Modeling Heterogeneity in Direct Infectious Disease Transmission in a Compartmental Model
Identifieur interne : 000692 ( Pmc/Checkpoint ); précédent : 000691; suivant : 000693Modeling Heterogeneity in Direct Infectious Disease Transmission in a Compartmental Model
Auteurs : Lingcai Kong [République populaire de Chine] ; Jinfeng Wang [République populaire de Chine] ; Weiguo Han ; Zhidong CaoSource :
- International Journal of Environmental Research and Public Health [ 1661-7827 ] ; 2016.
Abstract
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,
Url:
DOI: 10.3390/ijerph13030253
PubMed: 26927140
PubMed Central: 4808916
Affiliations:
Links toward previous steps (curation, corpus...)
Links to Exploration step
PMC:4808916Le document en format XML
<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 wicri:level="1"><nlm:aff id="af2-ijerph-13-00253">University of the Chinese Academy of Sciences, Beijing 100049, China</nlm:aff><country xml:lang="fr">République populaire de Chine</country><wicri:regionArea>University of the Chinese Academy of Sciences, Beijing 100049</wicri:regionArea><placeName><settlement type="city">Pékin</settlement></placeName></affiliation><affiliation wicri:level="1"><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><country xml:lang="fr">République populaire de Chine</country><wicri:regionArea>Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206</wicri:regionArea><placeName><settlement type="city">Pékin</settlement></placeName></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 wicri:level="1"><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><country xml:lang="fr">République populaire de Chine</country><wicri:regionArea>Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206</wicri:regionArea><placeName><settlement type="city">Pékin</settlement></placeName></affiliation><affiliation wicri:level="1"><nlm:aff id="af4-ijerph-13-00253">Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023, China</nlm:aff><country xml:lang="fr">République populaire de Chine</country><wicri:regionArea>Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023</wicri:regionArea><wicri:noRegion>Nanjing 210023</wicri:noRegion></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><idno type="wicri:Area/Pmc/Curation">000023</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Curation">000023</idno><idno type="wicri:Area/Pmc/Checkpoint">000692</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Checkpoint">000692</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 wicri:level="1"><nlm:aff id="af2-ijerph-13-00253">University of the Chinese Academy of Sciences, Beijing 100049, China</nlm:aff><country xml:lang="fr">République populaire de Chine</country><wicri:regionArea>University of the Chinese Academy of Sciences, Beijing 100049</wicri:regionArea><placeName><settlement type="city">Pékin</settlement></placeName></affiliation><affiliation wicri:level="1"><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><country xml:lang="fr">République populaire de Chine</country><wicri:regionArea>Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206</wicri:regionArea><placeName><settlement type="city">Pékin</settlement></placeName></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 wicri:level="1"><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><country xml:lang="fr">République populaire de Chine</country><wicri:regionArea>Key Laboratory of Surveillance and Early-Warning on Infectious Disease, Chinese Center for Disease Control and Prevention, Beijing 102206</wicri:regionArea><placeName><settlement type="city">Pékin</settlement></placeName></affiliation><affiliation wicri:level="1"><nlm:aff id="af4-ijerph-13-00253">Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023, China</nlm:aff><country xml:lang="fr">République populaire de Chine</country><wicri:regionArea>Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023</wicri:regionArea><wicri:noRegion>Nanjing 210023</wicri:noRegion></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><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><affiliations><list><country><li>République populaire de Chine</li></country><settlement><li>Pékin</li></settlement></list><tree><noCountry><name sortKey="Cao, Zhidong" sort="Cao, Zhidong" uniqKey="Cao Z" first="Zhidong" last="Cao">Zhidong Cao</name><name sortKey="Han, Weiguo" sort="Han, Weiguo" uniqKey="Han W" first="Weiguo" last="Han">Weiguo Han</name></noCountry><country name="République populaire de Chine"><noRegion><name sortKey="Kong, Lingcai" sort="Kong, Lingcai" uniqKey="Kong L" first="Lingcai" last="Kong">Lingcai Kong</name></noRegion><name sortKey="Kong, Lingcai" sort="Kong, Lingcai" uniqKey="Kong L" first="Lingcai" last="Kong">Lingcai Kong</name><name sortKey="Wang, Jinfeng" sort="Wang, Jinfeng" uniqKey="Wang J" first="Jinfeng" last="Wang">Jinfeng Wang</name><name sortKey="Wang, Jinfeng" sort="Wang, Jinfeng" uniqKey="Wang J" first="Jinfeng" last="Wang">Jinfeng Wang</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Sante/explor/SrasV1/Data/Pmc/Checkpoint
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000692 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Pmc/Checkpoint/biblio.hfd -nk 000692 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien
|wiki= Sante
|area= SrasV1
|flux= Pmc
|étape= Checkpoint
|type= RBID
|clé= PMC:4808916
|texte= Modeling Heterogeneity in Direct Infectious Disease Transmission in a Compartmental Model
}}
Pour générer des pages wiki
HfdIndexSelect -h $EXPLOR_AREA/Data/Pmc/Checkpoint/RBID.i -Sk "pubmed:26927140" \
| HfdSelect -Kh $EXPLOR_AREA/Data/Pmc/Checkpoint/biblio.hfd \
| NlmPubMed2Wicri -a SrasV1
| This area was generated with Dilib version V0.6.33. |  |