>+</mml:mo><mml:mi>ε</mml:mi><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">jk</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>j</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>where<disp-formula><mml:math id="M42" altimg="si42.gif" overflow="scroll"><mml:mo stretchy="false">〈</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:mi mathvariant="italic">kP</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>and<disp-formula><mml:math id="M43" altimg="si43.gif" overflow="scroll"><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mtext>.</mml:mtext></mml:math></disp-formula>Given an individual in the <italic>j</italic>th connectivity class, <italic>P</italic>(<italic>k</italic>∣<italic>j</italic>) is the conditional probability that he or she is linked to an individual in the <italic>k</italic>th connectivity class <xref rid="bib43" ref-type="bibr">[43]</xref>. Utilizing Eqs. <xref rid="fd56" ref-type="disp-formula">(32)</xref>, <xref rid="fd58" ref-type="disp-formula">(33)</xref> to replace the transmission matrix elements in Eq. <xref rid="fd55" ref-type="disp-formula">(31)</xref>, the average disease transmission rate and the new infection rate for the <italic>j</italic>th connectivity class then become<disp-formula id="fd81"><label>(35)</label><mml:math id="M44" altimg="si44.gif" overflow="scroll"><mml:mi>ω</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mfenced open="/"><mml:mrow><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mtext>,</mml:mtext><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi mathvariant="italic">jS</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:mfenced></mml:mrow></mml:math></disp-formula>and<disp-formula id="fd82"><label>(36)</label><mml:math id="M45" altimg="si45.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="italic">jS</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mfenced open="/"><mml:mrow><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mtext>,</mml:mtext><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi mathvariant="italic">jS</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:mfenced></mml:mrow><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p>Simply indexing Eqs. <xref rid="fd21" ref-type="disp-formula">(14)</xref>, <xref rid="fd22" ref-type="disp-formula">(15)</xref>, <xref rid="fd23" ref-type="disp-formula">(16)</xref> is problematic, essentially because this action would quarantine the same number of susceptible individuals from any connectivity class containing a newly identified case-patient, regardless of that group’s size.
<xref rid="fn6" ref-type="fn">6</xref> Over 150 000 susceptible people were quarantined during the 2003 SARS outbreak in Taiwan and equating connectivity class PDFs for the quarantined susceptible population and the overall susceptible population is a reasonable approximation. Thus, in the <italic>j</italic>th connectivity class, the approximate number of quarantined susceptible individuals per unit time is<disp-formula id="fd65"><label>(37)</label><mml:math id="M46" altimg="si46.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">S</mml:mi><mml:mtext>,</mml:mtext><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>Eqs. <xref rid="fd22" ref-type="disp-formula">(15)</xref>, <xref rid="fd23" ref-type="disp-formula">(16)</xref> still define the respective aggregate numbers of diagnosed initial and follow-on cases per unit time, <italic>q</italic><sub>1</sub> and <italic>q</italic><sub>2</sub>. Of course, summing over the index <italic>j</italic> in Eq. <xref rid="fd65" ref-type="disp-formula">(37)</xref> yields Eq. <xref rid="fd21" ref-type="disp-formula">(14)</xref>.</p><p>Indexed versions of Eqs. <xref rid="fd7" ref-type="disp-formula">(11)</xref>, <xref rid="fd8" ref-type="disp-formula">(12)</xref>, <xref rid="fd9" ref-type="disp-formula">(13)</xref> and <xref rid="fd37" ref-type="disp-formula">(17)</xref>, <xref rid="fd38" ref-type="disp-formula">(18)</xref>, <xref rid="fd39" ref-type="disp-formula">(19)</xref> entail nothing more than substituting <italic>E</italic><sub><italic>j</italic></sub>(0) for E(0) and <italic>y</italic><sub><italic>j</italic></sub>(<italic>u</italic>) for <italic>y</italic>(<italic>u</italic>). Such straightforward substitutions obviate the need for a full listing of every indexed equation in the network-based epidemic model. In summary, the model for a structured population encompasses Eqs. <xref rid="fd46" ref-type="disp-formula">(23)</xref>, <xref rid="fd47" ref-type="disp-formula">(24)</xref>, <xref rid="fd75" ref-type="disp-formula">(25)</xref>, <xref rid="fd76" ref-type="disp-formula">(26)</xref>, <xref rid="fd77" ref-type="disp-formula">(27)</xref>, <xref rid="fd78" ref-type="disp-formula">(28)</xref>, <xref rid="fd79" ref-type="disp-formula">(29)</xref>, <xref rid="fd80" ref-type="disp-formula">(30)</xref> and <xref rid="fd81" ref-type="disp-formula">(35)</xref>, <xref rid="fd82" ref-type="disp-formula">(36)</xref>, <xref rid="fd65" ref-type="disp-formula">(37)</xref> as well as the indexed versions of Eqs. <xref rid="fd7" ref-type="disp-formula">(11)</xref>, <xref rid="fd8" ref-type="disp-formula">(12)</xref>, <xref rid="fd9" ref-type="disp-formula">(13)</xref> and <xref rid="fd37" ref-type="disp-formula">(17)</xref>, <xref rid="fd38" ref-type="disp-formula">(18)</xref>, <xref rid="fd39" ref-type="disp-formula">(19)</xref>.</p><p>The last item of business with respect to the network-based model is a relationship between the combinative mixing parameter, <italic>ε</italic>, and the Pearson correlation coefficient, <italic>r</italic>. Consider two linked individuals in a heterogeneous network, one of which is in the <italic>j</italic>th connectivity class. Pastor-Satorras and Vespignani <xref rid="bib44" ref-type="bibr">[44]</xref> have shown that the average connectivity class (<inline-formula><mml:math id="M47" altimg="si47.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">nn</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>) for the remaining individual (nearest neighbor) is<disp-formula id="fd90"><label>(38)</label><mml:math id="M48" altimg="si48.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">nn</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:mi mathvariant="italic">kP</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula>and, in this paper,<disp-formula><label>(39)</label><mml:math id="M49" altimg="si49.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">nn</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>+</mml:mo><mml:mi>ε</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo>-</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>In addition, Ramasco, Dorogovtsev and Pastor-Satorras <xref rid="bib45" ref-type="bibr">[45]</xref> defined <italic>r</italic> as<disp-formula><label>(40)</label><mml:math id="M50" altimg="si50.gif" overflow="scroll"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo>-</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mfenced open="[" close="]"><mml:mrow><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msup><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">nn</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mtext>,</mml:mtext></mml:math></disp-formula>so that Eq. <xref rid="fd90" ref-type="disp-formula">(38)</xref> then allows us to write<disp-formula id="fd91"><label>(41)</label><mml:math id="M51" altimg="si51.gif" overflow="scroll"><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">〉</mml:mo><mml:mo>-</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">〉</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula></p><p>The preferred mixing concept is related to the assortivity <xref rid="bib46" ref-type="bibr">[46]</xref> of a contact network in that the combinative mixing parameter depends linearly on the Pearson correlation coefficient. A positive value of <italic>r</italic> (an assortative network) implies that individuals in higher connectivity classes tend to be linked to other individuals in high connectivity classes, while a negative value of <italic>r</italic> (a disassortative network) indicates individuals in lower connectivity classes are often linked to highly connected individuals. For our scale-free network with <italic>N</italic><sub>0</sub> individuals and <italic>N</italic><sub>0</sub> links, the second term on the right-hand side of Eq. <xref rid="fd91" ref-type="disp-formula">(41)</xref> is 0.047. A null value of the combinative mixing parameter means that <italic>r</italic> is negative and, therefore, purely proportionate mixing is slightly disassortative. Values of <italic>r</italic> for certain real-world collaboration networks range from about 0.1 up to 0.4.</p></sec><sec><label>4</label><title>Discussion of results</title><p>Wallinga and Teunis developed a likelihood-based algorithm that relies upon symptom onset data to determine the mean effective reproduction number, <italic>R</italic><sub>e</sub>, over time <xref rid="bib47" ref-type="bibr">[47]</xref>. Denoting <italic>L</italic><sub>ij</sub> as the relative likelihood that case <italic>j</italic> was the source of infection for case <italic>i</italic>, the Wallinga–Teunis algorithm for SARS outbreaks is comprised of<disp-formula id="fd92"><label>(42)</label><mml:math id="M52" altimg="si52.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">ij</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>8.4</mml:mn><mml:mtext>,</mml:mtext><mml:mn>3.8</mml:mn><mml:mtext>;</mml:mtext><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>≠</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:mi>W</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>8.4</mml:mn><mml:mtext>,</mml:mtext><mml:mn>3.8</mml:mn><mml:mtext>;</mml:mtext><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>and<disp-formula id="fd95"><label>(43)</label><mml:math id="M53" altimg="si53.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">ij</mml:mi></mml:mrow></mml:msub><mml:mtext>.</mml:mtext></mml:math></disp-formula>In Eq. <xref rid="fd92" ref-type="disp-formula">(42)</xref>, <italic>W</italic>(8.4, 3.8; <italic>x</italic>) is a Weibull PDF describing the random SARS generation interval (<italic>x</italic>) with a respective mean and standard deviation of 8.4 and 3.8 days. The generation interval is the onset-to-onset time for an infector and a corresponding ‘infectee.’ When this interval is negative in Eq. <xref rid="fd92" ref-type="disp-formula">(42)</xref>, Wallinga and Teunis require that the Weibull PDF be zero. Additionally, the specified mean (8.4 days) and standard deviation (3.8 days) are sample statistics from the 2003 SARS outbreak in Singapore.</p><p>Mean effective reproduction numbers in <xref rid="fig2" ref-type="fig">Fig. 2</xref>are from Eqs. <xref rid="fd92" ref-type="disp-formula">(42)</xref>, <xref rid="fd95" ref-type="disp-formula">(43)</xref>, with the symptom onset data in <xref rid="fig1" ref-type="fig">Fig. 1</xref> as input. (“D” in <xref rid="fig2" ref-type="fig">Fig. 2</xref> represents February 20, 2003 and, for example, D + 22 corresponds to March 14, 2003.). Our <italic>R</italic><sub>e</sub> results for the 2003 Taiwan outbreak are similar to the Wallinga–Teunis mean reproduction numbers for the 2003 Hong Kong outbreak, although the latter outbreak was somewhat longer. Outbreaks within two Taiwan hospitals appear to have been responsible for the largest reproduction numbers in <xref rid="fig2" ref-type="fig">Fig. 2</xref>.<fig id="fig2"><label>Fig. 2</label><caption><p>Effective reproduction number for the 2003 SARS outbreak in Taiwan with pertinent events and periods.</p></caption><graphic xlink:href="gr2"></graphic></fig></p><p>The mean basic reproduction number, <italic>R</italic><sub>0</sub>, for the 2003 Taiwan outbreak is related to (a) the total number (670) of follow-on infections, (b) reported daily onsets of illness in <xref rid="fig1" ref-type="fig">Fig. 1</xref> and (c) mean effective reproduction numbers in <xref rid="fig2" ref-type="fig">Fig. 2</xref>. If <italic>O</italic>(<italic>n</italic>) symbolizes the reported number of illness onsets on day number <italic>n</italic>, the defining relationship for <italic>R</italic><sub>0</sub> becomes<disp-formula><mml:math id="M54" altimg="si54.gif" overflow="scroll"><mml:mn>670</mml:mn><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:munderover><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>14</mml:mn></mml:mrow><mml:mrow><mml:mn>114</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula>where <italic>n</italic> = 4, 14 and 114 are the respective day numbers for the onset of the index case, the immediately following onsets and the final onset. The resulting value of <italic>R</italic><sub>0</sub> is 2.23, which conforms to comparable findings of Lipsitch and co-authors.</p><p>Since DEs (versus IDEs) represent a more conventional approach to the reconstruction of outbreak dynamics, a system of parallel or analogous DEs can serve as a baseline for assessing the merits of the basic and network models put forward here. The desired DEs encompass many of the parameters in <xref rid="tbl1" ref-type="table">Table 1</xref>, <xref rid="tbl2" ref-type="table">Table 2</xref> and the principal free parameter is still the SARS transmission rate. This rate is assumed to be a fixed non-zero constant within a time interval corresponding to the duration of SARS transmission (in <xref rid="fig2" ref-type="fig">Fig. 2</xref>) and its numerical value assures a total of 670 follow-on infections. Employing lower-case labels for populated compartments and sub-compartments, the initial conditions for the DEs are<disp-formula id="fd121"><label>(44)</label><mml:math id="M55" altimg="si55.gif" overflow="scroll"><mml:mi>s</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd122"><label>(45)</label><mml:math id="M56" altimg="si56.gif" overflow="scroll"><mml:mi>e</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd97"><label>(46)</label><mml:math id="M57" altimg="si57.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>and<disp-formula id="fd98"><label>(47)</label><mml:math id="M58" altimg="si58.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">iso</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mtext>.</mml:mtext></mml:math></disp-formula>Note that <italic>s</italic>, <italic>x</italic><sub>s</sub>, <italic>e</italic>, <italic>c</italic>, <italic>c</italic><sub>iso</sub>, <italic>r</italic>, <italic>h</italic><sub>r</sub> and <italic>h</italic><sub>f</sub> are, respectively, analogs of <italic>S</italic>, <italic>X</italic><sub>S</sub>, <italic>E</italic>, <italic>I</italic>, <italic>X</italic><sub>I</sub>, <italic>R</italic>, <italic>H</italic><sub>R</sub> and <italic>H</italic><sub>F</sub>.</p><p>A system of eight DEs is mathematically defined below:<disp-formula id="fd99"><label>(48)</label><mml:math id="M59" altimg="si59.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd100"><label>(49)</label><mml:math id="M60" altimg="si60.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi></mml:mrow></mml:msub><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd101"><label>(50)</label><mml:math id="M61" altimg="si61.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd102"><label>(51)</label><mml:math id="M62" altimg="si62.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd103"><label>(52)</label><mml:math id="M63" altimg="si63.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">iso</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>e</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">iso</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd104"><label>(53)</label><mml:math id="M64" altimg="si64.gif" overflow="scroll"><mml:mover accent="true"><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">iso</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">iso</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula><disp-formula id="fd105"><label>(54)</label><mml:math id="M65" altimg="si65.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></disp-formula>and<disp-formula id="fd106"><label>(55)</label><mml:math id="M66" altimg="si66.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>Although Eq. <xref rid="fd96" ref-type="disp-formula">(20)</xref> still defines the functional form of the new infection rate in the above equations, an epidemic curve no longer constrains the time history of <italic>p</italic>(<italic>t</italic>) or, for that matter, <italic>β</italic>(<italic>t</italic>). Of particular interest is the following specification of the disease transmission rate:<disp-formula><label>(56)</label><mml:math id="M67" altimg="si67.gif" overflow="scroll"><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">FIX</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">off</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mtext>,</mml:mtext></mml:math></disp-formula>where the adjusted value of <italic>β</italic><sub>FIX</sub> is 0.4899, <italic>U</italic> stands for the Heaviside step function and <italic>t</italic><sub>off</sub> equals 110 days. Lastly, since definitions of all parameters in Eqs. <xref rid="fd97" ref-type="disp-formula">(46)</xref>, <xref rid="fd98" ref-type="disp-formula">(47)</xref>, <xref rid="fd99" ref-type="disp-formula">(48)</xref>, <xref rid="fd100" ref-type="disp-formula">(49)</xref>, <xref rid="fd101" ref-type="disp-formula">(50)</xref>, <xref rid="fd102" ref-type="disp-formula">(51)</xref>, <xref rid="fd103" ref-type="disp-formula">(52)</xref>, <xref rid="fd104" ref-type="disp-formula">(53)</xref>, <xref rid="fd105" ref-type="disp-formula">(54)</xref>, <xref rid="fd106" ref-type="disp-formula">(55)</xref> are already at hand, the reasoning behind compartmental entry and exit rates (as well as delays) is evident.</p><p><xref rid="fig3" ref-type="fig">Fig. 3</xref>contains two sets of numerical results from the basic epidemic model (Eqs. <xref rid="fd11" ref-type="disp-formula">(2)</xref>, <xref rid="fd12" ref-type="disp-formula">(3)</xref>, <xref rid="fd13" ref-type="disp-formula">(4)</xref> and <xref rid="fd15" ref-type="disp-formula">(9)</xref>, <xref rid="fd16" ref-type="disp-formula">(10)</xref>, <xref rid="fd7" ref-type="disp-formula">(11)</xref>, <xref rid="fd8" ref-type="disp-formula">(12)</xref>, <xref rid="fd9" ref-type="disp-formula">(13)</xref>, <xref rid="fd21" ref-type="disp-formula">(14)</xref>, <xref rid="fd22" ref-type="disp-formula">(15)</xref>, <xref rid="fd23" ref-type="disp-formula">(16)</xref>, <xref rid="fd37" ref-type="disp-formula">(17)</xref>, <xref rid="fd38" ref-type="disp-formula">(18)</xref>, <xref rid="fd39" ref-type="disp-formula">(19)</xref>) and one set of numerical results from the system of parallel DEs (Eqs. <xref rid="fd121" ref-type="disp-formula">(44)</xref>, <xref rid="fd122" ref-type="disp-formula">(45)</xref>, <xref rid="fd97" ref-type="disp-formula">(46)</xref>, <xref rid="fd98" ref-type="disp-formula">(47)</xref>, <xref rid="fd99" ref-type="disp-formula">(48)</xref>, <xref rid="fd100" ref-type="disp-formula">(49)</xref>, <xref rid="fd101" ref-type="disp-formula">(50)</xref>, <xref rid="fd102" ref-type="disp-formula">(51)</xref>, <xref rid="fd103" ref-type="disp-formula">(52)</xref>, <xref rid="fd104" ref-type="disp-formula">(53)</xref>, <xref rid="fd105" ref-type="disp-formula">(54)</xref>, <xref rid="fd106" ref-type="disp-formula">(55)</xref>). Regarding the basic epidemic model, red curves signify that both the backtracking Monte Carlo algorithm and the IDEs relied upon Gamma PDFs, while green curves are connected with an input of Exponential PDFs. Differences between red and green curves for the same compartment or sub-compartment are generally quite small; the exceptions are results for <italic>X</italic><sub>I</sub>(<italic>n</italic>) and <italic>H</italic><sub>R</sub>(<italic>n</italic>). But even small differences between red and green curves for either <italic>p</italic>(<italic>n</italic>) or <italic>I</italic>(<italic>n</italic>) can lead to more noticeable differences between red and green curves for <italic>β</italic>(<italic>n</italic>). This is to say that <italic>β</italic>(<italic>n</italic>) tends to follow the ratio of <italic>p</italic>(<italic>n</italic>) over <italic>I</italic>(<italic>n</italic>), assuming a small outbreak in a large population.<fig id="fig3"><label>Fig. 3</label><caption><p>Calculated new infection rates, compartmental populations and transmission rates for the 2003 SARS outbreak in Taiwan using the basic epidemic model (red and green curves) and a parallel system of DEs (black curves).</p></caption><graphic xlink:href="gr3"></graphic></fig></p><p>Black curves in <xref rid="fig3" ref-type="fig">Fig. 3</xref> describe numerical results from the system of parallel DEs. The three curves for <italic>p</italic>(<italic>n</italic>) yield the same total number of follow-on infections, but the black curve is in stark contrast to the companion red and green curves. Of course, the slow rise and sharp decline of the black curve for <italic>p</italic>(<italic>n</italic>) are due to the assumed rectangular shape of the transmission rate. Slow rise times are likewise distinguishing aspects of the other black curves in <xref rid="fig3" ref-type="fig">Fig. 3</xref>.</p><p>Since the basic epidemic model and Gamma PDFs may have yielded the most realistic results in <xref rid="fig3" ref-type="fig">Fig. 3</xref>, some detailed aspects of the red curves are worth mentioning. As many as 3000 people were quarantined in a single day (∼<italic>D</italic> + 95) and the maximum number of people in quarantine (on any given day) was found to be 50 000 (∼<italic>D</italic> + 115). Similarly, the calculations show that nearly 175 (50) SARS case-patients were (were not) in hospital isolation on or about <italic>D</italic> + 85. And after <italic>D</italic> + 24 or so (20 days after onset of the index case), the calculated waveform of the time-varying transmission rate is close to the waveform in <xref rid="fig2" ref-type="fig">Fig. 2</xref>. In passing, note that red curves in <xref rid="fig3" ref-type="fig">Fig. 3</xref> and Eq. <xref rid="fd96" ref-type="disp-formula">(20)</xref> gave rise to the points (vice solid curve) in the upper graph of <xref rid="fig1" ref-type="fig">Fig. 1</xref>.</p><p>Focusing on the network-based model, the total number of new infections in the <italic>j</italic>th connectivity class is the subject of <xref rid="fig4" ref-type="fig">Fig. 4</xref>. The contents of this figure originate in Eq. <xref rid="fd82" ref-type="disp-formula">(36)</xref>, where the new infection rate, <italic>p</italic><sub><italic>j</italic></sub>(<italic>t</italic>), is summed over all time steps (<italic>t</italic> =
<italic>n</italic>Δ<italic>t</italic> and Δ<italic>t</italic> = 1 day). With regard to the upper graph of <xref rid="fig4" ref-type="fig">Fig. 4</xref>, the combinative mixing parameter is fixed at a value of 0.147 and the initial infection is either distributed across all connectivity classes (<italic>j</italic><sub>SEED</sub> = PDF
<xref rid="fn7" ref-type="fn">7</xref>) or localized within a single class (<italic>j</italic><sub>SEED</sub> = 3, 10, 30 or 100). In our network-based reconstruction of the Taiwan SARS outbreak, the disposition of the initial infection seldom influenced allocations of new (follow-on) infections to connectivity classes. But placing the initial infection in a higher connectivity class (specifically, <italic>j</italic><sub>SEED</sub> = 100) did significantly increase the number of new infections in that class
<xref rid="fn8" ref-type="fn">8</xref> (leading to the small blue spike in the upper graph of <xref rid="fig4" ref-type="fig">Fig. 4</xref>).<fig id="fig4"><label>Fig. 4</label><caption><p>Numerical results that describe total follow-on infections per connectivity class for: <italic>ε</italic> = 0.147 with several locations of the initial infection (upper graph) and <italic>j</italic><sub>SEED</sub> = 3 with several values of the combinative mixing parameter (lower graph).</p></caption><graphic xlink:href="gr4"></graphic></fig></p><p>The location of the initial infection is fixed (<italic>j</italic><sub>SEED</sub> = 3) and the combinative mixing parameter takes on different values (<italic>ε</italic> = 0.147, 0.247, 0.347 and 0.447) in the lower graph of <xref rid="fig4" ref-type="fig">Fig. 4</xref>. As the value of <italic>ε</italic> increases in this graph, ‘shoulder’ regions (between <italic>j</italic> ≅ 30 and <italic>j</italic> ≅ 400) of the curves become more pronounced; also, as <italic>j</italic> decreases from about 10 down to 1, smaller values of <italic>ε</italic> give rise to slightly larger numbers of new infections.</p><p>To better illustrate how the combinative mixing parameter affects the distribution of new infections within our contact network, consider the following definitions:<disp-formula><label>(57)</label><mml:math id="M68" altimg="si68.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">jk</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="italic">jS</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:math></disp-formula>and<disp-formula><label>(58)</label><mml:math id="M69" altimg="si69.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">jk</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">jk</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>.</mml:mtext></mml:math></disp-formula>The function <italic>p</italic><sub><italic>jk</italic></sub>(<italic>n</italic>) is the time-dependent new infection rate in the <italic>j</italic>th connectivity class due to infectors in the <italic>k</italic>th connectivity class, while <italic>η</italic><sub><italic>jk</italic></sub> is the resultant number of new infections.</p><p>Density plots in <xref rid="fig5" ref-type="fig">Fig. 5</xref>are associated with <italic>j</italic><sub>SEED</sub> = 3 and selected values of <italic>ε</italic>, and each of these plots shows variations of <italic>η</italic><sub><italic>jk</italic></sub> for <italic>j</italic> and <italic>k</italic> ranging from 1 to 50. The lower-right density plot contains the color legend for the entire figure; e.g., a dark blue pixel indicates <italic>η</italic><sub><italic>jk</italic></sub> ⩽ 1 and a pure red pixel means <italic>η</italic><sub><italic>jk</italic></sub> ⩾ 3. A ‘pool’ of red pixels (<italic>k</italic> ⩽ 2 and <italic>j</italic> ⩽ 12) and a few more red pixels along the diagonal (<italic>j</italic> =
<italic>k</italic>) are key features of the upper-left density plot for <italic>ε</italic> = 0.147. On the contrary, nearly all red and reddish orange pixels lie on the diagonal in the lower-right density plot (<italic>ε</italic> = 0.447). Net effects of raising the value of <italic>ε</italic> are: (a) reduced disease-causing contacts involving infectors in the lowest connectivity classes and (b) enhanced disease transmission within individual connectivity classes. To be sure, the latter effect is discernable from Eq. <xref rid="fd136" ref-type="disp-formula">(34)</xref>.<fig id="fig5"><label>Fig. 5</label><caption><p>Density plots, corresponding to <italic>j</italic><sub>SEED</sub> = 3 and several values of the combinative mixing parameter, that define total follow-on infections in the <italic>j</italic>th connectivity class due to contacts with infectors in the <italic>k</italic>th connectivity class.</p></caption><graphic xlink:href="gr5"></graphic></fig></p><p>Time histories of new infection rates in the 3rd, 10th, 30th and 100th connectivity classes are in <xref rid="fig6" ref-type="fig">Fig. 6</xref>. The value of <italic>ε</italic> is fixed at 0.147 in the upper four graphs and early (<italic>n</italic> ⩽ 25) spikes in new infection rates are associable with locations of the initial infection. If the size of the seeded connectivity class is relatively small, then the magnitude of the early spike is relatively large. Regarding the lower four graphs and a fixed location (<italic>j</italic><sub>SEED</sub> = 3) of the initial infection, previously-considered values of <italic>ε</italic> engender four time histories of the new infection rate in each connectivity class. In the 3rd (100th) connectivity class, lower (higher) values of the combinative mixing parameter produce higher (lower) peak values of the new infection rate. These influences of the combinative mixing parameter are consistent with curves in the lower graph of <xref rid="fig4" ref-type="fig">Fig. 4</xref>.<fig id="fig6"><label>Fig. 6</label><caption><p>Time-varying new infection rates in selected connectivity classes: four upper graphs related to <italic>ε</italic> = 0.147 and several locations of the initial infection and four lower graphs related to <italic>j</italic><sub>SEED</sub> = 3 and several values of the combinative mixing parameter.</p></caption><graphic xlink:href="gr6"></graphic></fig></p><p><xref rid="fig6" ref-type="fig">Fig. 6</xref> does not reveal a cascade of infection affecting lower and lower connectivity classes over time. However, <xref rid="fig7" ref-type="fig">Fig. 7</xref>displays new infection rates in very high connectivity classes (<italic>j</italic> = 300, 500 and 700) that are compatible with the concept of an infection cascade. More specifically, the maximum new infection rate in the 700th (500th) connectivity class precedes the peak rate in the 500th (300th) connectivity class. Perhaps it is worth noting again that sizes of very high connectivity classes are minuscule in our network-based model.<fig id="fig7"><label>Fig. 7</label><caption><p>Time-varying new infection rates in very high connectivity classes for <italic>ε</italic> = 0.147 and <italic>j</italic><sub>SEED</sub> = 3.</p></caption><graphic xlink:href="gr7"></graphic></fig></p><p>Numerical results for the time-varying rate of SARS transmission, <italic>ω</italic>(<italic>n</italic>), are in <xref rid="fig8" ref-type="fig">Fig. 8</xref>. Transmission rates for different values of <italic>j</italic><sub>SEED</sub> span two time frames (<italic>n</italic> ⩽ 25 and 25 <
<italic>n</italic> ⩽ 120) in the two upper graphs, and additional transmission rates for different values of <italic>ε</italic> are displayed similarly in the two lower graphs. The fixed value of <italic>ε</italic> for the upper graphs is 0.147 and the fixed value of <italic>j</italic><sub>SEED</sub> for the lower graphs is 3. The seeding assumption shapes the early transmission rate, whereas the combinative mixing parameter is most influential at later times. And because all complete time histories in <xref rid="fig8" ref-type="fig">Fig. 8</xref> produce the same number (670) of new infections, the lower graphs show that smaller <italic>ε</italic> values entail higher late-time transmission rates. As <italic>ε</italic> approaches a value of 0.047, the Pearson correlation coefficient vanishes and the connectivity classes of linked individuals become uncorrelated.<fig id="fig8"><label>Fig. 8</label><caption><p>Time-varying disease transmission rates: two upper graphs associated with <italic>ε</italic> = 0.147 and several locations of the initial infection and two lower graphs associated with <italic>j</italic><sub>SEED</sub> = 3 and several values of the combinative mixing parameter.</p></caption><graphic xlink:href="gr8"></graphic></fig></p><p>A network-based SARS modeling investigation by Meyers, Pourbohloul, Newman, Skowronski and Burnham <xref rid="bib48" ref-type="bibr">[48]</xref> warrants some discussion before considering further variations of the present author’s modeling parameters. Meyers and co-authors constructed a detailed contact network by focusing on households, schools, workplaces and hospitals and by drawing upon available data for 1000 households (∼2600 people) in the city of Vancouver, British Columbia. Percolation theory and the node degree distribution for this contact network (with node degree correlation) enabled the investigators to theoretically and numerically predict outcomes of outbreaks for various fixed values of the transmissibility (transmission rate). Similarly, they obtained contrasting results for Poisson and scale-free contact networks (without node degree correlation). It is also worth noting that the modeling approach of these investigators yielded no information about outbreak dynamics.</p><p>Revisiting the upper-left graph in <xref rid="fig8" ref-type="fig">Fig. 8</xref>, it’s apparent that the connectivity class of the seed can strongly influence the early (⩽ day number 13 or so) transmission rate for the selected historical outbreak; i.e., to attain the same basic reproduction number, the early transmission rate for a seed in the 3rd connectivity class generally dominates the accompanying early rate for a seed in the 100th connectivity class. The clear implication is that the basic reproduction number for an unconstrained outbreak in a contact network depends on the connectivity class of the initial infection. In this vein, <xref rid="fig8" ref-type="fig">Fig. 8</xref> complements and partially supports the impact of initial conditions as described by Meyers and co-authors.</p><p>Summing over connectivity classes in the network-based model generates aggregate populated compartments and sub-compartments that are identical to counterparts in the basic epidemic model. This confluence of epidemic models for structured and unstructured populations is due to a common modeling framework, common parameters and linked time-varying transmission rates (deriving from the same epidemic curve). Most importantly, under certain constraints, dynamical impacts of parameter variations in the basic epidemic model will also be aggregate dynamical impacts of the same parameter variations in the network-based model. The main constraint calls for maintaining the correspondence of transmission rates in the two models. A variation of parameters in <xref rid="tbl1" ref-type="table">Table 1</xref>, for example, may significantly alter the dynamics of the Taiwan outbreak through changes in the disease transmission rate, which could be derived anew using the basic or network-based model. Alternatively, variations of parameters in <xref rid="tbl2" ref-type="table">Table 2</xref> could be explored by utilizing either model with the appropriate previously-derived rate of disease transmission. In bringing this section of the paper to a close, quantitative results from the basic epidemic model elucidate changes in outbreak dynamics due to variations of selected parameters in <xref rid="tbl1" ref-type="table">Table 1</xref>, <xref rid="tbl2" ref-type="table">Table 2</xref>.</p><p>The onset-to-diagnosis PDF (<italic>G</italic><sub>3</sub>) governs the random time interval extending from the onset of illness up to a clinical diagnosis. In the models under consideration, this time interval is fundamentally the amount of time that a contagious individual is free to infect other people before he or she enters hospital isolation. It’s noteworthy that onset-to-diagnosis periods (<italic>μ</italic><sub>3</sub> = 4.75 and <italic>σ</italic><sub>3</sub> = 3.45) for the Hong Kong SARS outbreak of 2003 are substantially longer than those for either the Taiwan or Singapore (<italic>μ</italic><sub>3</sub> = 3.09 and <italic>σ</italic><sub>3</sub> = 2.50) outbreak. This information leads to the following question: How would longer (Hong Kong or Singapore) onset-to-diagnosis periods have affected the dynamics of the Taiwan outbreak?</p><p>The upper graph in <xref rid="fig9" ref-type="fig">Fig. 9</xref>shows onset-to-diagnosis PDFs and CDFs for Taiwan, Hong Kong and Singapore and the lower graph displays the resultant SARS transmission rates for the Taiwan outbreak. Furthermore, these transmission rates are connected with the new infection rates, populated compartments and sub-compartments in <xref rid="fig10" ref-type="fig">Fig. 10</xref>. If longer onset-to-diagnosis periods like those in the Hong Kong outbreak had been present in the Taiwan outbreak, there would have been a substantially lower rate of SARS transmission and a concomitant increase (decrease) in the maximum number of contagious individuals who were not (were) in hospital isolation.<fig id="fig9"><label>Fig. 9</label><caption><p>Onset-to-diagnosis PDFs and CDFs for the Taiwan, Hong Kong and Singapore SARS outbreaks (upper graph) and the resultant transmission rates (lower graph) utilizing the basic epidemic model and Taiwan epidemic curve.</p></caption><graphic xlink:href="gr9"></graphic></fig><fig id="fig10"><label>Fig. 10</label><caption><p>Calculated new infection rates and compartmental populations that come from three onset-to-diagnosis PDFs, the basic epidemic model and Taiwan epidemic curve.</p></caption><graphic xlink:href="gr10"></graphic></fig></p><p>One parameter that accounts for the effectiveness of both contact tracing activities and medical surveillance is the proportion (<italic>ϕ</italic><sub>2</sub>) of follow-on infections entering hospital isolation upon symptom onset. A reasonable estimate of <italic>ϕ</italic><sub>2</sub> for the Taiwan outbreak is 45/670, which appears in <xref rid="tbl2" ref-type="table">Table 2</xref> and comes from published information on probable cases and quarantine. To assess how the Taiwan outbreak would have been altered by larger values of <italic>ϕ</italic><sub>2</sub>, the basic epidemic model is implemented with a single ‘underlying’ transmission rate. (See the red curve in the lower graph of <xref rid="fig9" ref-type="fig">Fig. 9</xref>.) Calculated time histories of new infection rates and compartmental populations appear in <xref rid="fig11" ref-type="fig">Fig. 11</xref>for four values of <italic>ϕ</italic><sub>2</sub> (45/670, 90/670, 135/670 and 180/670). Unsurprisingly, more effective contact tracing and medical surveillance could have substantially reduced the total number of infections and the total number of deaths.<fig id="fig11"><label>Fig. 11</label><caption><p>Calculated new infection rates and compartmental populations stemming from four values of <italic>ϕ</italic><sub>2</sub> (proportion of follow-on infections entering hospital isolation upon symptom onset), the basic epidemic model and Taiwan epidemic curve.</p></caption><graphic xlink:href="gr11"></graphic></fig></p><p>The proportion (<italic>ϕ</italic><sub>1</sub>) of multiple initial infections entering hospital isolation upon symptom onset is another parameter dealing with contact tracing and medical surveillance. In the event of a single initial infection and a new emerging infectious disease, the prompt isolation of the index case-patient is unlikely and a null value of <italic>ϕ</italic><sub>1</sub> is appropriate. But multiple individuals who are simultaneously infected with a familiar pathogen would have different incubation periods and sufficient information may be at hand to promptly identify and isolate some of the index case-patients. The basic epidemic model with one underlying transmission rate is readily implemented for multiple initial infections and non-zero values of <italic>ϕ</italic><sub>1</sub>. <xref rid="fig12" ref-type="fig">Fig. 12</xref>contains time histories of new infection rates and compartmental populations for 10 initial infections and three non-zero <italic>ϕ</italic><sub>1</sub> values (0.2, 0.4 and 0.6). In terms of the total number of infections and the total number of deaths, the potential benefits of quickly finding initially infected individuals are also large.<fig id="fig12"><label>Fig. 12</label><caption><p>Calculated new infection rates and compartmental populations that derive from three non-zero values of <italic>ϕ</italic><sub>1</sub> (proportion of multiple initial infections entering hospital isolation upon symptom onset), ten initial infections, the basic epidemic model and Taiwan epidemic curve.</p></caption><graphic xlink:href="gr12"></graphic></fig></p></sec><sec><label>5</label><title>Concluding observations</title><p>The 2003 SARS outbreaks have yielded much epidemiological data covering both the course of the disease and modern implementations of traditional outbreak controls. The amount and quality of this data is unusual, spawning many simple and complex SARS outbreak models. Since few of those models were designed to exploit currently available data in a systematic manner, an epidemic modeling approach of moderate complexity has been described here that accommodates either an unstructured or a structured population.</p><p>A key thrust of this study is to describe the dynamics of an outbreak in new informative ways, especially when traditional outbreak controls are the only viable countermeasures. The epidemic models incorporate PDFs for a number of random time intervals that define the progression of initial and follow-on SARS infections as well as the implementation of traditional outbreak controls. Another significant dynamical aspect of these models is the semi-empirical time-varying rate of disease transmission. Using probabilistic reasoning to account for important random time intervals and deriving time-varying disease transmission rates from epidemic curves augment the realism and explanatory power of outbreak analyses.</p><p>Simpler SARS outbreak models in the literature often rely on a system of ordinary DEs with constant coefficients and models of this type have produced many important results concerning basic reproduction numbers and other dominant features of SARS outbreaks. To highlight the utility or value of the basic epidemic model, the present author formulated and evaluated a parallel simple system of DEs. The basic epidemic model and the parallel system of DEs generated (a) disparate new infection rates for the 2003 Taiwan outbreak (even though the total numbers of new infections were identical) and (b) very different time histories for each compartmental population. When disease progression and outbreak controls are well-characterized, the basic epidemic model seems better suited to the task of reconstructing the overall dynamics of a historical outbreak.</p><p>The mathematical theory of scale-free networks is a work in progress and the full epidemiological utility of these networks remains to be seen. To date, ideal scale-free networks (infinite number of nodes and/or continuous node degree distribution) have been incorporated into relatively simple epidemic models (SI, SIS and SIR). The present author chose a finite and discrete scale-free network (with a rigorous mathematical foundation), developed a network-based deterministic model (with a realistic progression of infections) and implemented the preferred mixing concept to account for correlated connectivity classes of linked individuals.</p><p>The matter of cascading infections within a scale-free contact network warrants further discussion. Even though the initial infection is in the 3rd connectivity class, <xref rid="fig7" ref-type="fig">Fig. 7</xref> displays new infection rates peaking at times that decrease (from ∼<italic>D</italic> + 75 down to ∼<italic>D</italic> + 53) as the number of links per infected individual increases (from 300 up to 700). <xref rid="fig7" ref-type="fig">Fig. 7</xref> thus implies that individuals in very high connectivity classes can become infected well before those individuals in somewhat lower connectivity classes. But the upper four graphs in <xref rid="fig6" ref-type="fig">Fig. 6</xref> suggest that the first generation of follow-on infections emerges by <italic>D</italic> + 25. (As the degree of the initial infection gets smaller and drops below 30 or so, the total number of infections in that connectivity class gets large and the first generation of follow-on infections becomes harder to identify.) After the first generation of follow-on infections, the upper half of <xref rid="fig6" ref-type="fig">Fig. 6</xref> reveals remarkably similar exponential phases and general wave shapes for all four locations of the initial infection. And lastly, keeping the initial infection in the 3rd connectivity class, the lower half of <xref rid="fig6" ref-type="fig">Fig. 6</xref> shows that different values of the combinative mixing parameter primarily affect magnitudes (rather than exponential phases) of new infection rates. In summary, our network-based analytical reconstruction of an actual outbreak exhibits no signs of cascading infections in connectivity classes containing more than about one individual.</p><p>Basic and network-based epidemic models under consideration here allow outbreak analyses at two levels of resolution, respectively, aggregate compartmental populations and connectivity classes. The common analytical framework for these two models assures congruence at the level of aggregate compartmental populations. Whereas the basic epidemic model is efficient in assessing the general dynamical impacts of disease progression and outbreak control parameter variations, the network-based model provides insights into disease transmission within and between connectivity classes. Both models were designed to support the development and/or evaluation of biodefense scenarios, requirements and investment alternatives. It’s also worth emphasizing that biodefense investment decisions must often be made years before the formulation and implementation of new outbreak control strategies.</p><p>Coherent analyses of historical outbreak dynamics at multiple levels of resolution can broaden our knowledge of both disease transmission and the effectiveness of outbreak control measures. Perhaps more importantly, in establishing biodefense requirements and making biodefense investment decisions, retrospective questions (‘what would have happened if…’) deserve as much attention as questions about possible future events (‘what will happen if…’). Historical outbreaks are obviously tied to particular settings and epidemiological circumstances that may or may not appear again in future outbreaks. Nevertheless, sound analyses and interpretations of historical outbreak dynamics could play a stronger supporting role in the biodefense planning process.</p><p>The epidemic modeling framework in this paper is somewhat flexible and adaptations of the SARS models for other diseases and/or other control measures (including prophylaxis and chemotherapy) are feasible. In reconstructing a primary pneumonic plague outbreak, for instance, log normal PDFs should be good choices for characterizing the progression of primary pneumonic plague infections and the insertion of these PDFs in the above models is an easy first step. On the other hand, determining a suitable scale-free contact network for primary pneumonic plague and getting enough data to evaluate the key parameters (<xref rid="tbl1" ref-type="table">Table 1</xref>, <xref rid="tbl2" ref-type="table">Table 2</xref>) could be more challenging tasks. It is interesting to note that the projection of respiratory droplets within a radius of 2 m or so is the main mechanism for transmitting both SARS and primary pneumonic plague from person to person. If the scale-free network in this paper is a reasonable vehicle for analyzing SARS transmission, then this same network might also be helpful in understanding primary pneumonic plague transmission.</p><p>The Corporate Research Program at the Institute for Defense Analyses funded this work and the author thanks Dr. Victor Utgoff (Institute for Defense Analyses) for his interest and encouragement. 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Biol.</source><volume>232</volume><issue>1</issue><year>2005</year></element-citation></ref></ref-list><fn-group><fn id="fn1"><label>1</label><p>Typical SEIR (susceptible-exposed-infectious-removed) models allocate the fixed total population among compartments in accordance with a constant disease transmission rate and mean incubation and contagious periods. In these models, ‘exposed’ individuals are infected and non-contagious, ‘infectious’ people shed the pathogen and can infect others, while those in the ‘removed’ category are immune and no longer sources of infection. Removed individuals can be alive or dead and, if alive, they may still need medical care.</p></fn><fn id="fn2"><label>2</label><p>The onset of illness in the index or initial case occurred on February 24, 2003. In this paper, the initial infection is assumed to have occurred on February 20, 2003 (day number 0).</p></fn><fn id="fn3"><label>3</label><p>Hospital isolation upon symptom onset is most likely to happen in conjunction with contact tracing activities (identifying, localizing and monitoring people who have been in contact with infected individuals). These activities are not modeled here, but an attempt is made to emulate their general dynamical impacts.</p></fn><fn id="fn4"><label>4</label><p>No links between new network nodes and the primordial node are allowed in this model.</p></fn><fn id="fn5"><label>5</label><p>A maximum node degree of 1482 implies that 1482 sets of IDEs must be solved numerically. Several hours are required for a 3-GHz Pentium PC and the Mathematica software to obtain numerical solutions for one set of parameter values; consequently, a parallel processing capability is desirable for a population much larger than 1 million.</p></fn><fn id="fn6"><label>6</label><p>Explicit algorithms for contact tracing activities in a network could improve our understanding of <italic>X</italic><sub>S,<italic>j</italic></sub>, but such algorithms would undoubtedly increase (perhaps substantially) the complexity of our network-based model.</p></fn><fn id="fn7"><label>7</label><p>The PDF for a distributed seed is <italic>P</italic>(<italic>j</italic>). See Eq. <xref rid="fd46" ref-type="disp-formula">(23)</xref>.</p></fn><fn id="fn8"><label>8</label><p>For our scale-free network with 1 000 000 nodes, sizes of 5 connectivity classes appear within parentheses in the following list: <italic>j</italic> = 3 (66,666.7), 10 (3,030.3), 30 (134.41), 100 (3.88) and 300 (0.15). Obviously, the vast majority of connectivity classes are too small to accommodate even a single initial or follow-on infection.</p></fn></fn-group></back></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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