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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Modeling Heterogeneity in Direct Infectious Disease Transmission in a Compartmental Model</title><author><name sortKey="Kong, Lingcai" sort="Kong, Lingcai" uniqKey="Kong L" first="Lingcai" last="Kong">Lingcai Kong</name><affiliation><nlm:aff id="af1-ijerph-13-00253">State Key Laboratory of Resources and Environmental Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China;<email>konglc@lreis.ac.cn</email></nlm:aff></affiliation><affiliation><nlm:aff id="af2-ijerph-13-00253">University of the Chinese Academy of Sciences, Beijing 100049, China</nlm:aff></affiliation><affiliation><nlm:aff id="af3-ijerph-13-00253">Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206, China</nlm:aff></affiliation></author><author><name sortKey="Wang, Jinfeng" sort="Wang, Jinfeng" uniqKey="Wang J" first="Jinfeng" last="Wang">Jinfeng Wang</name><affiliation><nlm:aff id="af1-ijerph-13-00253">State Key Laboratory of Resources and Environmental Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China;<email>konglc@lreis.ac.cn</email></nlm:aff></affiliation><affiliation><nlm:aff id="af3-ijerph-13-00253">Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206, China</nlm:aff></affiliation><affiliation><nlm:aff id="af4-ijerph-13-00253">Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023, China</nlm:aff></affiliation></author><author><name sortKey="Han, Weiguo" sort="Han, Weiguo" uniqKey="Han W" first="Weiguo" last="Han">Weiguo Han</name><affiliation><nlm:aff id="af5-ijerph-13-00253">University Corporation for Atmospheric Research/Visiting Scientist Programs, 3090 Center Green Drive, Boulder, CO 80301, USA;<email>hanwg.bj@gmail.com</email></nlm:aff></affiliation></author><author><name sortKey="Cao, Zhidong" sort="Cao, Zhidong" uniqKey="Cao Z" first="Zhidong" last="Cao">Zhidong Cao</name><affiliation><nlm:aff id="af6-ijerph-13-00253">State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Science, Beijing 100190, China;<email>zhidong.cao@ia.ac.cn</email></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">26927140</idno><idno type="pmc">4808916</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4808916</idno><idno type="RBID">PMC:4808916</idno><idno type="doi">10.3390/ijerph13030253</idno><date when="2016">2016</date><idno type="wicri:Area/Pmc/Corpus">000023</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000023</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Modeling Heterogeneity in Direct Infectious Disease Transmission in a Compartmental Model</title><author><name sortKey="Kong, Lingcai" sort="Kong, Lingcai" uniqKey="Kong L" first="Lingcai" last="Kong">Lingcai Kong</name><affiliation><nlm:aff id="af1-ijerph-13-00253">State Key Laboratory of Resources and Environmental Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China;<email>konglc@lreis.ac.cn</email></nlm:aff></affiliation><affiliation><nlm:aff id="af2-ijerph-13-00253">University of the Chinese Academy of Sciences, Beijing 100049, China</nlm:aff></affiliation><affiliation><nlm:aff id="af3-ijerph-13-00253">Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206, China</nlm:aff></affiliation></author><author><name sortKey="Wang, Jinfeng" sort="Wang, Jinfeng" uniqKey="Wang J" first="Jinfeng" last="Wang">Jinfeng Wang</name><affiliation><nlm:aff id="af1-ijerph-13-00253">State Key Laboratory of Resources and Environmental Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China;<email>konglc@lreis.ac.cn</email></nlm:aff></affiliation><affiliation><nlm:aff id="af3-ijerph-13-00253">Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206, China</nlm:aff></affiliation><affiliation><nlm:aff id="af4-ijerph-13-00253">Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023, China</nlm:aff></affiliation></author><author><name sortKey="Han, Weiguo" sort="Han, Weiguo" uniqKey="Han W" first="Weiguo" last="Han">Weiguo Han</name><affiliation><nlm:aff id="af5-ijerph-13-00253">University Corporation for Atmospheric Research/Visiting Scientist Programs, 3090 Center Green Drive, Boulder, CO 80301, USA;<email>hanwg.bj@gmail.com</email></nlm:aff></affiliation></author><author><name sortKey="Cao, Zhidong" sort="Cao, Zhidong" uniqKey="Cao Z" first="Zhidong" last="Cao">Zhidong Cao</name><affiliation><nlm:aff id="af6-ijerph-13-00253">State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Science, Beijing 100190, China;<email>zhidong.cao@ia.ac.cn</email></nlm:aff></affiliation></author></analytic><series><title level="j">International Journal of Environmental Research and Public Health</title><idno type="ISSN">1661-7827</idno><idno type="eISSN">1660-4601</idno><imprint><date when="2016">2016</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>Mathematical models have been used to understand the transmission dynamics of infectious diseases and to assess the impact of intervention strategies. Traditional mathematical models usually assume a homogeneous mixing in the population, which is rarely the case in reality. Here, we construct a new transmission function by using as the probability density function a negative binomial distribution, and we develop a compartmental model using it to model the heterogeneity of contact rates in the population. We explore the transmission dynamics of the developed model using numerical simulations with different parameter settings, which characterize different levels of heterogeneity. The results show that when the reproductive number, <inline-formula><mml:math id="mm1"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>, is larger than one, a low level of heterogeneity results in dynamics similar to those predicted by the homogeneous mixing model. As the level of heterogeneity increases, the dynamics become more different. As a test case, we calibrated the model with the case incidence data for severe acute respiratory syndrome (SARS) in Beijing in 2003, and the estimated parameters demonstrated the effectiveness of the control measures taken during that period.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Anderson, R" uniqKey="Anderson R">R. Anderson</name></author><author><name sortKey="May, R" uniqKey="May R">R. May</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hethcote, H" uniqKey="Hethcote H">H. Hethcote</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Grassly, N C" uniqKey="Grassly N">N.C. Grassly</name></author><author><name sortKey="Fraser, C" uniqKey="Fraser C">C. Fraser</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Keeling, M J" uniqKey="Keeling M">M.J. Keeling</name></author><author><name sortKey="Rohani, P" uniqKey="Rohani P">P. Rohani</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Rodriguez, D" uniqKey="Rodriguez D">D. Rodriguez</name></author><author><name sortKey="Torres Sorando, L" uniqKey="Torres Sorando L">L. Torres-Sorando</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Bansal, S" uniqKey="Bansal S">S. Bansal</name></author><author><name sortKey="Grenfell, B T" uniqKey="Grenfell B">B.T. Grenfell</name></author><author><name sortKey="Meyers, L A" uniqKey="Meyers L">L.A. Meyers</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Liu, W M" uniqKey="Liu W">W.M. Liu</name></author><author><name sortKey="Hethcote, H W" uniqKey="Hethcote H">H.W. Hethcote</name></author><author><name sortKey="Levin, S A" uniqKey="Levin S">S.A. Levin</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hochberg, M" uniqKey="Hochberg M">M. Hochberg</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Stroud, P D" uniqKey="Stroud P">P.D. Stroud</name></author><author><name sortKey="Sydoriak, S J" uniqKey="Sydoriak S">S.J. Sydoriak</name></author><author><name sortKey="Riese, J M" uniqKey="Riese J">J.M. Riese</name></author><author><name sortKey="Smith, J P" uniqKey="Smith J">J.P. Smith</name></author><author><name sortKey="Mniszewski, S M" uniqKey="Mniszewski S">S.M. Mniszewski</name></author><author><name sortKey="Romero, P R" uniqKey="Romero P">P.R. Romero</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="May, R" uniqKey="May R">R. May</name></author><author><name sortKey="Anderson, R" uniqKey="Anderson R">R. Anderson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Babad, H R" uniqKey="Babad H">H.R. Babad</name></author><author><name sortKey="Nokes, D J" uniqKey="Nokes D">D.J. Nokes</name></author><author><name sortKey="Gay, N J" uniqKey="Gay N">N.J. Gay</name></author><author><name sortKey="Miller, E" uniqKey="Miller E">E. Miller</name></author><author><name sortKey="Morgancapner, P" uniqKey="Morgancapner P">P. Morgancapner</name></author><author><name sortKey="Anderson, R M" uniqKey="Anderson R">R.M. Anderson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Schenzle, D" uniqKey="Schenzle D">D. Schenzle</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Keeling, M J" uniqKey="Keeling M">M.J. Keeling</name></author><author><name sortKey="Eames, K T" uniqKey="Eames K">K.T. Eames</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Danon, L" uniqKey="Danon L">L. Danon</name></author><author><name sortKey="Ford, A P" uniqKey="Ford A">A.P. Ford</name></author><author><name sortKey="House, T" uniqKey="House T">T. House</name></author><author><name sortKey="Jewell, C P" uniqKey="Jewell C">C.P. Jewell</name></author><author><name sortKey="Keeling, M J" uniqKey="Keeling M">M.J. Keeling</name></author><author><name sortKey="Roberts, G O" uniqKey="Roberts G">G.O. Roberts</name></author><author><name sortKey="Ross, J V" uniqKey="Ross J">J.V. Ross</name></author><author><name sortKey="Vernon, M C" uniqKey="Vernon M">M.C. Vernon</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Roche, B" uniqKey="Roche B">B. Roche</name></author><author><name sortKey="Drake, J M" uniqKey="Drake J">J.M. Drake</name></author><author><name sortKey="Rohani, P" uniqKey="Rohani P">P. Rohani</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Duan, W" uniqKey="Duan W">W. Duan</name></author><author><name sortKey="Cao, Z" uniqKey="Cao Z">Z. Cao</name></author><author><name sortKey="Ge, Y" uniqKey="Ge Y">Y. Ge</name></author><author><name sortKey="Qiu, X" uniqKey="Qiu X">X. Qiu</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Dunham, J B" uniqKey="Dunham J">J.B. Dunham</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Keeling, M" uniqKey="Keeling M">M. Keeling</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Roy, M" uniqKey="Roy M">M. Roy</name></author><author><name sortKey="Pascual, M" uniqKey="Pascual M">M. Pascual</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Aparicio, J P" uniqKey="Aparicio J">J.P. Aparicio</name></author><author><name sortKey="Pascual, M" uniqKey="Pascual M">M. Pascual</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Abbey, H" uniqKey="Abbey H">H. Abbey</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Vynnycky, E" uniqKey="Vynnycky E">E. Vynnycky</name></author><author><name sortKey="White, R" uniqKey="White R">R. White</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Bury, K" uniqKey="Bury K">K. Bury</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lloyd Smith, J" uniqKey="Lloyd Smith J">J. Lloyd-Smith</name></author><author><name sortKey="Schreiber, S" uniqKey="Schreiber S">S. Schreiber</name></author><author><name sortKey="Kopp, P" uniqKey="Kopp P">P. Kopp</name></author><author><name sortKey="Getz, W" uniqKey="Getz W">W. Getz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="May, R M" uniqKey="May R">R.M. May</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Godfray, H C J" uniqKey="Godfray H">H.C.J. Godfray</name></author><author><name sortKey="Hassell, M P" uniqKey="Hassell M">M.P. Hassell</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Briggs, C J" uniqKey="Briggs C">C.J. Briggs</name></author><author><name sortKey="Godfray, H C J" uniqKey="Godfray H">H.C.J. Godfray</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Barlow, N" uniqKey="Barlow N">N. Barlow</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hoch, T" uniqKey="Hoch T">T. Hoch</name></author><author><name sortKey="Fourichon, C" uniqKey="Fourichon C">C. Fourichon</name></author><author><name sortKey="Viet, A F" uniqKey="Viet A">A.F. Viet</name></author><author><name sortKey="Seegers, H" uniqKey="Seegers H">H. Seegers</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Allen, L J" uniqKey="Allen L">L.J. Allen</name></author><author><name sortKey="Lewis, T" uniqKey="Lewis T">T. Lewis</name></author><author><name sortKey="Martin, C F" uniqKey="Martin C">C.F. Martin</name></author><author><name sortKey="Jones, M A" uniqKey="Jones M">M.A. Jones</name></author><author><name sortKey="Lo, C K" uniqKey="Lo C">C.K. Lo</name></author><author><name sortKey="Stamp, M" uniqKey="Stamp M">M. Stamp</name></author><author><name sortKey="Mundel, G" uniqKey="Mundel G">G. Mundel</name></author><author><name sortKey="Way, A B" uniqKey="Way A">A.B. Way</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Al Showaikh, F" uniqKey="Al Showaikh F">F. Al-Showaikh</name></author><author><name sortKey="Twizell, E" uniqKey="Twizell E">E. Twizell</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Chen, S C" uniqKey="Chen S">S.C. Chen</name></author><author><name sortKey="Chang, C F" uniqKey="Chang C">C.F. Chang</name></author><author><name sortKey="Jou, L J" uniqKey="Jou L">L.J. Jou</name></author><author><name sortKey="Liao, C M" uniqKey="Liao C">C.M. Liao</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Gao, L" uniqKey="Gao L">L. Gao</name></author><author><name sortKey="Hethcote, H" uniqKey="Hethcote H">H. Hethcote</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Buonomo, B" uniqKey="Buonomo B">B. Buonomo</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Grais, R" uniqKey="Grais R">R. Grais</name></author><author><name sortKey="Ellis, J" uniqKey="Ellis J">J. Ellis</name></author><author><name sortKey="Glass, G" uniqKey="Glass G">G. Glass</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Chen, S C" uniqKey="Chen S">S.C. Chen</name></author><author><name sortKey="Liao, C M" uniqKey="Liao C">C.M. Liao</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lipsitch, M" uniqKey="Lipsitch M">M. Lipsitch</name></author><author><name sortKey="Cohen, T" uniqKey="Cohen T">T. Cohen</name></author><author><name sortKey="Cooper, B" uniqKey="Cooper B">B. Cooper</name></author><author><name sortKey="Robins, J" uniqKey="Robins J">J. Robins</name></author><author><name sortKey="Ma, S" uniqKey="Ma S">S. Ma</name></author><author><name sortKey="James, L" uniqKey="James L">L. James</name></author><author><name sortKey="Gopalakrishna, G" uniqKey="Gopalakrishna G">G. Gopalakrishna</name></author><author><name sortKey="Chew, S" uniqKey="Chew S">S. Chew</name></author><author><name sortKey="Tan, C" uniqKey="Tan C">C. Tan</name></author><author><name sortKey="Samore, M" uniqKey="Samore M">M. Samore</name></author><author><name sortKey="Fisman, D" uniqKey="Fisman D">D. Fisman</name></author><author><name sortKey="Murray, M" uniqKey="Murray M">M. Murray</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Wang, J" uniqKey="Wang J">J. Wang</name></author><author><name sortKey="Mcmichael, A J" uniqKey="Mcmichael A">A.J. McMichael</name></author><author><name sortKey="Meng, B" uniqKey="Meng B">B. Meng</name></author><author><name sortKey="Becker, N G" uniqKey="Becker N">N.G. Becker</name></author><author><name sortKey="Han, W" uniqKey="Han W">W. Han</name></author><author><name sortKey="Glass, K" uniqKey="Glass K">K. Glass</name></author><author><name sortKey="Wu, J" uniqKey="Wu J">J. Wu</name></author><author><name sortKey="Liu, X" uniqKey="Liu X">X. Liu</name></author><author><name sortKey="Liu, J" uniqKey="Liu J">J. Liu</name></author><author><name sortKey="Li, X" uniqKey="Li X">X. Li</name></author><author><name sortKey="Zheng, X" uniqKey="Zheng X">X. Zheng</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Van Den Driessche, P" uniqKey="Van Den Driessche P">P. Van den Driessche</name></author><author><name sortKey="Watmough, J" uniqKey="Watmough J">J. Watmough</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hagenaars, T" uniqKey="Hagenaars T">T. Hagenaars</name></author><author><name sortKey="Donnelly, C" uniqKey="Donnelly C">C. Donnelly</name></author><author><name sortKey="Ferguson, N" uniqKey="Ferguson N">N. Ferguson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Liang, W" uniqKey="Liang W">W. Liang</name></author><author><name sortKey="Zhu, Z" uniqKey="Zhu Z">Z. Zhu</name></author><author><name sortKey="Guo, J" uniqKey="Guo J">J. Guo</name></author><author><name sortKey="Liu, Z" uniqKey="Liu Z">Z. Liu</name></author><author><name sortKey="He, X" uniqKey="He X">X. He</name></author><author><name sortKey="Zhou, W" uniqKey="Zhou W">W. Zhou</name></author><author><name sortKey="Chin, D P" uniqKey="Chin D">D.P. Chin</name></author><author><name sortKey="Schuchat, A" uniqKey="Schuchat A">A. Schuchat</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Pang, X" uniqKey="Pang X">X. Pang</name></author><author><name sortKey="Zhu, Z" uniqKey="Zhu Z">Z. Zhu</name></author><author><name sortKey="Xu, F" uniqKey="Xu F">F. Xu</name></author><author><name sortKey="Guo, J" uniqKey="Guo J">J. Guo</name></author><author><name sortKey="Gong, X" uniqKey="Gong X">X. Gong</name></author><author><name sortKey="Liu, D" uniqKey="Liu D">D. Liu</name></author><author><name sortKey="Liu, Z" uniqKey="Liu Z">Z. Liu</name></author><author><name sortKey="Chin, D" uniqKey="Chin D">D. Chin</name></author><author><name sortKey="Feikin, D" uniqKey="Feikin D">D. Feikin</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Puska, S M" uniqKey="Puska S">S.M. Puska</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Ugray, Z" uniqKey="Ugray Z">Z. Ugray</name></author><author><name sortKey="Lasdon, L" uniqKey="Lasdon L">L. Lasdon</name></author><author><name sortKey="Plummer, J" uniqKey="Plummer J">J. Plummer</name></author><author><name sortKey="Glover, F" uniqKey="Glover F">F. Glover</name></author><author><name sortKey="Kelly, J" uniqKey="Kelly J">J. Kelly</name></author><author><name sortKey="Marti, R" uniqKey="Marti R">R. Marti</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Barthelemy, M" uniqKey="Barthelemy M">M. Barthelemy</name></author><author><name sortKey="Barrat, A" uniqKey="Barrat A">A. Barrat</name></author><author><name sortKey="Pastor Satorras, R" uniqKey="Pastor Satorras R">R. Pastor-Satorras</name></author><author><name sortKey="Vespignani, A" uniqKey="Vespignani A">A. Vespignani</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Volz, E" uniqKey="Volz E">E. Volz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Miller, J C" uniqKey="Miller J">J.C. Miller</name></author><author><name sortKey="Slim, A C" uniqKey="Slim A">A.C. Slim</name></author><author><name sortKey="Volz, E M" uniqKey="Volz E">E.M. Volz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Miller, J C" uniqKey="Miller J">J.C. Miller</name></author><author><name sortKey="Volz, E M" uniqKey="Volz E">E.M. Volz</name></author></analytic></biblStruct></listBibl></div1></back></TEI><pmc article-type="research-article"><pmc-dir>properties open_access</pmc-dir>
<front><journal-meta><journal-id journal-id-type="nlm-ta">Int J Environ Res Public Health</journal-id><journal-id journal-id-type="iso-abbrev">Int J Environ Res Public Health</journal-id><journal-id journal-id-type="publisher-id">ijerph</journal-id><journal-title-group><journal-title>International Journal of Environmental Research and Public Health</journal-title></journal-title-group><issn pub-type="ppub">1661-7827</issn><issn pub-type="epub">1660-4601</issn><publisher><publisher-name>MDPI</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">26927140</article-id><article-id pub-id-type="pmc">4808916</article-id><article-id pub-id-type="doi">10.3390/ijerph13030253</article-id><article-id pub-id-type="publisher-id">ijerph-13-00253</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>Modeling Heterogeneity in Direct Infectious Disease Transmission in a Compartmental Model</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Kong</surname><given-names>Lingcai</given-names></name><xref ref-type="aff" rid="af1-ijerph-13-00253">1</xref><xref ref-type="aff" rid="af2-ijerph-13-00253">2</xref><xref ref-type="aff" rid="af3-ijerph-13-00253">3</xref></contrib><contrib contrib-type="author"><name><surname>Wang</surname><given-names>Jinfeng</given-names></name><xref ref-type="aff" rid="af1-ijerph-13-00253">1</xref><xref ref-type="aff" rid="af3-ijerph-13-00253">3</xref><xref ref-type="aff" rid="af4-ijerph-13-00253">4</xref><xref rid="c1-ijerph-13-00253" ref-type="corresp">*</xref></contrib><contrib contrib-type="author"><name><surname>Han</surname><given-names>Weiguo</given-names></name><xref ref-type="aff" rid="af5-ijerph-13-00253">5</xref></contrib><contrib contrib-type="author"><name><surname>Cao</surname><given-names>Zhidong</given-names></name><xref ref-type="aff" rid="af6-ijerph-13-00253">6</xref></contrib></contrib-group><contrib-group><contrib contrib-type="editor"><name><surname>Burstyn</surname><given-names>Igor</given-names></name><role>Academic Editor</role></contrib><contrib contrib-type="editor"><name><surname>Luta</surname><given-names>Gheorghe</given-names></name><role>Academic Editor</role></contrib></contrib-group><aff id="af1-ijerph-13-00253"><label>1</label>State Key Laboratory of Resources and Environmental Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China;<email>konglc@lreis.ac.cn</email></aff><aff id="af2-ijerph-13-00253"><label>2</label>University of the Chinese Academy of Sciences, Beijing 100049, China</aff><aff id="af3-ijerph-13-00253"><label>3</label>Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206, China</aff><aff id="af4-ijerph-13-00253"><label>4</label>Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023, China</aff><aff id="af5-ijerph-13-00253"><label>5</label>University Corporation for Atmospheric Research/Visiting Scientist Programs, 3090 Center Green Drive, Boulder, CO 80301, USA;<email>hanwg.bj@gmail.com</email></aff><aff id="af6-ijerph-13-00253"><label>6</label>State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Science, Beijing 100190, China;<email>zhidong.cao@ia.ac.cn</email></aff><author-notes><corresp id="c1-ijerph-13-00253"><label>*</label>Correspondence: <email>wangjf@lreis.ac.cn</email>; Tel.: +86-010-64888965</corresp></author-notes><pub-date pub-type="epub"><day>24</day><month>2</month><year>2016</year></pub-date><pub-date pub-type="ppub"><month>3</month><year>2016</year></pub-date><volume>13</volume><issue>3</issue><elocation-id>253</elocation-id><history><date date-type="received"><day>07</day><month>6</month><year>2015</year></date><date date-type="accepted"><day>16</day><month>2</month><year>2016</year></date></history><permissions><copyright-statement>© 2016 by the authors; licensee MDPI, Basel, Switzerland.</copyright-statement><copyright-year>2016</copyright-year><license><license-p><pmc-comment>CREATIVE COMMONS</pmc-comment>This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (<ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">http://creativecommons.org/licenses/by/4.0/</ext-link>).</license-p></license></permissions><abstract><p>Mathematical models have been used to understand the transmission dynamics of infectious diseases and to assess the impact of intervention strategies. Traditional mathematical models usually assume a homogeneous mixing in the population, which is rarely the case in reality. Here, we construct a new transmission function by using as the probability density function a negative binomial distribution, and we develop a compartmental model using it to model the heterogeneity of contact rates in the population. We explore the transmission dynamics of the developed model using numerical simulations with different parameter settings, which characterize different levels of heterogeneity. The results show that when the reproductive number, <inline-formula><mml:math id="mm1"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>, is larger than one, a low level of heterogeneity results in dynamics similar to those predicted by the homogeneous mixing model. As the level of heterogeneity increases, the dynamics become more different. As a test case, we calibrated the model with the case incidence data for severe acute respiratory syndrome (SARS) in Beijing in 2003, and the estimated parameters demonstrated the effectiveness of the control measures taken during that period.</p></abstract><kwd-group><kwd>infectious diseases</kwd><kwd>mathematical models</kwd><kwd>homogeneous mixing</kwd><kwd>heterogeneity</kwd><kwd>negative binomial distribution</kwd></kwd-group></article-meta></front><body><sec id="sec1-ijerph-13-00253"><title>1. Introduction</title><p>Mathematical models play an important role in understanding epidemic spread patterns and designing public health intervention measures [<xref rid="B1-ijerph-13-00253" ref-type="bibr">1</xref>,<xref rid="B2-ijerph-13-00253" ref-type="bibr">2</xref>,<xref rid="B3-ijerph-13-00253" ref-type="bibr">3</xref>,<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>]. The traditional deterministic compartmental models usually assume homogeneous mixing, which means that each individual has the same probability of contact with all of the others in the population [<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>]. However, there is a growing awareness that this assumption is not the case in reality, because heterogeneity can arise due to many sources [<xref rid="B5-ijerph-13-00253" ref-type="bibr">5</xref>], including age, sex, susceptibility to disease, position in space and the activities and behaviors of individuals, among others [<xref rid="B6-ijerph-13-00253" ref-type="bibr">6</xref>]. Here, we will focus on the heterogeneity in host contact rates at the population level.</p><p>In recent years, scientists have developed different approaches to model heterogeneity in host contact rates. First, traditional compartmental models were extended: the infection term of the homogeneous mixing compartmental models was modified [<xref rid="B7-ijerph-13-00253" ref-type="bibr">7</xref>,<xref rid="B8-ijerph-13-00253" ref-type="bibr">8</xref>,<xref rid="B9-ijerph-13-00253" ref-type="bibr">9</xref>]. The compartments were further divided into multiple subgroups with similar behavioral characteristics (e.g., risk [<xref rid="B10-ijerph-13-00253" ref-type="bibr">10</xref>]) or demography (e.g., age [<xref rid="B11-ijerph-13-00253" ref-type="bibr">11</xref>,<xref rid="B12-ijerph-13-00253" ref-type="bibr">12</xref>]). Second, along with the rapid development in research on complex networks, a large body of literature has examined the effects of the heterogeneous contact structure on disease spread in networks [<xref rid="B13-ijerph-13-00253" ref-type="bibr">13</xref>,<xref rid="B14-ijerph-13-00253" ref-type="bibr">14</xref>]. The third type of modeling approach considering heterogeneity is agent-based modeling [<xref rid="B15-ijerph-13-00253" ref-type="bibr">15</xref>,<xref rid="B16-ijerph-13-00253" ref-type="bibr">16</xref>,<xref rid="B17-ijerph-13-00253" ref-type="bibr">17</xref>], which characterizes the heterogeneity in individual attributes and behaviors. Additionally, several researchers have attempted to bridge the gap between traditional compartmental models and individual-based models [<xref rid="B18-ijerph-13-00253" ref-type="bibr">18</xref>,<xref rid="B19-ijerph-13-00253" ref-type="bibr">19</xref>,<xref rid="B20-ijerph-13-00253" ref-type="bibr">20</xref>].</p><p>In this paper, we develop a new compartmental model to incorporate heterogeneous contact rates in disease transmission. First, by combining a Poisson distribution and a Gamma distribution, we derived a negative binomial distribution (NBD) transmission function, with which we developed a compartmental model. Then, we explored the influence of different levels of heterogeneity on the transmission dynamics of infectious diseases using numerical simulations. Finally, we calibrated the model with the number of daily cases of severe acute respiratory syndrome (SARS) in Beijing in 2003, and the estimated parameters show that the control measures taken at that time were effective.</p></sec><sec id="sec2-ijerph-13-00253"><title>2. Methods</title><sec id="sec2dot1-ijerph-13-00253"><title>2.1. NBD Transmission Function</title><p>The heterogeneity in transmission can be modeled by assuming that the number of contacts among individuals varies from person to person. Let <inline-formula><mml:math id="mm2"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> represent the number of effective contacts (the number of contacts that would be sufficient for transmitting the disease successfully, were it to occur between a susceptible individual and an infectious individual [<xref rid="B21-ijerph-13-00253" ref-type="bibr">21</xref>,<xref rid="B22-ijerph-13-00253" ref-type="bibr">22</xref>]) with infectious individuals of the <italic>i</italic>-th susceptible person per unit time. Assume that <inline-formula><mml:math id="mm3"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> has a Poisson distribution <inline-formula><mml:math id="mm4"><mml:mrow><mml:mi>π</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math id="mm5"><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> is the mean of the number of effective contacts that the <italic>i</italic>-th susceptible individual makes with infectious individuals per unit time. That <inline-formula><mml:math id="mm6"><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> are identical means that each individual has an equal chance of effective contact with infectious individuals and an equal chance of being infected, thereby resulting in a traditional homogeneous-mixing model. In reality, however, individuals typically come into contact with only a small, clustered, subpopulation [<xref rid="B20-ijerph-13-00253" ref-type="bibr">20</xref>]. Therefore, it is reasonable to assume that different individuals have different average effective numbers of contacts in a certain period of time; that is, <inline-formula><mml:math id="mm7"><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> is itself a random variable. The Gamma distribution is a good choice for describing <inline-formula><mml:math id="mm8"><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> for a variety of reasons: it is bounded on the left at zero (the numbers of contact must be non-negative), is positively skewed (it has non-zero probability of an extremely high number of contacts) and can represent a variety of distribution shapes [<xref rid="B23-ijerph-13-00253" ref-type="bibr">23</xref>]. It has been used to describe the expected number of secondary cases caused by a particular infected individual [<xref rid="B24-ijerph-13-00253" ref-type="bibr">24</xref>]. Therefore, we assume a Gamma distribution for <inline-formula><mml:math id="mm9"><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>, with shape parameter <italic>k</italic>, rate parameter <italic>m</italic> (or scale parameter <inline-formula><mml:math id="mm10"><mml:mfrac><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mfrac></mml:math></inline-formula>) and the following probability distribution function:
<disp-formula id="FD1-ijerph-13-00253"><label>(1)</label><mml:math id="mm11"><mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:msup><mml:mi>m</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mrow><mml:mo>Γ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mi>θ</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>m</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width="1.em"></mml:mspace><mml:mspace width="1.em"></mml:mspace><mml:mi>θ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></disp-formula></p><p>The conditional distribution of <inline-formula><mml:math id="mm12"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> given <inline-formula><mml:math id="mm13"><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:math></inline-formula> is:<disp-formula><mml:math id="mm14"><mml:mrow><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>θ</mml:mi><mml:mi>x</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width="1.em"></mml:mspace><mml:mspace width="1.em"></mml:mspace><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo></mml:mrow></mml:math></disp-formula></p><p>We obtain the marginal distribution of <inline-formula><mml:math id="mm15"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>:<disp-formula><mml:math id="mm16"><mml:mtable displaystyle="true"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>∞</mml:mi></mml:msubsup><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfenced separators="" open="(" close=")"><mml:mfrac linethickness="0pt"><mml:mrow><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>x</mml:mi></mml:mfrac></mml:mfenced><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mfrac><mml:mi>m</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>x</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mspace width="1.em"></mml:mspace><mml:mspace width="1.em"></mml:mspace><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula></p><p>This is the probability density function of an NBD with mean <inline-formula><mml:math id="mm17"><mml:mfrac><mml:mi>k</mml:mi><mml:mi>m</mml:mi></mml:mfrac></mml:math></inline-formula> and variance <inline-formula><mml:math id="mm18"><mml:mfrac><mml:mrow><mml:mi>k</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:math></inline-formula>. Then, the probability of a susceptible individual escaping from being infected can be represented by the zero term of the NBD:<disp-formula><mml:math id="mm19"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mfrac><mml:mi>m</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></disp-formula></p><p>Let the mean of the NBD be equal to the mean of the number of effective contacts of all susceptible individuals with infectious individuals, that is <inline-formula><mml:math id="mm20"><mml:mrow><mml:mfrac><mml:mi>k</mml:mi><mml:mi>m</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac></mml:mrow></mml:math></inline-formula>, where <italic>β</italic> denotes the transmission rate, defined as the per capita rate at which two specific individuals come into effective contact per unit time [<xref rid="B22-ijerph-13-00253" ref-type="bibr">22</xref>]; <italic>I</italic> denotes the number of infectious individuals; and <italic>N</italic> denotes the size of the total population. It follows that <inline-formula><mml:math id="mm21"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math></inline-formula>, and:<disp-formula><mml:math id="mm22"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></disp-formula></p><p>Consider a closed population (without births, deaths and migration into or out of the population). Let <inline-formula><mml:math id="mm23"><mml:msub><mml:mi>S</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math id="mm24"><mml:msub><mml:mi>I</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></inline-formula> denote the numbers of the susceptible and infectious individuals at time <italic>t</italic>, respectively. Then, the difference equation relating <inline-formula><mml:math id="mm25"><mml:msub><mml:mi>S</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math id="mm26"><mml:msub><mml:mi>I</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></inline-formula> at successive time steps <italic>t</italic> and <inline-formula><mml:math id="mm27"><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> is:<disp-formula><mml:math id="mm28"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:msub><mml:mi>I</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:msub><mml:mi>I</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></disp-formula></p><p>Here, <inline-formula><mml:math id="mm29"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:msub><mml:mi>I</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> is the risk of a susceptible individual becoming infected between time <italic>t</italic> and <inline-formula><mml:math id="mm30"><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>. Using the relationship between the risk and rate derived in [<xref rid="B22-ijerph-13-00253" ref-type="bibr">22</xref>], <inline-formula><mml:math id="mm31"><mml:mrow><mml:mi>r</mml:mi><mml:mi>i</mml:mi><mml:mi>s</mml:mi><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, we obtain the rate at which susceptible individuals become infected at time <italic>t</italic>:<disp-formula><mml:math id="mm32"><mml:mrow><mml:mi>λ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p>Therefore, the rate of change in the number of susceptible individuals can be represented by the differential equation representing:<disp-formula><mml:math id="mm33"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:math></disp-formula></p><p>We call <inline-formula><mml:math id="mm34"><mml:mrow><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> in the right side of this equation the NBD transmission function. A similar function, <inline-formula><mml:math id="mm35"><mml:mrow><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>a</mml:mi><mml:msub><mml:mi>P</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mi>k</mml:mi></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, and its discrete form, <inline-formula><mml:math id="mm36"><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>a</mml:mi><mml:msub><mml:mi>P</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mi>k</mml:mi></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, were first used in host-parasitoid models, where <italic>a</italic> denotes the per capita searching efficiency of the parasitoid and <inline-formula><mml:math id="mm37"><mml:msub><mml:mi>P</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></inline-formula> denotes the number of parasitoids [<xref rid="B25-ijerph-13-00253" ref-type="bibr">25</xref>,<xref rid="B26-ijerph-13-00253" ref-type="bibr">26</xref>]. Then, they were used in insect-pathogen models [<xref rid="B27-ijerph-13-00253" ref-type="bibr">27</xref>]. In [<xref rid="B28-ijerph-13-00253" ref-type="bibr">28</xref>], the author used the transmission function, <inline-formula><mml:math id="mm38"><mml:mrow><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mi>k</mml:mi></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, to model a possum-tuberculosis (TB) system. The influence of different transmission functions on a simulated pathogen spread was studied in [<xref rid="B29-ijerph-13-00253" ref-type="bibr">29</xref>]. Because:<disp-formula id="FD2-ijerph-13-00253"><label>(2)</label><mml:math id="mm39"><mml:mrow><mml:munder><mml:mo movablelimits="true" form="prefix">lim</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:munder><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac></mml:mrow></mml:math></disp-formula><disp-formula id="FD3-ijerph-13-00253"><label>(3)</label><mml:math id="mm40"><mml:mrow><mml:munder><mml:mo movablelimits="true" form="prefix">lim</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:munder><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mi>k</mml:mi></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:math></disp-formula>when <inline-formula><mml:math id="mm41"><mml:mrow><mml:mi>k</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></inline-formula>, the NBD transmission function we derived here approximates the frequency-dependent transmission function of the homogeneous-mixing model. Therefore, it can be regarded as a generalized frequency-dependent transmission function [<xref rid="B1-ijerph-13-00253" ref-type="bibr">1</xref>,<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>]. Similarly, the NBD transmission function used in [<xref rid="B28-ijerph-13-00253" ref-type="bibr">28</xref>] can be regarded as a generalized density-dependent transmission function [<xref rid="B1-ijerph-13-00253" ref-type="bibr">1</xref>,<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>].</p><p>Comparing the NBD transmission function with the density-dependent transmission function, <inline-formula><mml:math id="mm42"><mml:mrow><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:math></inline-formula>, and the frequency-dependent transmission function, <inline-formula><mml:math id="mm43"><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac></mml:math></inline-formula>, of the homogeneous-mixing model [<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>,<xref rid="B22-ijerph-13-00253" ref-type="bibr">22</xref>], we obtain one more parameter, <italic>k</italic>, which is the shape parameter of the Gamma distribution (Equation (<xref ref-type="disp-formula" rid="FD1-ijerph-13-00253">1</xref>)). Denote the mean of the Gamma distribution as <inline-formula><mml:math id="mm44"><mml:msub><mml:mi>μ</mml:mi><mml:mi>θ</mml:mi></mml:msub></mml:math></inline-formula>; then, the variance is <inline-formula><mml:math id="mm45"><mml:mfrac><mml:msubsup><mml:mi>μ</mml:mi><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mi>k</mml:mi></mml:mfrac></mml:math></inline-formula>. Setting the mean to be a constant and letting <inline-formula><mml:math id="mm46"><mml:mrow><mml:mi>k</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></inline-formula>, the variance goes to zero, resulting in homogeneous-mixing, just as shown in Equation (<xref ref-type="disp-formula" rid="FD2-ijerph-13-00253">2</xref>). In contrast, the variance increases as the value of <italic>k</italic> decreases, which indicates greater heterogeneity of the contact rates between the susceptible and infectious populations. Therefore, the parameter <italic>k</italic> characterizes the level of heterogeneity.</p></sec><sec id="sec2dot2-ijerph-13-00253"><title>2.2. NBD Compartmental Model</title><p>The standard susceptible-exposed-infectious-recovered (SEIR) model divides the total population into four compartments: susceptible (S, previously unexposed to the pathogen), exposed (E, infected, but not yet infectious), infected (I, infected and infectious) and recovered (R, recovered from infection and acquired lifelong immunity) [<xref rid="B1-ijerph-13-00253" ref-type="bibr">1</xref>,<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>,<xref rid="B22-ijerph-13-00253" ref-type="bibr">22</xref>]. The infection process is represented in <xref ref-type="fig" rid="ijerph-13-00253-f001">Figure 1</xref>. Children are born susceptible to the disease and enter the compartment S. A susceptible individual in compartment S is infected after effective contact with an infectious individual in compartment I and then enters the exposed compartment E. After the latent period ends, the individual enters the compartment I and becomes capable of transmitting the infection. When the infectious period ends, the individual enters the recovered class R and will never be infected again [<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>,<xref rid="B22-ijerph-13-00253" ref-type="bibr">22</xref>]. In each compartment, individual death occurs at a constant rate, <italic>μ</italic>, which is equal to the birth rate. Death induced by the disease is not considered here. Therefore, the total population size in the model, <italic>N</italic>, remains unchanged. The SEIR model and its extension have been used to model many infectious diseases, for example, measles [<xref rid="B30-ijerph-13-00253" ref-type="bibr">30</xref>,<xref rid="B31-ijerph-13-00253" ref-type="bibr">31</xref>,<xref rid="B32-ijerph-13-00253" ref-type="bibr">32</xref>], rubella [<xref rid="B33-ijerph-13-00253" ref-type="bibr">33</xref>,<xref rid="B34-ijerph-13-00253" ref-type="bibr">34</xref>], influenza [<xref rid="B35-ijerph-13-00253" ref-type="bibr">35</xref>,<xref rid="B36-ijerph-13-00253" ref-type="bibr">36</xref>] and SARS [<xref rid="B37-ijerph-13-00253" ref-type="bibr">37</xref>,<xref rid="B38-ijerph-13-00253" ref-type="bibr">38</xref>], among others.</p><p>Using the NBD transmission function, we set up a new SEIR model in a closed population, represented by a set of ordinary differential equations:<disp-formula id="FD4-ijerph-13-00253"><label>(4)</label><mml:math id="mm47"><mml:mfenced separators="" open="{" close=""><mml:mtable displaystyle="true"><mml:mtr><mml:mtd columnalign="right"><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi>μ</mml:mi><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo><mml:mi>S</mml:mi><mml:mo>-</mml:mo><mml:mi>μ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mi>E</mml:mi><mml:mo>-</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi>γ</mml:mi><mml:mi>I</mml:mi><mml:mo>-</mml:mo><mml:mi>μ</mml:mi><mml:mi>R</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mfenced></mml:math></disp-formula>where the parameter <italic>α</italic> is the rate at which individuals in the exposed category become infectious per unit time, and its reciprocal is the average latent period [<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>,<xref rid="B22-ijerph-13-00253" ref-type="bibr">22</xref>]; the parameter <italic>γ</italic> is the rate at which infectious individuals recover (become immune) per unit time, and its reciprocal is the average infectious period [<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>,<xref rid="B22-ijerph-13-00253" ref-type="bibr">22</xref>]; and the parameter <italic>μ</italic> refers to the birth and death rates.</p><p>Based on the next-generation matrix approach [<xref rid="B39-ijerph-13-00253" ref-type="bibr">39</xref>], we derive the basic reproductive number (see <xref ref-type="app" rid="app1-ijerph-13-00253">Appendix <bold>A</bold></xref> for further details),
<disp-formula id="FD5-ijerph-13-00253"><label>(5)</label><mml:math id="mm48"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula>which is identical to that of the homogeneous-mixing model with a frequency-dependent transmission function [<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>]. It is worth noting that it is irrelevant to <italic>k</italic>, which means that it does not depend on the level of heterogeneity. This can be explained by <inline-formula><mml:math id="mm49"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula> being an average quantity, which means that it does not consider the individual variance in infectiousness [<xref rid="B24-ijerph-13-00253" ref-type="bibr">24</xref>]. This result is in agreement with the conclusion made using a meta population version of the standard stochastic SIR model incorporating spatial heterogeneity [<xref rid="B40-ijerph-13-00253" ref-type="bibr">40</xref>].</p><p>We now determine the equilibrium states. Without much work, we can obtain the disease-free equilibrium <inline-formula><mml:math id="mm50"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>. We also derive the approximate size of the infectious compartment at the endemic equilibrium, <inline-formula><mml:math id="mm51"><mml:mrow><mml:msup><mml:mi>I</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>≈</mml:mo><mml:mfrac><mml:mrow><mml:mi>μ</mml:mi><mml:mi>N</mml:mi></mml:mrow><mml:mi>β</mml:mi></mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> when <inline-formula><mml:math id="mm52"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> (<xref ref-type="app" rid="app2-ijerph-13-00253">Appendix <bold>B</bold></xref>). This is identical to that of the homogeneous-mixing model with a frequency-dependent transmission function [<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>]. Similar to <inline-formula><mml:math id="mm53"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>, it does not depend on <italic>k</italic>. In other words, the contact heterogeneity does not influence the endemic equilibrium, although it does change the dynamics, which we demonstrate using numerical simulations in the next section.</p></sec></sec><sec id="sec3-ijerph-13-00253"><title>3. Results</title><sec id="sec3dot1-ijerph-13-00253"><title>3.1. Dynamics of the NBD Model</title><p>Using numerical simulations, we explore the influence of the heterogeneity level on the transmission dynamics, characterized by the parameter <italic>k</italic>. The results show that the infectious curves with fixed <italic>β</italic>, but different values of <italic>k</italic> achieve a peak after a period that is almost the same in duration (<xref ref-type="fig" rid="ijerph-13-00253-f002">Figure 2</xref>A). However, the transmission speed and, therefore, the peak size, as well as the dynamics after the peak are very different. A low level of heterogeneity results in dynamics similar to those predicted by the homogeneous-mixing model with a frequency-dependent transmission term, <inline-formula><mml:math id="mm54"><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac></mml:math></inline-formula>. This is consistent with the conclusion inferred in Equation (<xref ref-type="disp-formula" rid="FD2-ijerph-13-00253">2</xref>).</p><p>As the value of <italic>k</italic> decreases, that is the level of heterogeneity increases, the dynamics differ increasingly from those predicted by the homogeneous-mixing model. The greatest difference is that at the overall level, the heterogeneity slows the transmission speed and decreases the peak sizes, which means milder disease outbreaks, because in the scenario with a high level of heterogeneity, only a small proportion of susceptible individuals have chances of coming into contact with infectious individuals and becoming infected, which results in a slower increase of the infected population. Second, after the peak is attained, the infectious curves do not decline as rapidly as those predicted by the homogeneous-mixing model and the NBD models (Equation (<xref ref-type="disp-formula" rid="FD4-ijerph-13-00253">4</xref>)) with larger values of <italic>k</italic> (<xref ref-type="fig" rid="ijerph-13-00253-f002">Figure 2</xref>A), and the disease persists over a long term in the population (<xref ref-type="fig" rid="ijerph-13-00253-f002">Figure 2</xref>B). Compared to the homogeneous-mixing model or the NBD models with larger values of <italic>k</italic>, up to the peak time (almost the same), there are many more individuals who are still susceptible to the disease. A proportion of them come into contact with infectious individuals and become infected, and this process persists for a long period of time. Moreover, <xref ref-type="fig" rid="ijerph-13-00253-f002">Figure 2</xref>B shows that the endemic sizes of the two scenarios are approximately equal, just as noted in the previous section. In addition, when <italic>k</italic> drops to a very small value, there will be no disease outbreak, because almost none of the susceptible individuals have any chance of coming into contact with infectious individuals and becoming infected. It is shown that the contact patterns exhibit more heterogeneity than that assumed by homogeneous-mixing models, but they do not appear extremely heterogeneous [<xref rid="B6-ijerph-13-00253" ref-type="bibr">6</xref>].</p><p>We also simulate the dynamics with a fixed value of <italic>k</italic> and different values of <italic>β</italic>. Because the dynamics obtained with a large value of <italic>k</italic> are similar to those of the homogeneous-mixing model with a frequency-dependent transmission term, we only show the results for a relatively small value of <inline-formula><mml:math id="mm55"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> (<xref ref-type="fig" rid="ijerph-13-00253-f003">Figure 3</xref>). For larger values of <italic>β</italic>, the infectious curves reach their peaks earlier, and the peaks are higher than those obtained for smaller values of <italic>β</italic>. After the peak of the disease outbreak is achieved, the infectious curves decrease slowly and reach endemic equilibrium gradually (<xref ref-type="fig" rid="ijerph-13-00253-f003">Figure 3</xref>B). Additionally, for much smaller values of <italic>β</italic>, such that <inline-formula><mml:math id="mm56"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>, there will be no disease outbreak (here, for example, <inline-formula><mml:math id="mm57"><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>).</p></sec><sec id="sec3dot2-ijerph-13-00253"><title>3.2. Fitting the NBD Model to the 2003 Beijing SARS Outbreak Data</title><p>The SARS disease broke out in the beginning of March 2003 in Beijing, spread rapidly over the next six weeks and peaked during the third and fourth weeks of April [<xref rid="B41-ijerph-13-00253" ref-type="bibr">41</xref>]. In total, 2048 confirmed cases were reported during the entire outbreak period (the circle markers shown in <xref ref-type="fig" rid="ijerph-13-00253-f004">Figure 4</xref>; the data were provided by the Chinese Center for Disease Control and Prevention). Prompted by the rapid expansion of the epidemic, on 17 April, the Beijing municipal government established a Joint SARS Leading Group and deployed 10 task forces to oversee crisis management [<xref rid="B41-ijerph-13-00253" ref-type="bibr">41</xref>,<xref rid="B42-ijerph-13-00253" ref-type="bibr">42</xref>]. On 20 April, a larger number of cases was reported, and the Chinese government canceled the May Day holiday in an effort to reduce the mass movement of people [<xref rid="B43-ijerph-13-00253" ref-type="bibr">43</xref>]. Multiple measures were taken to control the spread of the disease, including the provision of personal protective equipment and training for healthcare workers [<xref rid="B41-ijerph-13-00253" ref-type="bibr">41</xref>]; introduction of community-based prevention and control through case detection, isolation, quarantine and community mobilization [<xref rid="B41-ijerph-13-00253" ref-type="bibr">41</xref>]; closure of the sites of public entertainment and schools [<xref rid="B42-ijerph-13-00253" ref-type="bibr">42</xref>]; and stopping the entry of all visitors or screening them for fever upon entry to universities and other places [<xref rid="B42-ijerph-13-00253" ref-type="bibr">42</xref>]. Additionally, a general increase in SARS awareness played an important role in controlling the outbreak [<xref rid="B42-ijerph-13-00253" ref-type="bibr">42</xref>]. The multiple measures implemented in Beijing likely led to the rapid resolution of the SARS outbreak [<xref rid="B42-ijerph-13-00253" ref-type="bibr">42</xref>].</p><p>To evaluate the effectiveness of the control measures taken in Beijing at that time, we calibrated the NBD model to the data of the SARS daily cases using the GlobalSearch algorithm in the MATLAB Global Optimization Toolbox [<xref rid="B44-ijerph-13-00253" ref-type="bibr">44</xref>,<xref rid="B45-ijerph-13-00253" ref-type="bibr">45</xref>] and estimated the parameters. We used two different values, <inline-formula><mml:math id="mm58"><mml:msub><mml:mi>k</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math id="mm59"><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>, to characterize the different levels of heterogeneity in contact in the population before and after 20 April [<xref rid="B38-ijerph-13-00253" ref-type="bibr">38</xref>]. We assumed a fixed value for <italic>β</italic> for simplicity (in reality, the value of <italic>β</italic> decreased along with the control strategies [<xref rid="B38-ijerph-13-00253" ref-type="bibr">38</xref>]; we mainly discuss the influence of the other parameter, <italic>k</italic>). We chose the normalized root mean square error (NRMSE) [<xref rid="B46-ijerph-13-00253" ref-type="bibr">46</xref>] as the goodness of fit between the model output and the daily case data, as well as the objective function of the calibration procedure. In order to compute the NRMSE, we solved the set of differential equations (Equation (<xref ref-type="disp-formula" rid="FD4-ijerph-13-00253">4</xref>)) with unknown parameters <inline-formula><mml:math id="mm60"><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="mm61"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> from 7 March to 20 April. The initial conditions were set as follows: <inline-formula><mml:math id="mm62"><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>.</mml:mo><mml:mn>4564</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>7</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula>, which was the size of the permanent population in Beijing in 2003 [<xref rid="B47-ijerph-13-00253" ref-type="bibr">47</xref>]; <inline-formula><mml:math id="mm63"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> corresponds to 7 March 2003; <inline-formula><mml:math id="mm64"><mml:mrow><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>0</mml:mn></mml:mrow></mml:math></inline-formula>; <inline-formula><mml:math id="mm65"><mml:mrow><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:mn>2</mml:mn></mml:mrow></mml:math></inline-formula>, which was the number of daily cases on 7 March 2003; and <inline-formula><mml:math id="mm66"><mml:mrow><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:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>. Then, the output of the model on 20 April was taken as the initial value to solve Equation (<xref ref-type="disp-formula" rid="FD4-ijerph-13-00253">4</xref>) with parameters <inline-formula><mml:math id="mm67"><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="mm68"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> from 20 April to 4 June. Finally, the two outputs were combined and used to calculate the goodness of fit to the SARS daily case data. The birth and death rate, <italic>μ</italic>, was assumed to be <inline-formula><mml:math id="mm69"><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>70</mml:mn></mml:mrow></mml:math></inline-formula><inline-formula><mml:math id="mm70"><mml:mrow><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>. In total, there were five unknown parameters to be estimated: <inline-formula><mml:math id="mm71"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:math></inline-formula> and <italic>γ</italic>.</p><p>The starting points of the parameters for the calibration procedure were selected randomly between the bounds of the parameters shown in <xref ref-type="table" rid="ijerph-13-00253-t001">Table 1</xref>.</p><p>Because of the stochasticity of the GlobalSearchalgorithm [<xref rid="B44-ijerph-13-00253" ref-type="bibr">44</xref>,<xref rid="B45-ijerph-13-00253" ref-type="bibr">45</xref>], the results varied slightly every time. We ran the procedure 100 times. <xref ref-type="table" rid="ijerph-13-00253-t002">Table 2</xref> presents the minimum, maximum, mean and standard variance of the results. The average latent and infectious periods are <inline-formula><mml:math id="mm72"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>α</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mn>6</mml:mn><mml:mo>.</mml:mo><mml:mn>8661</mml:mn></mml:mrow></mml:math></inline-formula> days and <inline-formula><mml:math id="mm73"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>γ</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mo>.</mml:mo><mml:mn>8439</mml:mn></mml:mrow></mml:math></inline-formula> days, respectively. The much smaller <inline-formula><mml:math id="mm74"><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula> value indicates that the control measures are extremely effective in controlling the SARS transmission in Beijing in 2003. This is in agreement with the result in [<xref rid="B38-ijerph-13-00253" ref-type="bibr">38</xref>]. <xref ref-type="fig" rid="ijerph-13-00253-f004">Figure 4</xref> shows the 100 fitted infectious curves and the daily cases.</p></sec></sec><sec id="sec4-ijerph-13-00253"><title>4. Discussion</title><p>In this paper, we aimed to study the influence of heterogeneity in the contact rates in disease transmission at the population level. The developed NBD model can be regarded as a generalized homogeneous-mixing model with a frequency-dependent transmission function. Our results show that, keeping other conditions identical, the higher is the level of heterogeneity in contact rates, the greater is the difference in the disease dynamics observed from those predicted using the homogeneous-mixing models.</p><p>It is worthwhile to compare our approach and results to previous approaches and results. To address heterogeneous-mixing within populations, the populations were further divided into multiple subgroups [<xref rid="B10-ijerph-13-00253" ref-type="bibr">10</xref>,<xref rid="B11-ijerph-13-00253" ref-type="bibr">11</xref>,<xref rid="B12-ijerph-13-00253" ref-type="bibr">12</xref>], and used the WAIFW matrix (“who acquires infection from whom” [<xref rid="B1-ijerph-13-00253" ref-type="bibr">1</xref>]), in which any individual is more likely to come into contact with other individuals from within the same subgroup than those outside. However, in this framework, contact rates within the subgroups are still homogeneous. A different class of approaches for extending the traditional compartmental models to incorporate heterogeneity involves modifying the transmission term; our approach belongs to this class. The work in [<xref rid="B7-ijerph-13-00253" ref-type="bibr">7</xref>,<xref rid="B8-ijerph-13-00253" ref-type="bibr">8</xref>,<xref rid="B19-ijerph-13-00253" ref-type="bibr">19</xref>] replaced the bilinear transmission term (<inline-formula><mml:math id="mm75"><mml:mrow><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:math></inline-formula>) in the homogeneous compartmental model with a nonlinear term <inline-formula><mml:math id="mm76"><mml:mrow><mml:mi>k</mml:mi><mml:msup><mml:mi>S</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:msup><mml:mi>I</mml:mi><mml:mi>q</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math id="mm77"><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula> are the “heterogeneity parameters”. Their results showed that the modified model was capable of predicting the disease transmission patterns in a clustered network [<xref rid="B19-ijerph-13-00253" ref-type="bibr">19</xref>]. Stroud <italic>et al.</italic> used a power-law scaling of the new infection rate <inline-formula><mml:math id="mm78"><mml:mrow><mml:mi>I</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>S</mml:mi><mml:mo>/</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>v</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>, with scaling power <italic>v</italic> greater than one, to relax the homogeneous-mixing assumption [<xref rid="B9-ijerph-13-00253" ref-type="bibr">9</xref>], and it was demonstrated that this power-law formulation leads to significantly lower predictions of the final epidemic size than the traditional linear formulation. Compared to these empirical or semi-empirical modifications [<xref rid="B7-ijerph-13-00253" ref-type="bibr">7</xref>,<xref rid="B8-ijerph-13-00253" ref-type="bibr">8</xref>,<xref rid="B9-ijerph-13-00253" ref-type="bibr">9</xref>,<xref rid="B19-ijerph-13-00253" ref-type="bibr">19</xref>], the NBD transmission function seems to agree more with the real transmission mechanics, in that it assumes that the mean of the number of effective contacts of the susceptible individuals with infectious individuals per unit time is different from individual to individual, and the choice of the Gamma distribution offers multiple advantages (see <xref ref-type="sec" rid="sec2dot1-ijerph-13-00253">Section 2.1</xref>).</p><p>In recent years, several network-based models have been developed to study the influence of contact heterogeneity on disease transmission. Keeling <italic>et al.</italic> reviewed multiple types of networks and the statistical and analytical approaches for the spread of infectious diseases [<xref rid="B13-ijerph-13-00253" ref-type="bibr">13</xref>,<xref rid="B14-ijerph-13-00253" ref-type="bibr">14</xref>]. In particular, Bansal <italic>et al.</italic> demonstrated that the high-level heterogeneous degree distributions generate an almost immediate expansion phase compared to homogeneous degree distributions, such as the Poisson distribution [<xref rid="B6-ijerph-13-00253" ref-type="bibr">6</xref>,<xref rid="B49-ijerph-13-00253" ref-type="bibr">49</xref>,<xref rid="B50-ijerph-13-00253" ref-type="bibr">50</xref>]. The NBD-SEIR model does not exhibit this feature. We suspect that this is because our approach belongs to the mean-field class of approaches and considers a large population at the overall level. In addition, it is possible to approximate the main features of disease spread in networks with compartmental models using an appropriate construction. The work in [<xref rid="B20-ijerph-13-00253" ref-type="bibr">20</xref>] used <inline-formula><mml:math id="mm79"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula> as a fundamental parameter to formulate a mean-field type model, which can implicitly capture some important effects of heterogeneous-mixing in contact networks. The work in [<xref rid="B51-ijerph-13-00253" ref-type="bibr">51</xref>,<xref rid="B52-ijerph-13-00253" ref-type="bibr">52</xref>] applied “edge-based compartmental modeling” (EBCM), which focuses on the status of a random partner rather than a random individual, to capture the heterogeneous contact rates in disease transmission.</p><p>Although it incorporates the heterogeneous contact rates in disease transmission in a tractable manner, the NBD model has some weaknesses. First, the parameter <italic>k</italic> characterizes the level of heterogeneity, which is difficult to measure directly, and this can be overcome by using contact tracing data. Second, some features cannot be recovered by the NBD model. In future research, it will be interesting to incorporate other factors that influence transmission dynamics, such as the migration of populations, seasonality and vaccinations, among others.</p></sec><sec id="sec5-ijerph-13-00253"><title>5. Conclusions</title><p>Using the probability density function for the negative binomial distribution, we constructed a NBD transmission function and further developed a compartmental model for direct infectious disease. The developed model considers the heterogeneity of contact rates in the population. The simulation results show that, at the population level, the dynamics vary widely according to the level of heterogeneity in contact rates. Once <inline-formula><mml:math id="mm80"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>, a low level of heterogeneity results in dynamics similar to those predicted by the homogeneous mixing models. Keeping other conditions identical, as the level of heterogeneity increases, the transmission speed becomes more and more slowly, the peak size becomes smaller and smaller. These results have implications for developing interventions, such as isolation, targeted vaccination, among others.</p></sec></body><back><ack><title>Acknowledgments</title><p>This study was supported by MOST (Nos. 2012CB955503, 2012ZX10004-201) and NSFC (No. 41271404, 41421001).</p></ack><notes><title>Author Contributions</title><p>The author for correspondence, Jinfeng Wang, designed the whole study, and Lingcai Kong implemented the method and drafted the manuscript. Weiguo Han and Zhidong Cao revised the manuscript critically and made constructive suggestions for the interpretation of the results.</p></notes><notes><title>Conflicts of Interest</title><p>There was no conflict of interest regarding the submission of this manuscript, and it was approved by all authors for publication.</p></notes><app-group><title>Appendix</title><app id="app1-ijerph-13-00253"><title>Appendix A. R 0 Expression for the Model</title><p>Using the next-generation operator approach [<xref rid="B39-ijerph-13-00253" ref-type="bibr">39</xref>], we compute the basic reproductive number <inline-formula><mml:math id="mm91"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>. First, we sort the compartments so that the first <italic>m</italic> compartments correspond to infected individuals: <inline-formula><mml:math id="mm92"><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>I</mml:mi><mml:mo>,</mml:mo><mml:mi>S</mml:mi><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>. Here, the infected compartments are E and I, yielding <inline-formula><mml:math id="mm93"><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></inline-formula>. Then, we decompose the components of the differential equations into <inline-formula><mml:math id="mm94"><mml:mi mathvariant="script">F</mml:mi></mml:math></inline-formula>, in which <inline-formula><mml:math id="mm95"><mml:msub><mml:mi mathvariant="script">F</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> is the rate of appearance of new infections in compartment <italic>i</italic>, and <inline-formula><mml:math id="mm96"><mml:mi mathvariant="script">V</mml:mi></mml:math></inline-formula>, in which <inline-formula><mml:math id="mm97"><mml:msub><mml:mi mathvariant="script">V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> is the rate of transfer of individuals into and out of compartment <italic>i</italic> by all other means:<disp-formula><mml:math id="mm98"><mml:mrow><mml:mi mathvariant="script">F</mml:mi><mml:mo>=</mml:mo><mml:mfenced separators="" open="(" close=")"><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo><mml:mi>S</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mo>,</mml:mo><mml:mspace width="1.em"></mml:mspace><mml:mi mathvariant="script">V</mml:mi><mml:mo>=</mml:mo><mml:mfenced separators="" open="(" close=")"><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>I</mml:mi><mml:mo>-</mml:mo><mml:mi>α</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mi>μ</mml:mi><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo><mml:mi>S</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mi>γ</mml:mi><mml:mi>I</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mi>R</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mfenced></mml:mrow></mml:math></disp-formula></p><p>The disease-free equilibrium (DFE) for this model is <inline-formula><mml:math id="mm99"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>. Then,
<disp-formula><mml:math id="mm100"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:mfenced separators="" open="[" close="]"><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:msub><mml:mi mathvariant="script">F</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced separators="" open="(" close=")"><mml:mtable><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mi>β</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mo>,</mml:mo><mml:mspace width="1.em"></mml:mspace><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mfenced separators="" open="[" close="]"><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:msub><mml:mi mathvariant="script">V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced separators="" open="(" close=")"><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mo>,</mml:mo><mml:mspace width="1.em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:math></disp-formula>giving:<disp-formula><mml:math id="mm101"><mml:mrow><mml:mi>F</mml:mi><mml:msup><mml:mi>V</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mfenced separators="" open="(" close=")"><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:mtd><mml:mtd><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mfenced></mml:mrow></mml:math></disp-formula></p><p>This is called the next-generation matrix for the model [<xref rid="B39-ijerph-13-00253" ref-type="bibr">39</xref>]. Finally, the basic reproductive number, <inline-formula><mml:math id="mm102"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>, is calculated using the spectral ratio:<disp-formula><mml:math id="mm103"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>F</mml:mi><mml:msup><mml:mi>V</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula></p></app><app id="app2-ijerph-13-00253"><title>Appendix B. Endemic Equilibrium</title><p>Because the total population size <italic>N</italic> is a constant and <inline-formula><mml:math id="mm81"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mi>S</mml:mi><mml:mo>-</mml:mo><mml:mi>E</mml:mi><mml:mo>-</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></inline-formula>, the last equation in Equation (<xref ref-type="disp-formula" rid="FD4-ijerph-13-00253">4</xref>) is redundant. To find the endemic equilibrium, we set the right side of the other three equations to zero. Then, <italic>S</italic> and <italic>E</italic> can be represented by <italic>I</italic>:<disp-formula><mml:math id="mm82"><mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>μ</mml:mi><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width="1.em"></mml:mspace><mml:mi>E</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:mfrac></mml:mrow></mml:math></disp-formula></p><p>Substituting them into <inline-formula><mml:math id="mm83"><mml:mrow><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>E</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> and after some algebraic manipulation, we obtain:<disp-formula><mml:math id="mm84"><mml:mrow><mml:mi>α</mml:mi><mml:mi>μ</mml:mi><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></disp-formula></p><p>Obviously, it is difficult and even impossible to find an explicit solution. We find an approximate solution using the first-degree Taylor polynomial of <inline-formula><mml:math id="mm85"><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> near <inline-formula><mml:math id="mm86"><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>, that is <inline-formula><mml:math id="mm87"><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≈</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:math></inline-formula>. It follows that,
<disp-formula><mml:math id="mm88"><mml:mrow><mml:mi>α</mml:mi><mml:mi>μ</mml:mi><mml:mi>β</mml:mi><mml:mi>I</mml:mi><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi><mml:mi>I</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mo>≈</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></disp-formula></p><p>We obtain the approximate solution for I:<disp-formula><mml:math id="mm89"><mml:mrow><mml:msup><mml:mi>I</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>≈</mml:mo><mml:mfrac><mml:mrow><mml:mi>μ</mml:mi><mml:mi>N</mml:mi></mml:mrow><mml:mi>β</mml:mi></mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="mm90"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula> is given in Equation (<xref ref-type="disp-formula" rid="FD5-ijerph-13-00253">5</xref>).</p></app></app-group><ref-list><title>References</title><ref id="B1-ijerph-13-00253"><label>1.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Anderson</surname><given-names>R.</given-names></name><name><surname>May</surname><given-names>R.</given-names></name></person-group><source>Infectious Diseases of Humans: Dynamics and Control</source><publisher-name>Oxford University Press</publisher-name><publisher-loc>New York, NY, USA</publisher-loc><year>1991</year></element-citation></ref><ref id="B2-ijerph-13-00253"><label>2.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hethcote</surname><given-names>H.</given-names></name></person-group><article-title>The mathematics of infectious diseases</article-title><source>SIAM Rev.</source><year>2000</year><volume>42</volume><fpage>599</fpage><lpage>653</lpage><pub-id pub-id-type="doi">10.1137/S0036144500371907</pub-id></element-citation></ref><ref id="B3-ijerph-13-00253"><label>3.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Grassly</surname><given-names>N.C.</given-names></name><name><surname>Fraser</surname><given-names>C.</given-names></name></person-group><article-title>Mathematical models of infectious disease transmission</article-title><source>Nat. Rev. Microbiol.</source><year>2008</year><volume>6</volume><fpage>477</fpage><lpage>487</lpage><pub-id pub-id-type="doi">10.1038/nrmicro1845</pub-id><pub-id pub-id-type="pmid">18533288</pub-id></element-citation></ref><ref id="B4-ijerph-13-00253"><label>4.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Keeling</surname><given-names>M.J.</given-names></name><name><surname>Rohani</surname><given-names>P.</given-names></name></person-group><source>Modeling Infectious Diseases in Humans and Animals</source><publisher-name>Princeton University Press</publisher-name><publisher-loc>Princeton, NJ, USA</publisher-loc><year>2008</year></element-citation></ref><ref id="B5-ijerph-13-00253"><label>5.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Rodriguez</surname><given-names>D.</given-names></name><name><surname>Torres-Sorando</surname><given-names>L.</given-names></name></person-group><article-title>Models of infectious diseases in spatially heterogeneous environments</article-title><source>Bull. Math. Biol.</source><year>2001</year><volume>63</volume><fpage>547</fpage><lpage>571</lpage><pub-id pub-id-type="doi">10.1006/bulm.2001.0231</pub-id><pub-id pub-id-type="pmid">11374305</pub-id></element-citation></ref><ref id="B6-ijerph-13-00253"><label>6.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Bansal</surname><given-names>S.</given-names></name><name><surname>Grenfell</surname><given-names>B.T.</given-names></name><name><surname>Meyers</surname><given-names>L.A.</given-names></name></person-group><article-title>When individual behaviour matters: Homogeneous and network models in epidemiology</article-title><source>J. R. Soc. Interface</source><year>2007</year><volume>4</volume><fpage>879</fpage><lpage>891</lpage><pub-id pub-id-type="doi">10.1098/rsif.2007.1100</pub-id><pub-id pub-id-type="pmid">17640863</pub-id></element-citation></ref><ref id="B7-ijerph-13-00253"><label>7.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Liu</surname><given-names>W.M.</given-names></name><name><surname>Hethcote</surname><given-names>H.W.</given-names></name><name><surname>Levin</surname><given-names>S.A.</given-names></name></person-group><article-title>Dynamical behavior of epidemiological models with nonlinear incidence rates</article-title><source>J. Math. Biol.</source><year>1987</year><volume>25</volume><fpage>359</fpage><lpage>380</lpage><pub-id pub-id-type="doi">10.1007/BF00277162</pub-id><pub-id pub-id-type="pmid">3668394</pub-id></element-citation></ref><ref id="B8-ijerph-13-00253"><label>8.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hochberg</surname><given-names>M.</given-names></name></person-group><article-title>Nonlinear transmission rates and the dynamics of infectious-disease</article-title><source>J. Theor. Biol.</source><year>1991</year><volume>153</volume><fpage>301</fpage><lpage>321</lpage><pub-id pub-id-type="doi">10.1016/S0022-5193(05)80572-7</pub-id><pub-id pub-id-type="pmid">1798335</pub-id></element-citation></ref><ref id="B9-ijerph-13-00253"><label>9.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Stroud</surname><given-names>P.D.</given-names></name><name><surname>Sydoriak</surname><given-names>S.J.</given-names></name><name><surname>Riese</surname><given-names>J.M.</given-names></name><name><surname>Smith</surname><given-names>J.P.</given-names></name><name><surname>Mniszewski</surname><given-names>S.M.</given-names></name><name><surname>Romero</surname><given-names>P.R.</given-names></name></person-group><article-title>Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing</article-title><source>Math. Biosci.</source><year>2006</year><volume>203</volume><fpage>301</fpage><lpage>318</lpage><pub-id pub-id-type="doi">10.1016/j.mbs.2006.01.007</pub-id><pub-id pub-id-type="pmid">16540129</pub-id></element-citation></ref><ref id="B10-ijerph-13-00253"><label>10.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>May</surname><given-names>R.</given-names></name><name><surname>Anderson</surname><given-names>R.</given-names></name></person-group><article-title>The transmission dynamics of human immunodeficiency virus (HIV)</article-title><source>Philos. Trans. R. Soc. B Bio. Sci.</source><year>1988</year><volume>321</volume><fpage>565</fpage><lpage>607</lpage><pub-id pub-id-type="doi">10.1098/rstb.1988.0108</pub-id></element-citation></ref><ref id="B11-ijerph-13-00253"><label>11.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Babad</surname><given-names>H.R.</given-names></name><name><surname>Nokes</surname><given-names>D.J.</given-names></name><name><surname>Gay</surname><given-names>N.J.</given-names></name><name><surname>Miller</surname><given-names>E.</given-names></name><name><surname>Morgancapner</surname><given-names>P.</given-names></name><name><surname>Anderson</surname><given-names>R.M.</given-names></name></person-group><article-title>Predicting the impact of measles vaccination in England and Wales: Model validation and analysis of policy options</article-title><source>Epidemiol. Infect.</source><year>1995</year><volume>114</volume><fpage>319</fpage><lpage>344</lpage><pub-id pub-id-type="doi">10.1017/S0950268800057976</pub-id><pub-id pub-id-type="pmid">7705494</pub-id></element-citation></ref><ref id="B12-ijerph-13-00253"><label>12.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Schenzle</surname><given-names>D.</given-names></name></person-group><article-title>An age-structured model of pre- and post-vaccination measles transmission</article-title><source>IMA J. Math. Appl. Med. Biol.</source><year>1984</year><volume>1</volume><fpage>169</fpage><lpage>191</lpage><pub-id pub-id-type="doi">10.1093/imammb/1.2.169</pub-id><pub-id pub-id-type="pmid">6600102</pub-id></element-citation></ref><ref id="B13-ijerph-13-00253"><label>13.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Keeling</surname><given-names>M.J.</given-names></name><name><surname>Eames</surname><given-names>K.T.</given-names></name></person-group><article-title>Networks and epidemic models</article-title><source>J. R. Soc. Interface</source><year>2005</year><volume>2</volume><fpage>295</fpage><lpage>307</lpage><pub-id pub-id-type="doi">10.1098/rsif.2005.0051</pub-id><pub-id pub-id-type="pmid">16849187</pub-id></element-citation></ref><ref id="B14-ijerph-13-00253"><label>14.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Danon</surname><given-names>L.</given-names></name><name><surname>Ford</surname><given-names>A.P.</given-names></name><name><surname>House</surname><given-names>T.</given-names></name><name><surname>Jewell</surname><given-names>C.P.</given-names></name><name><surname>Keeling</surname><given-names>M.J.</given-names></name><name><surname>Roberts</surname><given-names>G.O.</given-names></name><name><surname>Ross</surname><given-names>J.V.</given-names></name><name><surname>Vernon</surname><given-names>M.C.</given-names></name></person-group><article-title>Networks and the epidemiology of infectious disease</article-title><source>Interdiscip. Perspect. Infect. Dis.</source><year>2011</year><volume>2011</volume><pub-id pub-id-type="doi">10.1155/2011/284909</pub-id><pub-id pub-id-type="pmid">21437001</pub-id></element-citation></ref><ref id="B15-ijerph-13-00253"><label>15.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Roche</surname><given-names>B.</given-names></name><name><surname>Drake</surname><given-names>J.M.</given-names></name><name><surname>Rohani</surname><given-names>P.</given-names></name></person-group><article-title>An agent-based model to study the epidemiological and evolutionary dynamics of influenza viruses</article-title><source>BMC Bioinform.</source><year>2011</year><volume>12</volume><pub-id pub-id-type="doi">10.1186/1471-2105-12-87</pub-id><pub-id pub-id-type="pmid">21450071</pub-id></element-citation></ref><ref id="B16-ijerph-13-00253"><label>16.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Duan</surname><given-names>W.</given-names></name><name><surname>Cao</surname><given-names>Z.</given-names></name><name><surname>Ge</surname><given-names>Y.</given-names></name><name><surname>Qiu</surname><given-names>X.</given-names></name></person-group><article-title>Modeling and simulation for the spread of H1N1 influenza in school using artificial societies</article-title><source>Intelligence and Security Informatics</source><person-group person-group-type="editor"><name><surname>Chau</surname><given-names>M.</given-names></name><name><surname>Wang</surname><given-names>G.</given-names></name><name><surname>Zheng</surname><given-names>X.</given-names></name><name><surname>Chen</surname><given-names>H.</given-names></name><name><surname>Zeng</surname><given-names>D.</given-names></name><name><surname>Mao</surname><given-names>W.</given-names></name></person-group><comment>Lecture Notes in Computer Science</comment><publisher-name>Springer</publisher-name><publisher-loc>Berlin, Germany; Heidelberg, Germany</publisher-loc><year>2011</year><volume>Volume 6749</volume><fpage>121</fpage><lpage>129</lpage></element-citation></ref><ref id="B17-ijerph-13-00253"><label>17.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Dunham</surname><given-names>J.B.</given-names></name></person-group><article-title>An agent-based spatially explicit epidemiological model in MASON</article-title><source>J. Artif. Soc. Soc. Simul.</source><year>2006</year><volume>9</volume><elocation-id>253</elocation-id></element-citation></ref><ref id="B18-ijerph-13-00253"><label>18.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Keeling</surname><given-names>M.</given-names></name></person-group><article-title>The implications of network structure for epidemic dynamics</article-title><source>Theor. Popul. Biol.</source><year>2005</year><volume>67</volume><fpage>1</fpage><lpage>8</lpage><pub-id pub-id-type="doi">10.1016/j.tpb.2004.08.002</pub-id><pub-id pub-id-type="pmid">15649519</pub-id></element-citation></ref><ref id="B19-ijerph-13-00253"><label>19.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Roy</surname><given-names>M.</given-names></name><name><surname>Pascual</surname><given-names>M.</given-names></name></person-group><article-title>On representing network heterogeneities in the incidence rate of simple epidemic models</article-title><source>Ecol. Complex.</source><year>2006</year><volume>3</volume><fpage>80</fpage><lpage>90</lpage><pub-id pub-id-type="doi">10.1016/j.ecocom.2005.09.001</pub-id></element-citation></ref><ref id="B20-ijerph-13-00253"><label>20.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Aparicio</surname><given-names>J.P.</given-names></name><name><surname>Pascual</surname><given-names>M.</given-names></name></person-group><article-title>Building epidemiological models from R-0: An implicit treatment of transmission in networks</article-title><source>Proc. R. Soc. B Biol. Sci.</source><year>2007</year><volume>274</volume><fpage>505</fpage><lpage>512</lpage><pub-id pub-id-type="doi">10.1098/rspb.2006.0057</pub-id></element-citation></ref><ref id="B21-ijerph-13-00253"><label>21.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Abbey</surname><given-names>H.</given-names></name></person-group><article-title>An examination of the Reed-Frost theory of epidemics</article-title><source>Hum. Biol.</source><year>1952</year><volume>24</volume><fpage>201</fpage><lpage>233</lpage><pub-id pub-id-type="pmid">12990130</pub-id></element-citation></ref><ref id="B22-ijerph-13-00253"><label>22.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Vynnycky</surname><given-names>E.</given-names></name><name><surname>White</surname><given-names>R.</given-names></name></person-group><source>An Introduction to Infectious Disease Modelling</source><publisher-name>Oxford University Press</publisher-name><publisher-loc>Oxford, UK</publisher-loc><year>2010</year></element-citation></ref><ref id="B23-ijerph-13-00253"><label>23.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Bury</surname><given-names>K.</given-names></name></person-group><source>Statistical Distributions in Engineering</source><publisher-name>Cambridge University Press</publisher-name><publisher-loc>Cambridge, UK</publisher-loc><year>1999</year></element-citation></ref><ref id="B24-ijerph-13-00253"><label>24.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Lloyd-Smith</surname><given-names>J.</given-names></name><name><surname>Schreiber</surname><given-names>S.</given-names></name><name><surname>Kopp</surname><given-names>P.</given-names></name><name><surname>Getz</surname><given-names>W.</given-names></name></person-group><article-title>Superspreading and the effect of individual variation on disease emergence</article-title><source>Nature</source><year>2005</year><volume>438</volume><fpage>355</fpage><lpage>359</lpage><pub-id pub-id-type="doi">10.1038/nature04153</pub-id><pub-id pub-id-type="pmid">16292310</pub-id></element-citation></ref><ref id="B25-ijerph-13-00253"><label>25.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>May</surname><given-names>R.M.</given-names></name></person-group><article-title>Host-parasitoid systems in patchy environments: A phenomenological model</article-title><source>J. Anim. Ecol.</source><year>1978</year><volume>47</volume><fpage>833</fpage><lpage>844</lpage><pub-id pub-id-type="doi">10.2307/3674</pub-id></element-citation></ref><ref id="B26-ijerph-13-00253"><label>26.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Godfray</surname><given-names>H.C.J.</given-names></name><name><surname>Hassell</surname><given-names>M.P.</given-names></name></person-group><article-title>Discrete and continuous insect populations in tropical environments</article-title><source>J. Anim. Ecol.</source><year>1989</year><volume>58</volume><fpage>153</fpage><lpage>174</lpage><pub-id pub-id-type="doi">10.2307/4992</pub-id></element-citation></ref><ref id="B27-ijerph-13-00253"><label>27.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Briggs</surname><given-names>C.J.</given-names></name><name><surname>Godfray</surname><given-names>H.C.J.</given-names></name></person-group><article-title>The dynamics of insect-pathogen interactions in stage-structured populations</article-title><source>Am. Nat.</source><year>1995</year><volume>145</volume><fpage>855</fpage><lpage>887</lpage><pub-id pub-id-type="doi">10.1086/285774</pub-id></element-citation></ref><ref id="B28-ijerph-13-00253"><label>28.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Barlow</surname><given-names>N.</given-names></name></person-group><article-title>Non-linear transmission and simple models for bovine tuberculosis</article-title><source>J. Anim. Ecol.</source><year>2000</year><volume>69</volume><fpage>703</fpage><lpage>713</lpage><pub-id pub-id-type="doi">10.1046/j.1365-2656.2000.00428.x</pub-id></element-citation></ref><ref id="B29-ijerph-13-00253"><label>29.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hoch</surname><given-names>T.</given-names></name><name><surname>Fourichon</surname><given-names>C.</given-names></name><name><surname>Viet</surname><given-names>A.F.</given-names></name><name><surname>Seegers</surname><given-names>H.</given-names></name></person-group><article-title>Influence of the transmission function on a simulated pathogen spread within a population</article-title><source>Epidemiol. Infect.</source><year>2007</year><volume>136</volume><fpage>1374</fpage><lpage>1382</lpage><pub-id pub-id-type="doi">10.1017/S095026880700979X</pub-id><pub-id pub-id-type="pmid">18062825</pub-id></element-citation></ref><ref id="B30-ijerph-13-00253"><label>30.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Allen</surname><given-names>L.J.</given-names></name><name><surname>Lewis</surname><given-names>T.</given-names></name><name><surname>Martin</surname><given-names>C.F.</given-names></name><name><surname>Jones</surname><given-names>M.A.</given-names></name><name><surname>Lo</surname><given-names>C.K.</given-names></name><name><surname>Stamp</surname><given-names>M.</given-names></name><name><surname>Mundel</surname><given-names>G.</given-names></name><name><surname>Way</surname><given-names>A.B.</given-names></name></person-group><article-title>Analysis of a measles epidemic</article-title><source>Stat. Med.</source><year>1993</year><volume>12</volume><fpage>229</fpage><lpage>239</lpage><pub-id pub-id-type="doi">10.1002/sim.4780120307</pub-id><pub-id pub-id-type="pmid">8456208</pub-id></element-citation></ref><ref id="B31-ijerph-13-00253"><label>31.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Al-Showaikh</surname><given-names>F.</given-names></name><name><surname>Twizell</surname><given-names>E.</given-names></name></person-group><article-title>One-dimensional measles dynamics</article-title><source>Appl. Math. Comput.</source><year>2004</year><volume>152</volume><fpage>169</fpage><lpage>194</lpage><pub-id pub-id-type="doi">10.1016/S0096-3003(03)00554-X</pub-id></element-citation></ref><ref id="B32-ijerph-13-00253"><label>32.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Chen</surname><given-names>S.C.</given-names></name><name><surname>Chang</surname><given-names>C.F.</given-names></name><name><surname>Jou</surname><given-names>L.J.</given-names></name><name><surname>Liao</surname><given-names>C.M.</given-names></name></person-group><article-title>Modelling vaccination programmes against measles in Taiwan</article-title><source>Epidemiol. Infect.</source><year>2007</year><volume>135</volume><fpage>775</fpage><lpage>786</lpage><pub-id pub-id-type="doi">10.1017/S0950268806007369</pub-id><pub-id pub-id-type="pmid">17064459</pub-id></element-citation></ref><ref id="B33-ijerph-13-00253"><label>33.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Gao</surname><given-names>L.</given-names></name><name><surname>Hethcote</surname><given-names>H.</given-names></name></person-group><article-title>Simulations of rubella vaccination strategies in China</article-title><source>Math. Biosci.</source><year>2006</year><volume>202</volume><fpage>371</fpage><lpage>385</lpage><pub-id pub-id-type="doi">10.1016/j.mbs.2006.02.005</pub-id><pub-id pub-id-type="pmid">16603207</pub-id></element-citation></ref><ref id="B34-ijerph-13-00253"><label>34.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Buonomo</surname><given-names>B.</given-names></name></person-group><article-title>A simple analysis of vaccination strategies for rubella</article-title><source>Math. Biosci. Eng.</source><year>2011</year><volume>8</volume><fpage>677</fpage><lpage>687</lpage><pub-id pub-id-type="doi">10.3934/mbe.2011.8.677</pub-id><pub-id pub-id-type="pmid">21675803</pub-id></element-citation></ref><ref id="B35-ijerph-13-00253"><label>35.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Grais</surname><given-names>R.</given-names></name><name><surname>Ellis</surname><given-names>J.</given-names></name><name><surname>Glass</surname><given-names>G.</given-names></name></person-group><article-title>Assessing the impact of airline travel on the geographic spread of pandemic influenza</article-title><source>Eur. J. Epidemiol.</source><year>2003</year><volume>18</volume><fpage>1065</fpage><lpage>1072</lpage><pub-id pub-id-type="doi">10.1023/A:1026140019146</pub-id><pub-id pub-id-type="pmid">14620941</pub-id></element-citation></ref><ref id="B36-ijerph-13-00253"><label>36.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Chen</surname><given-names>S.C.</given-names></name><name><surname>Liao</surname><given-names>C.M.</given-names></name></person-group><article-title>Modelling control measures to reduce the impact of pandemic influenza among schoolchildren</article-title><source>Epidemiol. Infect.</source><year>2008</year><volume>136</volume><fpage>1035</fpage><lpage>1045</lpage><pub-id pub-id-type="doi">10.1017/S0950268807009284</pub-id><pub-id pub-id-type="pmid">17850689</pub-id></element-citation></ref><ref id="B37-ijerph-13-00253"><label>37.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Lipsitch</surname><given-names>M.</given-names></name><name><surname>Cohen</surname><given-names>T.</given-names></name><name><surname>Cooper</surname><given-names>B.</given-names></name><name><surname>Robins</surname><given-names>J.</given-names></name><name><surname>Ma</surname><given-names>S.</given-names></name><name><surname>James</surname><given-names>L.</given-names></name><name><surname>Gopalakrishna</surname><given-names>G.</given-names></name><name><surname>Chew</surname><given-names>S.</given-names></name><name><surname>Tan</surname><given-names>C.</given-names></name><name><surname>Samore</surname><given-names>M.</given-names></name><name><surname>Fisman</surname><given-names>D.</given-names></name><name><surname>Murray</surname><given-names>M.</given-names></name></person-group><article-title>Transmission dynamics and control of severe acute respiratory syndrome</article-title><source>Science</source><year>2003</year><volume>300</volume><fpage>1966</fpage><lpage>1970</lpage><pub-id pub-id-type="doi">10.1126/science.1086616</pub-id><pub-id pub-id-type="pmid">12766207</pub-id></element-citation></ref><ref id="B38-ijerph-13-00253"><label>38.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Wang</surname><given-names>J.</given-names></name><name><surname>McMichael</surname><given-names>A.J.</given-names></name><name><surname>Meng</surname><given-names>B.</given-names></name><name><surname>Becker</surname><given-names>N.G.</given-names></name><name><surname>Han</surname><given-names>W.</given-names></name><name><surname>Glass</surname><given-names>K.</given-names></name><name><surname>Wu</surname><given-names>J.</given-names></name><name><surname>Liu</surname><given-names>X.</given-names></name><name><surname>Liu</surname><given-names>J.</given-names></name><name><surname>Li</surname><given-names>X.</given-names></name><name><surname>Zheng</surname><given-names>X.</given-names></name></person-group><article-title>Spatial dynamics of an epidemic of severe acute respiratory syndrome in an urban area</article-title><source>Bull. World Health Organ.</source><year>2006</year><volume>84</volume><fpage>965</fpage><lpage>968</lpage><pub-id pub-id-type="doi">10.2471/BLT.06.030247</pub-id><pub-id pub-id-type="pmid">17242832</pub-id></element-citation></ref><ref id="B39-ijerph-13-00253"><label>39.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Van den Driessche</surname><given-names>P.</given-names></name><name><surname>Watmough</surname><given-names>J.</given-names></name></person-group><article-title>Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission</article-title><source>Math. Biosci.</source><year>2002</year><volume>180</volume><fpage>29</fpage><lpage>48</lpage><pub-id pub-id-type="doi">10.1016/S0025-5564(02)00108-6</pub-id><pub-id pub-id-type="pmid">12387915</pub-id></element-citation></ref><ref id="B40-ijerph-13-00253"><label>40.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hagenaars</surname><given-names>T.</given-names></name><name><surname>Donnelly</surname><given-names>C.</given-names></name><name><surname>Ferguson</surname><given-names>N.</given-names></name></person-group><article-title>Spatial heterogeneity and the persistence of infectious diseases</article-title><source>J. Theor. Biol.</source><year>2004</year><volume>229</volume><fpage>349</fpage><lpage>359</lpage><pub-id pub-id-type="doi">10.1016/j.jtbi.2004.04.002</pub-id><pub-id pub-id-type="pmid">15234202</pub-id></element-citation></ref><ref id="B41-ijerph-13-00253"><label>41.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Liang</surname><given-names>W.</given-names></name><name><surname>Zhu</surname><given-names>Z.</given-names></name><name><surname>Guo</surname><given-names>J.</given-names></name><name><surname>Liu</surname><given-names>Z.</given-names></name><name><surname>He</surname><given-names>X.</given-names></name><name><surname>Zhou</surname><given-names>W.</given-names></name><name><surname>Chin</surname><given-names>D.P.</given-names></name><name><surname>Schuchat</surname><given-names>A.</given-names></name></person-group><article-title>Severe acute respiratory syndrome, Beijing, 2003</article-title><source>Emerg. Infect. Dis. J.</source><year>2004</year><volume>10</volume><fpage>25</fpage><lpage>31</lpage><pub-id pub-id-type="doi">10.3201/eid1001.030553</pub-id><pub-id pub-id-type="pmid">15078593</pub-id></element-citation></ref><ref id="B42-ijerph-13-00253"><label>42.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Pang</surname><given-names>X.</given-names></name><name><surname>Zhu</surname><given-names>Z.</given-names></name><name><surname>Xu</surname><given-names>F.</given-names></name><name><surname>Guo</surname><given-names>J.</given-names></name><name><surname>Gong</surname><given-names>X.</given-names></name><name><surname>Liu</surname><given-names>D.</given-names></name><name><surname>Liu</surname><given-names>Z.</given-names></name><name><surname>Chin</surname><given-names>D.</given-names></name><name><surname>Feikin</surname><given-names>D.</given-names></name></person-group><article-title>Evaluation of control measures implemented in the severe acute respiratory syndrome outbreak in Beijing, 2003</article-title><source>J. Am. Med. Assoc.</source><year>2003</year><volume>290</volume><fpage>3215</fpage><lpage>3221</lpage><pub-id pub-id-type="doi">10.1001/jama.290.24.3215</pub-id><pub-id pub-id-type="pmid">14693874</pub-id></element-citation></ref><ref id="B43-ijerph-13-00253"><label>43.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Puska</surname><given-names>S.M.</given-names></name></person-group><source>Chinese National Security: Decisionmaking Under Stress</source><publisher-name>Strategic Studies Institute of the US Army War College (SSI)</publisher-name><publisher-loc>Carlisle, PA, USA</publisher-loc><year>2005</year><fpage>85</fpage><lpage>133</lpage></element-citation></ref><ref id="B44-ijerph-13-00253"><label>44.</label><element-citation publication-type="webpage"><person-group person-group-type="author"><collab>MathWorks</collab></person-group><article-title>Global Optimization Toolbox</article-title><comment>Available online: <ext-link ext-link-type="uri" xlink:href="http://cn.mathworks.com/help/gads/index.html">http://cn.mathworks.com/help/gads/index.html</ext-link></comment><date-in-citation>(accessed on 16 February 2016)</date-in-citation></element-citation></ref><ref id="B45-ijerph-13-00253"><label>45.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Ugray</surname><given-names>Z.</given-names></name><name><surname>Lasdon</surname><given-names>L.</given-names></name><name><surname>Plummer</surname><given-names>J.</given-names></name><name><surname>Glover</surname><given-names>F.</given-names></name><name><surname>Kelly</surname><given-names>J.</given-names></name><name><surname>Marti</surname><given-names>R.</given-names></name></person-group><article-title>Scatter search and local NLP solvers: A multistart framework for global optimization</article-title><source>Inf. J. Comput.</source><year>2007</year><volume>19</volume><fpage>328</fpage><lpage>340</lpage><pub-id pub-id-type="doi">10.1287/ijoc.1060.0175</pub-id></element-citation></ref><ref id="B46-ijerph-13-00253"><label>46.</label><element-citation publication-type="webpage"><person-group person-group-type="author"><collab>MathWorks</collab></person-group><article-title>Goodness of Fit between Test and Reference Data</article-title><comment>Available online: <ext-link ext-link-type="uri" xlink:href="http://cn.mathworks.com/help/ident/ref/goodnessoffit.html">http://cn.mathworks.com/help/ident/ref/goodnessoffit.html</ext-link></comment><date-in-citation>(accessed on 16 February 2016)</date-in-citation></element-citation></ref><ref id="B47-ijerph-13-00253"><label>47.</label><element-citation publication-type="book"><person-group person-group-type="author"><collab>Beijing Municipal Bureau of Statistics NBS Survey Office in Beijing</collab></person-group><source>Beijing Statistical Yearbook 2014</source><publisher-name>China Statistics Press</publisher-name><publisher-loc>Beijing, China</publisher-loc><year>2014</year></element-citation></ref><ref id="B48-ijerph-13-00253"><label>48.</label><element-citation publication-type="gov"><person-group person-group-type="author"><collab>Centers for Disease Control and Prevention (America)</collab></person-group><article-title>Frequently Asked Questions about SARS</article-title><comment>Available online: <ext-link ext-link-type="uri" xlink:href="http://www.cdc.gov/sars/about/faq.html">http://www.cdc.gov/sars/about/faq.html</ext-link></comment><date-in-citation>(accessed on 16 February 2016)</date-in-citation></element-citation></ref><ref id="B49-ijerph-13-00253"><label>49.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Barthelemy</surname><given-names>M.</given-names></name><name><surname>Barrat</surname><given-names>A.</given-names></name><name><surname>Pastor-Satorras</surname><given-names>R.</given-names></name><name><surname>Vespignani</surname><given-names>A.</given-names></name></person-group><article-title>Dynamical patterns of epidemic outbreaks in complex heterogeneous networks</article-title><source>J. Theor. Biol.</source><year>2005</year><volume>235</volume><fpage>275</fpage><lpage>288</lpage><pub-id pub-id-type="doi">10.1016/j.jtbi.2005.01.011</pub-id><pub-id pub-id-type="pmid">15862595</pub-id></element-citation></ref><ref id="B50-ijerph-13-00253"><label>50.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Volz</surname><given-names>E.</given-names></name></person-group><article-title>SIR dynamics in random networks with heterogeneous connectivity</article-title><source>J. Math. Biol.</source><year>2008</year><volume>56</volume><fpage>293</fpage><lpage>310</lpage><pub-id pub-id-type="doi">10.1007/s00285-007-0116-4</pub-id><pub-id pub-id-type="pmid">17668212</pub-id></element-citation></ref><ref id="B51-ijerph-13-00253"><label>51.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Miller</surname><given-names>J.C.</given-names></name><name><surname>Slim</surname><given-names>A.C.</given-names></name><name><surname>Volz</surname><given-names>E.M.</given-names></name></person-group><article-title>Edge-based compartmental modelling for infectious disease spread</article-title><source>J. R. Soc. Interface</source><year>2011</year><volume>9</volume><fpage>890</fpage><lpage>906</lpage><pub-id pub-id-type="doi">10.1098/rsif.2011.0403</pub-id><pub-id pub-id-type="pmid">21976638</pub-id></element-citation></ref><ref id="B52-ijerph-13-00253"><label>52.</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Miller</surname><given-names>J.C.</given-names></name><name><surname>Volz</surname><given-names>E.M.</given-names></name></person-group><article-title>Incorporating disease and population structure into models of SIR disease in contact networks</article-title><source>PLoS ONE</source><year>2013</year><volume>8</volume><elocation-id>253</elocation-id><pub-id pub-id-type="doi">10.1371/journal.pone.0069162</pub-id><pub-id pub-id-type="pmid">23990880</pub-id></element-citation></ref></ref-list></back><floats-group><fig id="ijerph-13-00253-f001" position="float"><label>Figure 1</label><caption><p>Structure of a susceptible-exposed-infectious-recovered (SEIR) model.</p></caption><graphic xlink:href="ijerph-13-00253-g001"></graphic></fig><fig id="ijerph-13-00253-f002" position="float"><label>Figure 2</label><caption><p>Infectious curves for different values of <italic>k</italic> and fixed <italic>β</italic> for the negative binomial distribution (NBD) model (Equation (<xref ref-type="disp-formula" rid="FD4-ijerph-13-00253">4</xref>)). The values of <italic>k</italic> are shown in the legend. The other parameters are as follows: <inline-formula><mml:math id="mm104"><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="mm105"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>α</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:mrow></mml:math></inline-formula> days, <inline-formula><mml:math id="mm106"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>γ</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></inline-formula> days and <inline-formula><mml:math id="mm107"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>μ</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mn>70</mml:mn></mml:mrow></mml:math></inline-formula> years. The initial conditions are <inline-formula><mml:math id="mm108"><mml:mrow><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:mn>99</mml:mn><mml:mo>,</mml:mo><mml:mn>999</mml:mn><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:mn>0</mml:mn><mml:mo>,</mml:mo><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:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="mm109"><mml:mrow><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:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>. The top curve in (<bold>A</bold>) is the infectious curve of the homogeneous-mixing model with a frequency-dependent transmission term [<xref rid="B4-ijerph-13-00253" ref-type="bibr">4</xref>]; it is compared to the infectious curves of the NBD model; (<bold>B</bold>) The long trend of the infectious curves of the NBD model with <inline-formula><mml:math id="mm110"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>0001</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="mm111"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>001</mml:mn></mml:mrow></mml:math></inline-formula>.</p></caption><graphic xlink:href="ijerph-13-00253-g002"></graphic></fig><fig id="ijerph-13-00253-f003" position="float"><label>Figure 3</label><caption><p>Infectious curves for different values of <italic>β</italic> and fixed <italic>k</italic> for the NBD model (Equation (<xref ref-type="disp-formula" rid="FD4-ijerph-13-00253">4</xref>)). The values of <italic>β</italic> are shown in the legend. The other parameters are as follows: <inline-formula><mml:math id="mm112"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="mm113"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>α</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:mrow></mml:math></inline-formula> days, <inline-formula><mml:math id="mm114"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>γ</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></inline-formula> days and <inline-formula><mml:math id="mm115"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>μ</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:mn>70</mml:mn></mml:mrow></mml:math></inline-formula> years. The initial conditions are <inline-formula><mml:math id="mm116"><mml:mrow><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:mn>99</mml:mn><mml:mo>,</mml:mo><mml:mn>999</mml:mn><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:mn>0</mml:mn><mml:mo>,</mml:mo><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:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="mm117"><mml:mrow><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:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>. (<bold>A</bold>) The infectious curves around the peak; (<bold>B</bold>) The long trend of the infectious curves of the NBD model with the same parameters.</p></caption><graphic xlink:href="ijerph-13-00253-g003"></graphic></fig><fig id="ijerph-13-00253-f004" position="float"><label>Figure 4</label><caption><p>Infectious curves for the fitting procedure of the NBD model to the SARS outbreak in Beijing in 2003. The circle markers denote the daily reported SARS cases; the parts of the curve to the left and right of the vertical line are the infectious curves before and after the control strategies were taken, respectively.</p></caption><graphic xlink:href="ijerph-13-00253-g004"></graphic></fig><table-wrap id="ijerph-13-00253-t001" position="float"><object-id pub-id-type="pii">ijerph-13-00253-t001_Table 1</object-id><label>Table 1</label><caption><p>Parameter notations, biological meanings, values and sources.</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Parameter</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Biological Meaning</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Bound/Value</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Source</th></tr></thead><tbody><tr><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm118"><mml:msub><mml:mi>k</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1">Heterogeneity level before intervention</td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm119"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>12</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1">Assumed</td></tr><tr><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm120"><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1">Heterogeneity level after intervention</td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm121"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>12</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1">Assumed</td></tr><tr><td align="center" valign="middle" rowspan="1" colspan="1"><italic>β</italic></td><td align="center" valign="middle" rowspan="1" colspan="1">Transmission rate</td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm122"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>10</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1">Assumed</td></tr><tr><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm123"><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1">Latent period</td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm124"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> days</td><td align="center" valign="middle" rowspan="1" colspan="1">[<xref rid="B48-ijerph-13-00253" ref-type="bibr">48</xref>]</td></tr><tr><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm125"><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>γ</mml:mi></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1">Infectious period</td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm126"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>10</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> days</td><td align="center" valign="middle" rowspan="1" colspan="1">[<xref rid="B48-ijerph-13-00253" ref-type="bibr">48</xref>]</td></tr><tr><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm127"><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Expected human lifetime</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">70 years</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Assumed</td></tr></tbody></table></table-wrap><table-wrap id="ijerph-13-00253-t002" position="float"><object-id pub-id-type="pii">ijerph-13-00253-t002_Table 2</object-id><label>Table 2</label><caption><p>Descriptive statistics of the fitted parameters.</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Parameter</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Minimum</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Maximum</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Mean</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Standard Variance</th></tr></thead><tbody><tr><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm128"><mml:msub><mml:mi>k</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm129"><mml:mrow><mml:mn>8</mml:mn><mml:mo>.</mml:mo><mml:mn>4123</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm130"><mml:mrow><mml:mn>6</mml:mn><mml:mo>.</mml:mo><mml:mn>1781</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm131"><mml:mrow><mml:mn>1</mml:mn><mml:mo>.</mml:mo><mml:mn>1882</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm132"><mml:mrow><mml:mn>5</mml:mn><mml:mo>.</mml:mo><mml:mn>75</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td></tr><tr><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm133"><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm134"><mml:mrow><mml:mn>1</mml:mn><mml:mo>.</mml:mo><mml:mn>0130</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm135"><mml:mrow><mml:mn>1</mml:mn><mml:mo>.</mml:mo><mml:mn>1585</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>9</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm136"><mml:mrow><mml:mn>2</mml:mn><mml:mo>.</mml:mo><mml:mn>6311</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>11</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm137"><mml:mrow><mml:mn>1</mml:mn><mml:mo>.</mml:mo><mml:mn>4077</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td></tr><tr><td align="center" valign="middle" rowspan="1" colspan="1"><italic>β</italic></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm138"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>3525</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm139"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>6109</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm140"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>5459</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm141"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>0335</mml:mn></mml:mrow></mml:math></inline-formula></td></tr><tr><td align="center" valign="middle" rowspan="1" colspan="1"><italic>α</italic></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm142"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1429</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm143"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>2130</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm144"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1456</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm145"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>0095</mml:mn></mml:mrow></mml:math></inline-formula></td></tr><tr><td align="center" valign="middle" rowspan="1" colspan="1"><italic>γ</italic></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm146"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1407</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm147"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>2366</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm148"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>2064</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" rowspan="1" colspan="1"><inline-formula><mml:math id="mm149"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>0118</mml:mn></mml:mrow></mml:math></inline-formula></td></tr><tr><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">NRMSE</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm150"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>8005</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm151"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>8041</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm152"><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>8037</mml:mn></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm153"><mml:mrow><mml:mn>7</mml:mn><mml:mo>.</mml:mo><mml:mn>2604</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></td></tr></tbody></table></table-wrap></floats-group></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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