Stochastic modelling of infectious diseases for heterogeneous populations
Identifieur interne : 000389 ( Pmc/Corpus ); précédent : 000388; suivant : 000390Stochastic modelling of infectious diseases for heterogeneous populations
Auteurs : Rui-Xing Ming ; Jiming Liu ; William K. W. Cheung ; Xiang WanSource :
- Infectious Diseases of Poverty [ 2095-5162 ] ; 2016.
Abstract
Infectious diseases such as SARS and H1N1 can significantly impact people’s lives and cause severe social and economic damages. Recent outbreaks have stressed the urgency of effective research on the dynamics of infectious disease spread. However, it is difficult to predict when and where outbreaks may emerge and how infectious diseases spread because many factors affect their transmission, and some of them may be unknown.
One feasible means to promptly detect an outbreak and track the progress of disease spread is to implement surveillance systems in regional or national health and medical centres. The accumulated surveillance data, including temporal, spatial, clinical, and demographic information can provide valuable information that can be exploited to better understand and model the dynamics of infectious disease spread. The aim of this work is to develop and empirically evaluate a stochastic model that allows the investigation of transmission patterns of infectious diseases in heterogeneous populations.
We test the proposed model on simulation data and apply it to the surveillance data from the 2009 H1N1 pandemic in Hong Kong. In the simulation experiment, our model achieves high accuracy in parameter estimation (less than 10.0
We propose a stochastic model to study the dynamics of infectious disease spread in heterogeneous populations from temporal-spatial surveillance data. The proposed model is evaluated using both simulated data and the real data from the 2009 H1N1 epidemic in Hong Kong and achieves acceptable prediction accuracy. We believe that our model can provide valuable insights for public health authorities to predict the effect of disease spread and analyse its underlying factors and to guide new control efforts.
The online version of this article (doi:10.1186/s40249-016-0199-5) contains supplementary material, which is available to authorized users.
Url:
DOI: 10.1186/s40249-016-0199-5
PubMed: 28003016
PubMed Central: 5178099
Links to Exploration step
PMC:5178099Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Stochastic modelling of infectious diseases for heterogeneous populations</title><author><name sortKey="Ming, Rui Xing" sort="Ming, Rui Xing" uniqKey="Ming R" first="Rui-Xing" last="Ming">Rui-Xing Ming</name><affiliation><nlm:aff id="Aff1"><institution-wrap><institution-id institution-id-type="GRID">grid.413072.3</institution-id><institution-id institution-id-type="ISNI">0000000122297034</institution-id><institution>School of Statistics & Mathematics,</institution><institution>Zhejiang Gongshang University,</institution></institution-wrap>Hangzhou, China</nlm:aff></affiliation></author><author><name sortKey="Liu, Jiming" sort="Liu, Jiming" uniqKey="Liu J" first="Jiming" last="Liu">Jiming Liu</name><affiliation><nlm:aff id="Aff2"><institution-wrap><institution-id institution-id-type="GRID">grid.221309.b</institution-id><institution-id institution-id-type="ISNI">0000000417645980</institution-id><institution>Department of Computer Science,</institution><institution>Hong Kong Baptist University,</institution></institution-wrap>Kowloon Tong, Hong Kong</nlm:aff></affiliation></author><author><name sortKey="Cheung, William K W" sort="Cheung, William K W" uniqKey="Cheung W" first="William K. W." last="Cheung">William K. W. Cheung</name><affiliation><nlm:aff id="Aff2"><institution-wrap><institution-id institution-id-type="GRID">grid.221309.b</institution-id><institution-id institution-id-type="ISNI">0000000417645980</institution-id><institution>Department of Computer Science,</institution><institution>Hong Kong Baptist University,</institution></institution-wrap>Kowloon Tong, Hong Kong</nlm:aff></affiliation></author><author><name sortKey="Wan, Xiang" sort="Wan, Xiang" uniqKey="Wan X" first="Xiang" last="Wan">Xiang Wan</name><affiliation><nlm:aff id="Aff3"><institution-wrap><institution-id institution-id-type="GRID">grid.221309.b</institution-id><institution-id institution-id-type="ISNI">0000000417645980</institution-id><institution>Department of Computer Science & Institute of Theoretical and Computational Study,</institution><institution>Hong Kong Baptist University,</institution></institution-wrap>Kowloon Tong, Hong Kong</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">28003016</idno><idno type="pmc">5178099</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5178099</idno><idno type="RBID">PMC:5178099</idno><idno type="doi">10.1186/s40249-016-0199-5</idno><date when="2016">2016</date><idno type="wicri:Area/Pmc/Corpus">000389</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000389</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Stochastic modelling of infectious diseases for heterogeneous populations</title><author><name sortKey="Ming, Rui Xing" sort="Ming, Rui Xing" uniqKey="Ming R" first="Rui-Xing" last="Ming">Rui-Xing Ming</name><affiliation><nlm:aff id="Aff1"><institution-wrap><institution-id institution-id-type="GRID">grid.413072.3</institution-id><institution-id institution-id-type="ISNI">0000000122297034</institution-id><institution>School of Statistics & Mathematics,</institution><institution>Zhejiang Gongshang University,</institution></institution-wrap>Hangzhou, China</nlm:aff></affiliation></author><author><name sortKey="Liu, Jiming" sort="Liu, Jiming" uniqKey="Liu J" first="Jiming" last="Liu">Jiming Liu</name><affiliation><nlm:aff id="Aff2"><institution-wrap><institution-id institution-id-type="GRID">grid.221309.b</institution-id><institution-id institution-id-type="ISNI">0000000417645980</institution-id><institution>Department of Computer Science,</institution><institution>Hong Kong Baptist University,</institution></institution-wrap>Kowloon Tong, Hong Kong</nlm:aff></affiliation></author><author><name sortKey="Cheung, William K W" sort="Cheung, William K W" uniqKey="Cheung W" first="William K. W." last="Cheung">William K. W. Cheung</name><affiliation><nlm:aff id="Aff2"><institution-wrap><institution-id institution-id-type="GRID">grid.221309.b</institution-id><institution-id institution-id-type="ISNI">0000000417645980</institution-id><institution>Department of Computer Science,</institution><institution>Hong Kong Baptist University,</institution></institution-wrap>Kowloon Tong, Hong Kong</nlm:aff></affiliation></author><author><name sortKey="Wan, Xiang" sort="Wan, Xiang" uniqKey="Wan X" first="Xiang" last="Wan">Xiang Wan</name><affiliation><nlm:aff id="Aff3"><institution-wrap><institution-id institution-id-type="GRID">grid.221309.b</institution-id><institution-id institution-id-type="ISNI">0000000417645980</institution-id><institution>Department of Computer Science & Institute of Theoretical and Computational Study,</institution><institution>Hong Kong Baptist University,</institution></institution-wrap>Kowloon Tong, Hong Kong</nlm:aff></affiliation></author></analytic><series><title level="j">Infectious Diseases of Poverty</title><idno type="ISSN">2095-5162</idno><idno type="eISSN">2049-9957</idno><imprint><date when="2016">2016</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><sec><title>Background</title><p>Infectious diseases such as SARS and H1N1 can significantly impact people’s lives and cause severe social and economic damages. Recent outbreaks have stressed the urgency of effective research on the dynamics of infectious disease spread. However, it is difficult to predict when and where outbreaks may emerge and how infectious diseases spread because many factors affect their transmission, and some of them may be unknown.</p></sec><sec><title>Methods</title><p>One feasible means to promptly detect an outbreak and track the progress of disease spread is to implement surveillance systems in regional or national health and medical centres. The accumulated surveillance data, including temporal, spatial, clinical, and demographic information can provide valuable information that can be exploited to better understand and model the dynamics of infectious disease spread. The aim of this work is to develop and empirically evaluate a stochastic model that allows the investigation of transmission patterns of infectious diseases in heterogeneous populations.</p></sec><sec><title>Results</title><p>We test the proposed model on simulation data and apply it to the surveillance data from the 2009 H1N1 pandemic in Hong Kong. In the simulation experiment, our model achieves high accuracy in parameter estimation (less than 10.0 <italic>%</italic> mean absolute percentage error). In terms of the forward prediction of case incidence, the mean absolute percentage errors are 17.3 <italic>%</italic> for the simulation experiment and 20.0 <italic>%</italic> for the experiment on the real surveillance data.</p></sec><sec><title>Conclusion</title><p>We propose a stochastic model to study the dynamics of infectious disease spread in heterogeneous populations from temporal-spatial surveillance data. The proposed model is evaluated using both simulated data and the real data from the 2009 H1N1 epidemic in Hong Kong and achieves acceptable prediction accuracy. We believe that our model can provide valuable insights for public health authorities to predict the effect of disease spread and analyse its underlying factors and to guide new control efforts.</p></sec><sec><title>Electronic supplementary material</title><p>The online version of this article (doi:10.1186/s40249-016-0199-5) contains supplementary material, which is available to authorized users.</p></sec></div></front><back><div1 type="bibliography"><listBibl><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Cohen, J" uniqKey="Cohen J">J Cohen</name></author><author><name sortKey="Enserink, M" uniqKey="Enserink M">M Enserink</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Ferguson, Nm" uniqKey="Ferguson N">NM Ferguson</name></author><author><name sortKey="Cummings, Da" uniqKey="Cummings D">DA Cummings</name></author><author><name sortKey="Fraser, C" uniqKey="Fraser C">C Fraser</name></author><author><name sortKey="Cajka, Jc" uniqKey="Cajka J">JC Cajka</name></author><author><name sortKey="Cooley, Pc" uniqKey="Cooley P">PC Cooley</name></author><author><name sortKey="Burke, Ds" uniqKey="Burke D">DS Burke</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Germann, Tc" uniqKey="Germann T">TC Germann</name></author><author><name sortKey="Kadau, K" uniqKey="Kadau K">K Kadau</name></author><author><name sortKey="Longini, Im" uniqKey="Longini I">IM Longini</name></author><author><name sortKey="Macken, Ca" uniqKey="Macken C">CA Macken</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Bailey, Ntj" uniqKey="Bailey N">NTJ Bailey</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Anderson, Rm" uniqKey="Anderson R">RM Anderson</name></author><author><name sortKey="May, Rm" uniqKey="May R">RM May</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Heesterbeek, J" uniqKey="Heesterbeek J">J Heesterbeek</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Li, My" uniqKey="Li M">MY Li</name></author><author><name sortKey="Muldowney, Js" uniqKey="Muldowney J">JS Muldowney</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Kuznetsov, Ya" uniqKey="Kuznetsov Y">YA Kuznetsov</name></author><author><name sortKey="Piccardi, C" uniqKey="Piccardi C">C Piccardi</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hethcote, Hw" uniqKey="Hethcote H">HW Hethcote</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Becker, Ng" uniqKey="Becker N">NG Becker</name></author><author><name sortKey="Britton, T" uniqKey="Britton T">T Britton</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Ferrari, Mj" uniqKey="Ferrari M">MJ Ferrari</name></author><author><name sortKey="Bj Rnstad, On" uniqKey="Bj Rnstad O">ON Bjørnstad</name></author><author><name sortKey="Dobson, Ap" uniqKey="Dobson A">AP Dobson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Allen, L" uniqKey="Allen L">L Allen</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Cooper, Bs" uniqKey="Cooper B">BS Cooper</name></author><author><name sortKey="Pitman, Rj" uniqKey="Pitman R">RJ Pitman</name></author><author><name sortKey="Edmunds, Wj" uniqKey="Edmunds W">WJ Edmunds</name></author><author><name sortKey="Gay, Nj" uniqKey="Gay N">NJ Gay</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hufnagel, L" uniqKey="Hufnagel L">L Hufnagel</name></author><author><name sortKey="Brockmann, D" uniqKey="Brockmann D">D Brockmann</name></author><author><name sortKey="Geisel, T" uniqKey="Geisel T">T Geisel</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hollingsworth, Td" uniqKey="Hollingsworth T">TD Hollingsworth</name></author><author><name sortKey="Ferguson, Nm" uniqKey="Ferguson N">NM Ferguson</name></author><author><name sortKey="Anderson, Rm" uniqKey="Anderson R">RM Anderson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Keeling, Mj" uniqKey="Keeling M">MJ Keeling</name></author><author><name sortKey="Woolhouse, Me" uniqKey="Woolhouse M">ME Woolhouse</name></author><author><name sortKey="Shaw, Dj" uniqKey="Shaw D">DJ Shaw</name></author><author><name sortKey="Matthews, L" uniqKey="Matthews L">L Matthews</name></author><author><name sortKey="Chase Topping, M" uniqKey="Chase Topping M">M Chase-Topping</name></author><author><name sortKey="Haydon, Dt" uniqKey="Haydon D">DT Haydon</name></author><author><name sortKey="Cornell, Sj" uniqKey="Cornell S">SJ Cornell</name></author><author><name sortKey="Kappey, J" uniqKey="Kappey J">J Kappey</name></author><author><name sortKey="Wilesmith, J" uniqKey="Wilesmith J">J Wilesmith</name></author><author><name sortKey="Grenfell, Bt" uniqKey="Grenfell B">BT Grenfell</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Ferguson, Nm" uniqKey="Ferguson N">NM Ferguson</name></author><author><name sortKey="Cummings, Da" uniqKey="Cummings D">DA Cummings</name></author><author><name sortKey="Cauchemez, S" uniqKey="Cauchemez S">S Cauchemez</name></author><author><name sortKey="Fraser, C" uniqKey="Fraser C">C Fraser</name></author><author><name sortKey="Riley, S" uniqKey="Riley S">S Riley</name></author><author><name sortKey="Meeyai, A" uniqKey="Meeyai A">A Meeyai</name></author><author><name sortKey="Iamsirithaworn, S" uniqKey="Iamsirithaworn S">S Iamsirithaworn</name></author><author><name sortKey="Burke, Ds" uniqKey="Burke D">DS Burke</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Longini, Im" uniqKey="Longini I">IM Longini</name></author><author><name sortKey="Nizam, A" uniqKey="Nizam A">A Nizam</name></author><author><name sortKey="Xu, S" uniqKey="Xu S">S Xu</name></author><author><name sortKey="Ungchusak, K" uniqKey="Ungchusak K">K Ungchusak</name></author><author><name sortKey="Hanshaoworakul, W" uniqKey="Hanshaoworakul W">W Hanshaoworakul</name></author><author><name sortKey="Cummings, Da" uniqKey="Cummings D">DA Cummings</name></author><author><name sortKey="Halloran, Me" uniqKey="Halloran M">ME Halloran</name></author></analytic></biblStruct><biblStruct><analytic><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></biblStruct><biblStruct><analytic><author><name sortKey="Kuperman, M" uniqKey="Kuperman M">M Kuperman</name></author><author><name sortKey="Abramson, G" uniqKey="Abramson G">G Abramson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Newman, Me" uniqKey="Newman M">ME Newman</name></author><author><name sortKey="Jensen, I" uniqKey="Jensen I">I Jensen</name></author><author><name sortKey="Ziff, R" uniqKey="Ziff R">R Ziff</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Boguna, M" uniqKey="Boguna M">M Boguná</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="Grassberger, P" uniqKey="Grassberger P">P Grassberger</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Watts, Dj" uniqKey="Watts D">DJ Watts</name></author><author><name sortKey="Strogatz, Sh" uniqKey="Strogatz S">SH Strogatz</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Barabasi, Al" uniqKey="Barabasi A">AL Barabási</name></author><author><name sortKey="Albert, R" uniqKey="Albert R">R Albert</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><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="Rogers, Em" uniqKey="Rogers E">EM Rogers</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Leskovec, J" uniqKey="Leskovec J">J Leskovec</name></author><author><name sortKey="Adamic, La" uniqKey="Adamic L">LA Adamic</name></author><author><name sortKey="Huberman, Ba" uniqKey="Huberman B">BA Huberman</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Kumar, R" uniqKey="Kumar R">R Kumar</name></author><author><name sortKey="Novak, J" uniqKey="Novak J">J Novak</name></author><author><name sortKey="Raghavan, P" uniqKey="Raghavan P">P Raghavan</name></author><author><name sortKey="Tomkins, A" uniqKey="Tomkins A">A Tomkins</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Salathe, M" uniqKey="Salathe M">M Salathé</name></author><author><name sortKey="Kazandjieva, M" uniqKey="Kazandjieva M">M Kazandjieva</name></author><author><name sortKey="Lee, Jw" uniqKey="Lee J">JW Lee</name></author><author><name sortKey="Levis, P" uniqKey="Levis P">P Levis</name></author><author><name sortKey="Feldman, Mw" uniqKey="Feldman M">MW Feldman</name></author><author><name sortKey="Jones, Jh" uniqKey="Jones J">JH Jones</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Stehle, J" uniqKey="Stehle J">J Stehlé</name></author><author><name sortKey="Voirin, N" uniqKey="Voirin N">N Voirin</name></author><author><name sortKey="Barrat, A" uniqKey="Barrat A">A Barrat</name></author><author><name sortKey="Cattuto, C" uniqKey="Cattuto C">C Cattuto</name></author><author><name sortKey="Isella, L" uniqKey="Isella L">L Isella</name></author><author><name sortKey="Pinton, Jf" uniqKey="Pinton J">JF Pinton</name></author><author><name sortKey="Quaggiotto, M" uniqKey="Quaggiotto M">M Quaggiotto</name></author><author><name sortKey="Van Den Broeck, W" uniqKey="Van Den Broeck W">W Van den Broeck</name></author><author><name sortKey="Regis, C" uniqKey="Regis C">C Régis</name></author><author><name sortKey="Lina, B" uniqKey="Lina B">B Lina</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Dickison, M" uniqKey="Dickison M">M Dickison</name></author><author><name sortKey="Havlin, S" uniqKey="Havlin S">S Havlin</name></author><author><name sortKey="Stanley, He" uniqKey="Stanley H">HE Stanley</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Hancock, K" uniqKey="Hancock K">K Hancock</name></author><author><name sortKey="Veguilla, V" uniqKey="Veguilla V">V Veguilla</name></author><author><name sortKey="Lu, X" uniqKey="Lu X">X Lu</name></author><author><name sortKey="Zhong, W" uniqKey="Zhong W">W Zhong</name></author><author><name sortKey="Butler, En" uniqKey="Butler E">EN Butler</name></author><author><name sortKey="Sun, H" uniqKey="Sun H">H Sun</name></author><author><name sortKey="Liu, F" uniqKey="Liu F">F Liu</name></author><author><name sortKey="Dong, L" uniqKey="Dong L">L Dong</name></author><author><name sortKey="Devos, Jr" uniqKey="Devos J">JR DeVos</name></author><author><name sortKey="Gargiullo, Pm" uniqKey="Gargiullo P">PM Gargiullo</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Riley, S" uniqKey="Riley S">S Riley</name></author><author><name sortKey="Fraser, C" uniqKey="Fraser C">C Fraser</name></author><author><name sortKey="Donnelly, Ca" uniqKey="Donnelly C">CA Donnelly</name></author><author><name sortKey="Ghani, Ac" uniqKey="Ghani A">AC Ghani</name></author><author><name sortKey="Abu Raddad, Lj" uniqKey="Abu Raddad L">LJ Abu-Raddad</name></author><author><name sortKey="Hedley, Aj" uniqKey="Hedley A">AJ Hedley</name></author><author><name sortKey="Leung, Gm" uniqKey="Leung G">GM Leung</name></author><author><name sortKey="Ho, Lm" uniqKey="Ho L">LM Ho</name></author><author><name sortKey="Lam, Th" uniqKey="Lam T">TH Lam</name></author><author><name sortKey="Thach, Tq" uniqKey="Thach T">TQ Thach</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Mills, Ce" uniqKey="Mills C">CE Mills</name></author><author><name sortKey="Robins, Jm" uniqKey="Robins J">JM Robins</name></author><author><name sortKey="Lipsitch, M" uniqKey="Lipsitch M">M Lipsitch</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Birrell, Pj" uniqKey="Birrell P">PJ Birrell</name></author><author><name sortKey="Ketsetzis, G" uniqKey="Ketsetzis G">G Ketsetzis</name></author><author><name sortKey="Gay, Nj" uniqKey="Gay N">NJ Gay</name></author><author><name sortKey="Cooper, Bs" uniqKey="Cooper B">BS Cooper</name></author><author><name sortKey="Presanis, Am" uniqKey="Presanis A">AM Presanis</name></author><author><name sortKey="Harris, Rj" uniqKey="Harris R">RJ Harris</name></author><author><name sortKey="Charlett, A" uniqKey="Charlett A">A Charlett</name></author><author><name sortKey="Zhang, Xs" uniqKey="Zhang X">XS Zhang</name></author><author><name sortKey="White, Pj" uniqKey="White P">PJ White</name></author><author><name sortKey="Pebody, Rg" uniqKey="Pebody R">RG Pebody</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Towers, S" uniqKey="Towers S">S Towers</name></author><author><name sortKey="Geisse, Kv" uniqKey="Geisse K">KV Geisse</name></author><author><name sortKey="Zheng, Y" uniqKey="Zheng Y">Y Zheng</name></author><author><name sortKey="Feng, Z" uniqKey="Feng Z">Z Feng</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Cauchemez, S" uniqKey="Cauchemez S">S Cauchemez</name></author><author><name sortKey="Valleron, Aj" uniqKey="Valleron A">AJ Valleron</name></author><author><name sortKey="Boelle, Py" uniqKey="Boelle P">PY Boelle</name></author><author><name sortKey="Flahault, A" uniqKey="Flahault A">A Flahault</name></author><author><name sortKey="Ferguson, Nm" uniqKey="Ferguson N">NM Ferguson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Cauchemez, S" uniqKey="Cauchemez S">S Cauchemez</name></author><author><name sortKey="Donnelly, Ca" uniqKey="Donnelly C">CA Donnelly</name></author><author><name sortKey="Reed, C" uniqKey="Reed C">C Reed</name></author><author><name sortKey="Ghani, Ac" uniqKey="Ghani A">AC Ghani</name></author><author><name sortKey="Fraser, C" uniqKey="Fraser C">C Fraser</name></author><author><name sortKey="Kent, Ck" uniqKey="Kent C">CK Kent</name></author><author><name sortKey="Finelli, L" uniqKey="Finelli L">L Finelli</name></author><author><name sortKey="Ferguson, Nm" uniqKey="Ferguson N">NM Ferguson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lessler, J" uniqKey="Lessler J">J Lessler</name></author><author><name sortKey="Reich, Ng" uniqKey="Reich N">NG Reich</name></author><author><name sortKey="Cummings, Da" uniqKey="Cummings D">DA Cummings</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Klick, B" uniqKey="Klick B">B Klick</name></author><author><name sortKey="Nishiura, H" uniqKey="Nishiura H">H Nishiura</name></author><author><name sortKey="Ng, S" uniqKey="Ng S">S Ng</name></author><author><name sortKey="Fang, Vj" uniqKey="Fang V">VJ Fang</name></author><author><name sortKey="Leung, Gm" uniqKey="Leung G">GM Leung</name></author><author><name sortKey="Peiris, Jm" uniqKey="Peiris J">JM Peiris</name></author><author><name sortKey="Cowling, Bj" uniqKey="Cowling B">BJ Cowling</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Yang, Y" uniqKey="Yang Y">Y Yang</name></author><author><name sortKey="Sugimoto, Jd" uniqKey="Sugimoto J">JD Sugimoto</name></author><author><name sortKey="Halloran, Me" uniqKey="Halloran M">ME Halloran</name></author><author><name sortKey="Basta, Ne" uniqKey="Basta N">NE Basta</name></author><author><name sortKey="Chao, Dl" uniqKey="Chao D">DL Chao</name></author><author><name sortKey="Matrajt, L" uniqKey="Matrajt L">L Matrajt</name></author><author><name sortKey="Potter, G" uniqKey="Potter G">G Potter</name></author><author><name sortKey="Kenah, E" uniqKey="Kenah E">E Kenah</name></author><author><name sortKey="Longini, Im" uniqKey="Longini I">IM Longini</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Jin, Z" uniqKey="Jin Z">Z Jin</name></author><author><name sortKey="Zhang, J" uniqKey="Zhang J">J Zhang</name></author><author><name sortKey="Song, Lp" uniqKey="Song L">LP Song</name></author><author><name sortKey="Sun, Gq" uniqKey="Sun G">GQ Sun</name></author><author><name sortKey="Kan, J" uniqKey="Kan J">J Kan</name></author><author><name sortKey="Zhu, H" uniqKey="Zhu H">H Zhu</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Camacho, A" uniqKey="Camacho A">A Camacho</name></author><author><name sortKey="Kucharski, A" uniqKey="Kucharski A">A Kucharski</name></author><author><name sortKey="Aki Sawyerr, Y" uniqKey="Aki Sawyerr Y">Y Aki-Sawyerr</name></author><author><name sortKey="White, Ma" uniqKey="White M">MA White</name></author><author><name sortKey="Flasche, S" uniqKey="Flasche S">S Flasche</name></author><author><name sortKey="Baguelin, M" uniqKey="Baguelin M">M Baguelin</name></author><author><name sortKey="Pollington, T" uniqKey="Pollington T">T Pollington</name></author><author><name sortKey="Carney, Jr" uniqKey="Carney J">JR Carney</name></author><author><name sortKey="Glover, R" uniqKey="Glover R">R Glover</name></author><author><name sortKey="Smout, E" uniqKey="Smout E">E Smout</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Chowell, D" uniqKey="Chowell D">D Chowell</name></author><author><name sortKey="Castillo Chavez, C" uniqKey="Castillo Chavez C">C Castillo-Chavez</name></author><author><name sortKey="Krishna, S" uniqKey="Krishna S">S Krishna</name></author><author><name sortKey="Qiu, X" uniqKey="Qiu X">X Qiu</name></author><author><name sortKey="Anderson, Ks" uniqKey="Anderson K">KS Anderson</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Carroll, Mw" uniqKey="Carroll M">MW Carroll</name></author><author><name sortKey="Matthews, Da" uniqKey="Matthews D">DA Matthews</name></author><author><name sortKey="Hiscox, Ja" uniqKey="Hiscox J">JA Hiscox</name></author><author><name sortKey="Elmore, Mj" uniqKey="Elmore M">MJ Elmore</name></author><author><name sortKey="Pollakis, G" uniqKey="Pollakis G">G Pollakis</name></author><author><name sortKey="Rambaut, A" uniqKey="Rambaut A">A Rambaut</name></author><author><name sortKey="Hewson, R" uniqKey="Hewson R">R Hewson</name></author><author><name sortKey="Garcia Dorival, I" uniqKey="Garcia Dorival I">I García-Dorival</name></author><author><name sortKey="Bore, Ja" uniqKey="Bore J">JA Bore</name></author><author><name sortKey="Koundouno, R" uniqKey="Koundouno R">R Koundouno</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Chowell, G" uniqKey="Chowell G">G Chowell</name></author><author><name sortKey="Nishiura, H" uniqKey="Nishiura H">H Nishiura</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Van Dijk, A" uniqKey="Van Dijk A">A van Dijk</name></author><author><name sortKey="Aramini, J" uniqKey="Aramini J">J Aramini</name></author><author><name sortKey="Edge, G" uniqKey="Edge G">G Edge</name></author><author><name sortKey="Moore, Km" uniqKey="Moore K">KM Moore</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">Infect Dis Poverty</journal-id><journal-id journal-id-type="iso-abbrev">Infect Dis Poverty</journal-id><journal-title-group><journal-title>Infectious Diseases of Poverty</journal-title></journal-title-group><issn pub-type="ppub">2095-5162</issn><issn pub-type="epub">2049-9957</issn><publisher><publisher-name>BioMed Central</publisher-name><publisher-loc>London</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">28003016</article-id><article-id pub-id-type="pmc">5178099</article-id><article-id pub-id-type="publisher-id">199</article-id><article-id pub-id-type="doi">10.1186/s40249-016-0199-5</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group></article-categories><title-group><article-title>Stochastic modelling of infectious diseases for heterogeneous populations</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Ming</surname><given-names>Rui-Xing</given-names></name><address><email>rxming@ustc.edu.cn</email></address><xref ref-type="aff" rid="Aff1">1</xref></contrib><contrib contrib-type="author"><name><surname>Liu</surname><given-names>Jiming</given-names></name><address><email>jiming@comp.hkbu.edu.hk</email></address><xref ref-type="aff" rid="Aff2">2</xref></contrib><contrib contrib-type="author"><name><surname>Cheung</surname><given-names>William K. W.</given-names></name><address><email>william@comp.hkbu.edu.hk</email></address><xref ref-type="aff" rid="Aff2">2</xref></contrib><contrib contrib-type="author" corresp="yes"><name><surname>Wan</surname><given-names>Xiang</given-names></name><address><email>xwan@comp.hkbu.edu.hk</email></address><xref ref-type="aff" rid="Aff3">3</xref></contrib><aff id="Aff1"><label>1</label><institution-wrap><institution-id institution-id-type="GRID">grid.413072.3</institution-id><institution-id institution-id-type="ISNI">0000000122297034</institution-id><institution>School of Statistics & Mathematics,</institution><institution>Zhejiang Gongshang University,</institution></institution-wrap>Hangzhou, China</aff><aff id="Aff2"><label>2</label><institution-wrap><institution-id institution-id-type="GRID">grid.221309.b</institution-id><institution-id institution-id-type="ISNI">0000000417645980</institution-id><institution>Department of Computer Science,</institution><institution>Hong Kong Baptist University,</institution></institution-wrap>Kowloon Tong, Hong Kong</aff><aff id="Aff3"><label>3</label><institution-wrap><institution-id institution-id-type="GRID">grid.221309.b</institution-id><institution-id institution-id-type="ISNI">0000000417645980</institution-id><institution>Department of Computer Science & Institute of Theoretical and Computational Study,</institution><institution>Hong Kong Baptist University,</institution></institution-wrap>Kowloon Tong, Hong Kong</aff></contrib-group><pub-date pub-type="epub"><day>22</day><month>12</month><year>2016</year></pub-date><pub-date pub-type="pmc-release"><day>22</day><month>12</month><year>2016</year></pub-date><pub-date pub-type="collection"><year>2016</year></pub-date><volume>5</volume><elocation-id>107</elocation-id><history><date date-type="received"><day>21</day><month>12</month><year>2015</year></date><date date-type="accepted"><day>10</day><month>10</month><year>2016</year></date></history><permissions><copyright-statement>© The Author(s) 2016</copyright-statement><license license-type="OpenAccess"><license-p><bold>Open Access</bold> This article is distributed under the terms of the Creative Commons Attribution 4.0 International 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>), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(<ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/publicdomain/zero/1.0/">http://creativecommons.org/publicdomain/zero/1.0/</ext-link>) applies to the data made available in this article, unless otherwise stated.</license-p></license></permissions><abstract id="Abs1"><sec><title>Background</title><p>Infectious diseases such as SARS and H1N1 can significantly impact people’s lives and cause severe social and economic damages. Recent outbreaks have stressed the urgency of effective research on the dynamics of infectious disease spread. However, it is difficult to predict when and where outbreaks may emerge and how infectious diseases spread because many factors affect their transmission, and some of them may be unknown.</p></sec><sec><title>Methods</title><p>One feasible means to promptly detect an outbreak and track the progress of disease spread is to implement surveillance systems in regional or national health and medical centres. The accumulated surveillance data, including temporal, spatial, clinical, and demographic information can provide valuable information that can be exploited to better understand and model the dynamics of infectious disease spread. The aim of this work is to develop and empirically evaluate a stochastic model that allows the investigation of transmission patterns of infectious diseases in heterogeneous populations.</p></sec><sec><title>Results</title><p>We test the proposed model on simulation data and apply it to the surveillance data from the 2009 H1N1 pandemic in Hong Kong. In the simulation experiment, our model achieves high accuracy in parameter estimation (less than 10.0 <italic>%</italic> mean absolute percentage error). In terms of the forward prediction of case incidence, the mean absolute percentage errors are 17.3 <italic>%</italic> for the simulation experiment and 20.0 <italic>%</italic> for the experiment on the real surveillance data.</p></sec><sec><title>Conclusion</title><p>We propose a stochastic model to study the dynamics of infectious disease spread in heterogeneous populations from temporal-spatial surveillance data. The proposed model is evaluated using both simulated data and the real data from the 2009 H1N1 epidemic in Hong Kong and achieves acceptable prediction accuracy. We believe that our model can provide valuable insights for public health authorities to predict the effect of disease spread and analyse its underlying factors and to guide new control efforts.</p></sec><sec><title>Electronic supplementary material</title><p>The online version of this article (doi:10.1186/s40249-016-0199-5) contains supplementary material, which is available to authorized users.</p></sec></abstract><kwd-group xml:lang="en"><title>Keywords</title><kwd>Epidemiology</kwd><kwd>Stochastic model</kwd><kwd>Surveillance system</kwd><kwd>Spread pattern</kwd></kwd-group><custom-meta-group><custom-meta><meta-name>issue-copyright-statement</meta-name><meta-value>© The Author(s) 2016</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="Sec1"><title>Multilingual abstracts</title><p>Please see Additional file <xref rid="MOESM1" ref-type="media">1</xref> for translations of the abstract into the five official working languages of the United Nations.</p></sec><sec id="Sec2"><title>Background</title><p>Infectious diseases remain a major cause of morbidity and mortality worldwide, triggering immeasurable loss in many societies. Most people may still have a fresh memory of the H1N1 outbreak in 2009, which brought pictures of empty streets and people wearing face masks and collectively caused at least 12799 deaths according to the World Health Organization (WHO) report [<xref ref-type="bibr" rid="CR1">1</xref>]. The H1N1 pandemic calls for research on accurately modelling the spread dynamics of an infectious disease, which offers a practically useful means for policy makers to evaluate the potential effects of intervention strategies [<xref ref-type="bibr" rid="CR2">2</xref>–<xref ref-type="bibr" rid="CR4">4</xref>].</p><p>Mathematical models of the spread of infectious diseases are an important tool for investigating and quantifying the spread dynamics because direct experimental study on the spread of disease among humans is not ethical. Although the subjects involved in different epidemics may be different, many can be modeled by the popular Susceptible-Infected-Recovered (SIR) models [<xref ref-type="bibr" rid="CR5">5</xref>–<xref ref-type="bibr" rid="CR7">7</xref>], which study the spread of infectious diseases by tracking the number (S) of people susceptible to the disease, the number (I) of people infected with the disease, and the number (R) of people who have recovered from the disease. Three assumptions are made: (1) the total population <italic>N</italic>=<italic>S</italic>(<italic>t</italic>)+<italic>I</italic>(<italic>t</italic>)+<italic>R</italic>(<italic>t</italic>) is fixed at any time <italic>t</italic>; (2) those who have recovered from the disease are forever immune; and (3) those who have not had the disease are equally susceptible, and the probability of their contracting the disease at time <italic>t</italic> is proportional to the product of <italic>S</italic>(<italic>t</italic>) and <italic>I</italic>(<italic>t</italic>). Based on these assumptions, the SIR model defines a set of three ordinary differential equations for S(t), I(t), and R(t):
<disp-formula id="Equ1"><label>1</label><alternatives><tex-math id="M1">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} {dS}/{dt} & = & - \beta S(t) I(t) \\ {dI}/{dt} & = & \beta S(t)I(t) - k I(t) \\ {dR}/{dt} & = & k I(t). \end{array} $$ \end{document}</tex-math><mml:math id="M2"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mtext mathvariant="italic">dS</mml:mtext><mml:mo>/</mml:mo><mml:mtext mathvariant="italic">dt</mml:mtext></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mo>−</mml:mo><mml:mi>βS</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi>I</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"><mml:mtext mathvariant="italic">dI</mml:mtext><mml:mo>/</mml:mo><mml:mtext mathvariant="italic">dt</mml:mtext></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mi>βS</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi>I</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mtext mathvariant="italic">kI</mml:mtext><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"><mml:mtext mathvariant="italic">dR</mml:mtext><mml:mo>/</mml:mo><mml:mtext mathvariant="italic">dt</mml:mtext></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mtext mathvariant="italic">kI</mml:mtext><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ1.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>Here, <italic>β</italic>≥0 is the effective transmission rate and <italic>k</italic>≥0 is the recovery rate. Because the SIR-based models are well presented in the literature, herein, we omit a verbose introduction of these models. Readers with an interest in such a topic can find the details in [<xref ref-type="bibr" rid="CR5">5</xref>–<xref ref-type="bibr" rid="CR7">7</xref>].</p><p>The SIR-based models and its variants have proven to be quite useful in the study of the spread dynamics of infectious diseases [<xref ref-type="bibr" rid="CR8">8</xref>–<xref ref-type="bibr" rid="CR10">10</xref>]. In [<xref ref-type="bibr" rid="CR11">11</xref>–<xref ref-type="bibr" rid="CR13">13</xref>], the progression of disease spread is characterized by tracking the number of <italic>S</italic><sub><italic>t</italic></sub> with a chain binomial model. The number of susceptible members <italic>S</italic><sub><italic>t</italic>+△<italic>t</italic></sub> (△<italic>t</italic> represents the infectious period of the disease and is always chosen to be 1/<italic>k</italic>) at time <italic>t</italic>+△<italic>t</italic> is a binomial random variable that depends on <italic>S</italic><sub><italic>t</italic></sub> and <italic>I</italic><sub><italic>t</italic></sub><italic>α</italic>, <italic>S</italic><sub><italic>t</italic>+△<italic>t</italic></sub>∼<italic>B</italic><italic>i</italic><italic>n</italic>(<italic>S</italic><sub><italic>t</italic></sub>,1−<italic>I</italic><sub><italic>t</italic></sub><italic>α</italic>), which provides a recursive relationship between <italic>S</italic><sub><italic>t</italic>+△<italic>t</italic></sub> and <italic>S</italic><sub><italic>t</italic></sub> and produces a formal stochastic process. However, the power of these models is mainly limited to uniform and homogeneous populations or populations with infinite size and homogeneous interactions. In many cases, the actual spread of infectious diseases occurs in a diverse or dispersed population. To study the spread of infectious diseases in heterogeneous populations, people usually divide a population into subpopulations that differ from each other. Sub-populations can be determined on the basis of social, cultural, economic, demographic, and geographic factors. Next, besides the dynamics of the internal spread within a subpopulation, the transmission dynamics between subpopulations should also be considered in the study of epidemic spreading.</p><p>Network-based epidemic modelling represents a popular approach for heterogeneous populations in which the nodes in the network correspond to sub-populations, and the links indicate the neighboring relationships. Many network-based models have been proposed, including patch models [<xref ref-type="bibr" rid="CR14">14</xref>–<xref ref-type="bibr" rid="CR16">16</xref>], distance-transmission models [<xref ref-type="bibr" rid="CR17">17</xref>], and multi-group models [<xref ref-type="bibr" rid="CR18">18</xref>, <xref ref-type="bibr" rid="CR19">19</xref>]. However, these models require knowledge of every individual (or host) and all relationships between individuals, which may be not achievable due to information privacy-related restrictions and the high cost of subject recruitment. To overcome the difficulties of collecting data, researchers have investigated several types of computer-generated networks in the context of disease spread in population-scale studies [<xref ref-type="bibr" rid="CR20">20</xref>–<xref ref-type="bibr" rid="CR24">24</xref>]. Grassberger first studied the dynamics of infectious diseases that propagate on regular networks using the percolation theory [<xref ref-type="bibr" rid="CR25">25</xref>]. Recent studies have revealed that many real-world networks, including social networks in which infectious diseases propagate, are either small-world [<xref ref-type="bibr" rid="CR26">26</xref>] or scale-free [<xref ref-type="bibr" rid="CR27">27</xref>] rather than regular or random, as thought previously [<xref ref-type="bibr" rid="CR28">28</xref>]. Because the underlying structures of networks will influence the effect that the dynamics of epidemics will have on them, researchers, such as Pastor-Satorras and Vespignani, have made many contributions to critical value analysis of typical epidemics on different types of complex network [<xref ref-type="bibr" rid="CR23">23</xref>, <xref ref-type="bibr" rid="CR24">24</xref>, <xref ref-type="bibr" rid="CR29">29</xref>]. On the basis of the mean-field theory, they found that compared with homogeneous networks, scale-free networks are fragile to the invasion of infectious diseases, computer viruses, or any other type of negative epidemics.</p><p>Epidemics have also been studied in various disciplines. Sociologists are concerned with the diffusion of rumors or innovation on social networks [<xref ref-type="bibr" rid="CR30">30</xref>]; economists have studied viral marketing and recommendation strategies by considering both cascading dynamics and the network effects of vital nodes [<xref ref-type="bibr" rid="CR31">31</xref>]; and computer scientists are interested in how some topics can quickly cascade in virtual blog spaces and how their propagation trends [<xref ref-type="bibr" rid="CR32">32</xref>, <xref ref-type="bibr" rid="CR33">33</xref>].</p><p>Although network-based studies have contributed to the modelling of disease and/or information dynamics, some models make a strong assumption that the structures of underlying networks over which epidemics spread are known beforehand. In the real world, however, the structures of underlying diffusion networks are not known directly. Many others assume the availability of information about the interactions occurring between individuals [<xref ref-type="bibr" rid="CR34">34</xref>–<xref ref-type="bibr" rid="CR37">37</xref>] that are often not valid in the context of disease spread. What may be obtained is only the time at which particular sub-populations become infected, but not how they become infected, nor how they affect their neighboring areas. Moreover, the underlying structures of networks will greatly influence the dynamics of infectious disease spread.</p><p>Since the emergence of the H1N1 influenza pandemic in April 2009, its underlying dynamics have been of great public health interest, and many approaches for its study have been proposed [<xref ref-type="bibr" rid="CR14">14</xref>, <xref ref-type="bibr" rid="CR38">38</xref>–<xref ref-type="bibr" rid="CR41">41</xref>]. Most of them are based on the classic SIR model. For example, Birrell et al. [<xref ref-type="bibr" rid="CR40">40</xref>] provided an age structure-based compartmental model with a Bayesian synthesis of multiple evidence sources to reveal substantial changes in contact patterns throughout the epidemic. Besides of the compartmental models, other mathematical models are also used to describe the transmission dynamics [<xref ref-type="bibr" rid="CR3">3</xref>, <xref ref-type="bibr" rid="CR42">42</xref>–<xref ref-type="bibr" rid="CR47">47</xref>]. The chain binomial model was used to calculate the household secondary attack rates to measure the transmissibility of the 2009 H1N1 influenza pandemic by Lessler et al. [<xref ref-type="bibr" rid="CR44">44</xref>] and Klick et al. [<xref ref-type="bibr" rid="CR45">45</xref>]. Yang et al. [<xref ref-type="bibr" rid="CR46">46</xref>] constructed a model based on chains of infections and used the infection hazard function and survival function to study the 2009 H1N1 influenza pandemic. Ferguson et al. [<xref ref-type="bibr" rid="CR3">3</xref>] and Cauchemez et al. [<xref ref-type="bibr" rid="CR42">42</xref>, <xref ref-type="bibr" rid="CR43">43</xref>] incorporated other factors, such as household risk, within-school risk, and community risk, in the study of infection spread and found out that younger age groups under 19 years old were more susceptible than older age groups. Jin et al. [<xref ref-type="bibr" rid="CR47">47</xref>] formulated an epidemic model of influenza A based on networks and calculated the basic reproduction number and studied the effects of various immunization schemes. However, this work required that the individual contact pattern be provided. Nonetheless, none of the aforementioned approaches takes spatial heterogeneity into consideration in the study of disease spread.</p><p>Recently, an outbreak of Ebola virus disease (EVD) swept across parts of West Africa from March 2014 to April 2015. By June 10, 2015, WHO had reported 27,237 confirmed, probable, or suspected cases in three countries with 11,158 deaths [<xref ref-type="bibr" rid="CR48">48</xref>]. This epidemic received extensive research attention on its dynamics of spread [<xref ref-type="bibr" rid="CR49">49</xref>–<xref ref-type="bibr" rid="CR57">57</xref>] (for further references in the review article [<xref ref-type="bibr" rid="CR58">58</xref>]). To name a few, Chowell et al. found that district-level Ebola virus disease outbreaks in West Africa follow polynomial-based growth in time instead of the exponential growth that describes the progress of many infectious disease epidemics [<xref ref-type="bibr" rid="CR52">52</xref>]. Fisman et al. used a simple, two parameter mathematical model to characterize epidemic growth patterns in the 2014 Ebola outbreak [<xref ref-type="bibr" rid="CR53">53</xref>]. Webb et al. proposed a variant of the classic SIR model with three extra groups, incubating, contaminated and isolated, which can provide a more accurate prediction for the future incidences [<xref ref-type="bibr" rid="CR56">56</xref>]. Carroll et al. used a deep sequencing approach to gain insight into the evolution of the Ebola virus (EBOV) in Guinea from the ongoing West African outbreak. The viral sequence data can be combined with epidemiological information to retrospectively test the effectiveness of control measures, and provides an unprecedented window into the evolution of an ongoing outbreak of viral haemorrhagic fever [<xref ref-type="bibr" rid="CR57">57</xref>].</p><p>To accurately predict when and where outbreaks will occur, a feasible means is to deploy manual or electronic surveillance systems through regional or national public health and medical organizations [<xref ref-type="bibr" rid="CR59">59</xref>]. Most of the surveillance data accumulated from such systems contains temporal, spatial, clinical, and demographic information. For instance, Telehealth Ontario is a teletriage helpline that is available free to all Ontario residents, which allows those with suspected infections to connect with experts who can assess their symptoms. The records of such calls provide valuable information on which individual from where was possibly infected and by which type of disease at what time. In this paper, we address the problem of modelling disease spread dynamics in heterogeneous populations from temporal-spatial surveillance data. We analyse the role of heterogeneity in a stochastic epidemic model on a two-dimensional lattice. Within a particular sub-population, the speed of spread is controlled by a single parameter, the transmissibility of the pathogen between individuals. Between sub-populations, the transmissibility becomes a random variable drawn from a probability distribution. Our work differs from existing studies in some fundamental ways, in light of the unique nature of infectious disease diffusion dynamics. Our results have practical implications for the analysis of disease control strategies in realistic heterogeneous epidemic systems.</p></sec><sec id="Sec3"><title>Methods</title><p>In this work, we propose a stochastic model to study the dynamics of infectious disease spread in heterogeneous populations from temporal-spatial surveillance data. We divide the whole population into <italic>m</italic> sub-populations on the basis of geographic regions. In the following, we use <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>),<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>), and <italic>R</italic><sub><italic>i</italic></sub>(<italic>t</italic>) to denote the number of susceptible, infected, and recovered people, respectively, at time <italic>t</italic> in region <italic>i</italic>, <italic>i</italic>=1,2,⋯,<italic>m</italic> and <italic>t</italic>∈[0,<italic>T</italic>].</p><sec id="Sec4"><title>Stochastic model</title><p>Classic SIR-based modelling of infectious diseases assumes that the population is well-mixed. To take the role of heterogeneity into consideration, we use an alternative approach to model the dynamics of infectious disease spread. First, the classic SIR model (Eq. (<xref rid="Equ1" ref-type="">1</xref>)) studies the change in the numbers of peoples in the three groups. In reality, the change in the number of the infected people is the major concern of society. Second, in many epidemics or pandemics such as H1N1 and SARS, the number of infected people <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>) is relatively small compared to the whole subpopulation <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>). Therefore, we may consider <italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>) as a constant to simplify the modelling of the change in the number of infected people <italic>I</italic>(<italic>t</italic>), for which we propose the following stochastic differential equation:
<disp-formula id="Equ2"><label>2</label><alternatives><tex-math id="M3">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \mathrm{d}I_{i}(t)=(\alpha+\delta_{i}I_{i}(t))\mathrm{d}t+\sigma_{i}\mathrm{d}B_{i}(t), \end{array} $$ \end{document}</tex-math><mml:math id="M4"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ2.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>where <italic>α</italic> is a parameter that measures the auto-recovery rate of one particular infectious disease, which is usually considered as a constant among sub-populations, <italic>δ</italic><sub><italic>i</italic></sub> is the parameter that measures the different disease transmissibility in different subpopulations, <italic>σ</italic><sub><italic>i</italic></sub>>0 is the diffusion parameter that measures the disease spread from neighbors, and <italic>B</italic><sub><italic>i</italic></sub>(<italic>t</italic>) is a standard Brownian motion. It is worth noting that we assume the parameter <italic>δ</italic><sub><italic>i</italic></sub>≠0 for technical purposes, and the results in the case of <italic>δ</italic><sub><italic>i</italic></sub>=0 can be achieved with <italic>δ</italic><sub><italic>i</italic></sub>→0.</p><p>Comparing our model in Eq. (<xref rid="Equ2" ref-type="">2</xref>) with the classic model in Eq. (<xref rid="Equ1" ref-type="">1</xref>), we can see that they both capture the situation in which the change in the number of infected people has a positive relationship with the total number of infected people, which means that the more infected people there are, the more people will get infected. There are two key differences between these two models: first, the key factor (<italic>β</italic><italic>S</italic><sub><italic>i</italic></sub>(<italic>t</italic>)−<italic>k</italic>) associated with the disease spread in Eq. (<xref rid="Equ1" ref-type="">1</xref>) is replaced with a single parameter <italic>δ</italic><sub><italic>i</italic></sub> in Eq. (<xref rid="Equ2" ref-type="">2</xref>), which can be used to analyse the role of heterogeneity in the disease spread; and second, Eq. (<xref rid="Equ2" ref-type="">2</xref>) takes the neighboring relationships into consideration to study the dynamics of the disease spread among different sub-populations.</p><p>By Ito formula, the solution of Eq. (<xref rid="Equ2" ref-type="">2</xref>) is given by
<disp-formula id="Equ3"><label>3</label><alternatives><tex-math id="M5">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} &I_{i}(t)=I_{i}(0)e^{\delta_{i}t}+\frac{\alpha}{\delta_{i}}(e^{\delta_{i}t}-1)\\ &\quad+\sigma_{i} e^{\delta_{i}t}{\int_{0}^{t}}e^{-\delta_{i}s}\mathrm{d}B_{i}(s). \end{array} $$ \end{document}</tex-math><mml:math id="M6"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"><mml:mspace width="1em"></mml:mspace><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:munderover><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mi>.</mml:mi></mml:mtd><mml:mtd class="eqnarray-3"></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ3.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>Notice that for any fixed <italic>t</italic>, <inline-formula id="IEq1"><alternatives><tex-math id="M7">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}${\int _{0}^{t}}e^{-\delta _{i}s}\mathrm {d}B_{i}(s)$\end{document}</tex-math><mml:math id="M8"><mml:munderover><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula> is a normal random variable with
<disp-formula id="Equ4"><label>4</label><alternatives><tex-math id="M9">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} E\left[\int_{0}^{t}e^{-\delta_{i}s}\mathrm{d}B_{i}(s)\right]=0, \\ Var\left[\int_{0}^{t}e^{-\delta_{i}s}\mathrm{d}B_{i}(s)\right]=\frac{1-e^{-2\delta_{i}t}}{2\delta_{i}}. \end{array} $$ \end{document}</tex-math><mml:math id="M10"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mi>E</mml:mi><mml:mfenced close="]" open="[" separators=""><mml:mrow><mml:munderover><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"><mml:mtext mathvariant="italic">Var</mml:mtext><mml:mfenced close="]" open="[" separators=""><mml:mrow><mml:munderover><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ4.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>Thus, for any fixed <italic>t</italic>, <italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>) is a normal random variable with
<disp-formula id="Equ5"><label>5</label><alternatives><tex-math id="M11">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} E[I_{i}(t)]=I_{i}(0)e^{\delta_{i}t}+\frac{\alpha}{\delta_{i}}\left(e^{\delta_{i}t}-1\right) \end{array} $$ \end{document}</tex-math><mml:math id="M12"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mi>E</mml:mi><mml:mo>[</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>]</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfenced close=")" open="(" separators=""><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ5.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>and
<disp-formula id="Equ6"><label>6</label><alternatives><tex-math id="M13">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} Var[I_{i}(t)]=\frac{{\sigma_{i}^{2}}}{2\delta_{i}}\left(e^{2\delta_{i}t}-1\right). \end{array} $$ \end{document}</tex-math><mml:math id="M14"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mtext mathvariant="italic">Var</mml:mtext><mml:mo>[</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>]</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfenced close=")" open="(" separators=""><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ6.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>There are three cases of being interested for parameter <italic>α</italic>:
<list list-type="bullet"><list-item><p><italic>α</italic>>−<italic>I</italic><sub><italic>i</italic></sub>(0)<italic>δ</italic><sub><italic>i</italic></sub></p><p>In this case, <italic>E</italic>[<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>)] tends to infinity as <italic>t</italic> goes to infinity, which implies that all people in that region will be infected if the time is long enough.</p></list-item><list-item><p><italic>α</italic>=−<italic>I</italic><sub><italic>i</italic></sub>(0)<italic>δ</italic><sub><italic>i</italic></sub></p><p>In this case, the pandemic or epidemic will reach a state of equilibrium.</p></list-item><list-item><p><italic>α</italic><−<italic>I</italic><sub><italic>i</italic></sub>(0)<italic>δ</italic><sub><italic>i</italic></sub></p><p>In this case, <italic>E</italic>[<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>)] will reach 0 at some time <inline-formula id="IEq2"><alternatives><tex-math id="M15">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$t = \hat {t}$\end{document}</tex-math><mml:math id="M16"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula> and go to negative infinity as <italic>t</italic> goes to infinity, which implies the pandemic or epidemic will end at time <inline-formula id="IEq3"><alternatives><tex-math id="M17">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\hat {t}$\end{document}</tex-math><mml:math id="M18"><mml:mover accent="true"><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula>.</p></list-item></list></p></sec><sec id="Sec5"><title>Parameter estimation</title><p>To estimate the parameters in our proposed stochastic model from the surveillance data, we need to divide the interval [0,<italic>T</italic>] into <italic>n</italic> subintervals, [<italic>t</italic><sub>0</sub>,<italic>t</italic><sub>1</sub>], [<italic>t</italic><sub>1</sub>,<italic>t</italic><sub>2</sub>], ⋯, [<italic>t</italic><sub><italic>n</italic>−1</sub>,<italic>t</italic><sub><italic>n</italic></sub>], where 0=<italic>t</italic><sub>0</sub><<italic>t</italic><sub>1</sub><<italic>t</italic><sub>2</sub><⋯<<italic>t</italic><sub><italic>n</italic></sub>=<italic>T</italic>. Denote △<italic>t</italic>(<italic>k</italic>)=<italic>t</italic><sub><italic>k</italic>+1</sub>−<italic>t</italic><sub><italic>k</italic></sub>, △<italic>B</italic><sub><italic>i</italic></sub>(<italic>k</italic>)=<italic>B</italic><sub><italic>i</italic></sub>(<italic>t</italic><sub><italic>k</italic>+1</sub>)−<italic>B</italic><sub><italic>i</italic></sub>(<italic>t</italic><sub><italic>k</italic></sub>), △<italic>I</italic><sub><italic>i</italic></sub>(<italic>k</italic>)=<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic><sub><italic>k</italic>+1</sub>)−<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic><sub><italic>k</italic></sub>), <italic>k</italic>=0,1,⋯,<italic>n</italic>−1. Then Eq. (<xref rid="Equ2" ref-type="">2</xref>) is rewritten as
<disp-formula id="Equ7"><label>7</label><alternatives><tex-math id="M19">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \triangle I_{i}(k)=(\alpha+\delta_{i}I_{i}(t_{k}))\triangle t(k)+\sigma_{i}\triangle B_{i}(k). \end{array} $$ \end{document}</tex-math><mml:math id="M20"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mo>△</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>△</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ7.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>It is easy to see that <inline-formula id="IEq4"><alternatives><tex-math id="M21">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\triangle I_{i}(k)|I_{i}(t_{k})\sim N\big ((\alpha +\delta _{i}I_{i}(t_{k}))\triangle t(k),{\sigma _{i}^{2}}\triangle t(k)\big)$\end{document}</tex-math><mml:math id="M22"><mml:mo>△</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mstyle mathsize="1.19em"><mml:mfenced close="" open="(" separators=""><mml:mrow></mml:mrow></mml:mfenced></mml:mstyle><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mstyle mathsize="1.19em"><mml:mfenced close="" open=")" separators=""><mml:mrow></mml:mrow></mml:mfenced></mml:mstyle></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula>. Let <italic>θ</italic><sub><italic>i</italic></sub>=(<italic>α</italic>,<italic>δ</italic><sub><italic>i</italic></sub>,<italic>σ</italic><sub><italic>i</italic></sub>). Then the transition density of the process {<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>);<italic>t</italic>≥0} is
<disp-formula id="Equ8"><label>8</label><alternatives><tex-math id="M23">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} &&p_{\theta_{i}}(s+t,y|s,x)\\ &=&\frac{1}{\sqrt{2\pi t{\sigma_{i}^{2}}}}\exp\left\{-\frac{(y-x-(\alpha+\delta_{i}x)t)^{2}}{2t{\sigma_{i}^{2}}}\right\}. \end{array} $$ \end{document}</tex-math><mml:math id="M24"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>|</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>πt</mml:mi><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mo>exp</mml:mo><mml:mfenced close="}" open="{" separators=""><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mo>−</mml:mo><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>t</mml:mi><mml:munderover><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:munderover></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ8.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>Hence, the likelihood function is given by
<disp-formula id="Equ9"><label>9</label><alternatives><tex-math id="M25">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} &&f(\theta_{i}|I_{i})\triangleq f\left(\theta_{i}|I_{i}(t_{k}),0\leq k\leq n-1\right)\\&=&I_{i}(0)\left(\frac{1}{2{\pi\sigma_{i}^{2}}}\right)^{\frac{n}{2}}\prod\limits_{k=0}^{n-1}\frac{1}{\triangle t(k)} \\ &&\exp\left\{-\frac{(\triangle I_{i}(k)-(\alpha+\delta_{i}I_{i}(t_{k}))\triangle t(k))^{2}}{2\triangle t(k){\sigma_{i}^{2}}}\right\}.\\ \end{array} $$ \end{document}</tex-math><mml:math id="M26"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>≜</mml:mo><mml:mi>f</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mrow><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:msup><mml:mrow><mml:mfenced close=")" open="(" separators=""><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∏</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfrac></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:mo>exp</mml:mo><mml:mfenced close="}" open="{" separators=""><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>△</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:munderover><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:munderover></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mi>.</mml:mi></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ9.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>Consequently, the log-likelihood function is
<disp-formula id="Equ10"><label>10</label><alternatives><tex-math id="M27">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} &&\log f(\theta_{i}|I_{i})\varpropto-\frac{n}{2}{\log\sigma_{i}^{2}}\\ &&-\sum\limits_{k=0}^{n-1}\frac{\left(\triangle I_{i}(k)-(\alpha+\delta_{i}I_{i}(t_{k}))\triangle t(k)\right)^{2}}{2\triangle t(k){\sigma_{i}^{2}}}. \end{array} $$ \end{document}</tex-math><mml:math id="M28"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:mo>log</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>∝</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>log</mml:mo><mml:munderover><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:munderover></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:mo>−</mml:mo><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mfenced close=")" open="(" separators=""><mml:mrow><mml:mo>△</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ10.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>Let
<disp-formula id="Equ11"><label>11</label><alternatives><tex-math id="M29">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i1}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}I_{i}(t_{k}), \end{array} $$ \end{document}</tex-math><mml:math id="M30"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ11.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ12"><label>12</label><alternatives><tex-math id="M31">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i2}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}I_{i}(t_{k+1}), \end{array} $$ \end{document}</tex-math><mml:math id="M32"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ12.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ13"><label>13</label><alternatives><tex-math id="M33">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i11}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}{I_{i}^{2}}(t_{k}), \end{array} $$ \end{document}</tex-math><mml:math id="M34"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ13.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ14"><label>14</label><alternatives><tex-math id="M35">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i12}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}I_{i}(t_{k})I_{i}(t_{k+1}), \end{array} $$ \end{document}</tex-math><mml:math id="M36"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>12</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ14.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ15"><label>15</label><alternatives><tex-math id="M37">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i1\Delta}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}I_{i}(t_{k})\triangle t(k), \end{array} $$ \end{document}</tex-math><mml:math id="M38"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>Δ</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ15.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ16"><label>16</label><alternatives><tex-math id="M39">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i11\Delta}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}{I_{i}^{2}}(t_{k})\triangle t(k), \end{array} $$ \end{document}</tex-math><mml:math id="M40"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>11</mml:mn><mml:mi>Δ</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ16.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ17"><label>17</label><alternatives><tex-math id="M41">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i11\Delta^{-1}}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}{I_{i}^{2}}(t_{k})(\triangle t(k))^{-1}, \end{array} $$ \end{document}</tex-math><mml:math id="M42"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>11</mml:mn><mml:msup><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ17.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ18"><label>18</label><alternatives><tex-math id="M43">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i12\Delta^{-1}}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}I_{i}(t_{k})I_{i}(t_{k+1})(\triangle t(k))^{-1}, \end{array} $$ \end{document}</tex-math><mml:math id="M44"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>12</mml:mn><mml:msup><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ18.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ19"><label>19</label><alternatives><tex-math id="M45">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} u_{i22\Delta^{-1}}&=&\frac{1}{T}\sum\limits_{k=0}^{n-1}{I_{i}^{2}}(t_{k+1})(\triangle t(k))^{-1}. \end{array} $$ \end{document}</tex-math><mml:math id="M46"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>22</mml:mn><mml:msup><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ19.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>We have the estimator of <italic>θ</italic><sub><italic>i</italic></sub> as follows:
<disp-formula id="Equ20"><label>20</label><alternatives><tex-math id="M47">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \widehat{\delta}_{i}&=&\frac{u_{i12}-u_{i11}-u_{i2}u_{i1\Delta}+u_{i1}u_{i1\Delta}}{u_{i11\Delta}-u_{i1\Delta}^{2}}, \end{array} $$ \end{document}</tex-math><mml:math id="M48"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>Δ</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>Δ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>11</mml:mn><mml:mi>Δ</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>Δ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ20.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ21"><label>21</label><alternatives><tex-math id="M49">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \widehat{\alpha}&=&u_{i2}-u_{i1}-\widehat{\delta}_{i}u_{i1\Delta}, \end{array} $$ \end{document}</tex-math><mml:math id="M50"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mover accent="false"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>Δ</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ21.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ22"><label>22</label><alternatives><tex-math id="M51">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \widehat{{\sigma^{2}_{i}}}&=&Tn^{-1}\left\{u_{i22\Delta^{-1}}-2u_{i12\Delta^{-1}}+u_{i11\Delta^{-1}}\right.\\ && -(u_{i2}-u_{i1})^{2}+\left[({u_{i2}}-{u_{i1}})u_{i1\Delta}\right.\\ && \left.\left.-(u_{i12}-u_{i11})\right]\widehat{\delta}_{i}\right\}. \end{array} $$ \end{document}</tex-math><mml:math id="M52"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mover accent="false"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mi>T</mml:mi><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close="" open="{" separators=""><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>22</mml:mn><mml:msup><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>12</mml:mn><mml:msup><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>11</mml:mn><mml:msup><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mfenced close="" open="[" separators=""><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>Δ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfenced close="}" open="" separators=""><mml:mrow><mml:mfenced close="]" open="" separators=""><mml:mrow><mml:mo>−</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mfenced><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ22.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>It is obviously to see that <inline-formula id="IEq5"><alternatives><tex-math id="M53">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\widehat {\alpha }$\end{document}</tex-math><mml:math id="M54"><mml:mover accent="false"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq5.gif"></inline-graphic></alternatives></inline-formula> is not a bona fide estimator of <italic>α</italic>, because only the information of {<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>);0≤<italic>t</italic>≤<italic>T</italic>} is used to estimate <italic>α</italic>. A good estimator should pool all the information {<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>);0≤<italic>t</italic>≤<italic>T</italic>} (<italic>i</italic>=1,2,⋯,<italic>m</italic>). There are two ways to find the pool estimator. The first way is to approximate <italic>α</italic> by pooling all <inline-formula id="IEq6"><alternatives><tex-math id="M55">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\widehat {\alpha }_{i}$\end{document}</tex-math><mml:math id="M56"><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq6.gif"></inline-graphic></alternatives></inline-formula> as follows:
<disp-formula id="Equ23"><label>23</label><alternatives><tex-math id="M57">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \widehat{\alpha}&=&m^{-1}\sum\limits_{i=1}^{m}\widehat{\alpha_{i}}, \end{array} $$ \end{document}</tex-math><mml:math id="M58"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mover accent="false"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:munderover><mml:mover accent="false"><mml:mrow><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>^</mml:mo></mml:mover><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ23.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ24"><label>24</label><alternatives><tex-math id="M59">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \widehat{\alpha_{i}}&=& u_{i2}-u_{i1}-\widehat{\delta}_{i}u_{i1\Delta}. \end{array} $$ \end{document}</tex-math><mml:math id="M60"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mover accent="false"><mml:mrow><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>Δ</mml:mi></mml:mrow></mml:msub><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ24.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>But the issue in Eq. (<xref rid="Equ23" ref-type="">23</xref>) is that <italic>m</italic> must be very large in order to achieve the accurate estimate of <italic>α</italic>. In this work, we choose the second way, which is the maximum likelihood estimation. To do so, we need to assume that the processes {<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>);0≤<italic>t</italic>≤<italic>T</italic>} (<italic>i</italic>=1,2,⋯,<italic>m</italic>) are mutually independent. Then the log-likelihood function of {<italic>I</italic><sub><italic>i</italic></sub>(<italic>t</italic>);0≤<italic>t</italic>≤<italic>T</italic>} (<italic>i</italic>=1,2,⋯,<italic>m</italic>) is given by
<disp-formula id="Equ25"><label>25</label><alternatives><tex-math id="M61">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} &&\sum\limits_{i=1}^{m}\log f(\alpha|I_{i})\varpropto\\ &&-\sum\limits_{i=1}^{m}\sum\limits_{k=0}^{n-1}\frac{(\triangle I_{i}(k)-(\alpha+\widehat{\delta}_{i}I_{i}(t_{k}))\triangle t(k))^{2}}{2\triangle t(k)\widehat{{\sigma_{i}^{2}}}}. \end{array} $$ \end{document}</tex-math><mml:math id="M62"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:munderover><mml:mo>log</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>∝</mml:mo></mml:mtd><mml:mtd><mml:mtext></mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd class="eqnarray-1"></mml:mtd><mml:mtd class="eqnarray-2"></mml:mtd><mml:mtd class="eqnarray-3"><mml:mo>−</mml:mo><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:munderover><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>△</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mover accent="false"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow></mml:mfrac><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ25.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>The maximum likelihood estimate is
<disp-formula id="Equ26"><label>26</label><alternatives><tex-math id="M63">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \widetilde{\alpha}=\sum\limits_{i=1}^{m}\omega_{i}\widehat{\alpha}_{i}, \end{array} $$ \end{document}</tex-math><mml:math id="M64"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:mover accent="false"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:munderover><mml:msub><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ26.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>where
<disp-formula id="Equ27"><label>27</label><alternatives><tex-math id="M65">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} \omega_{j}&=&\frac{\widehat{{\sigma_{j}^{2}}}^{-1}}{\sum\limits_{i=1}^{m}\sum\limits_{k=0}^{n-1}\widehat{{\sigma_{i}^{2}}}^{-1}},\,\,j=1,2,\cdots,m. \end{array} $$ \end{document}</tex-math><mml:math id="M66"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd class="eqnarray-2"><mml:mo>=</mml:mo></mml:mtd><mml:mtd class="eqnarray-3"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mover accent="false"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:munderover><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mover accent="false"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width="0.3em"></mml:mspace><mml:mspace width="0.3em"></mml:mspace><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mspace width="0.3em"></mml:mspace><mml:mo>,</mml:mo><mml:mi>m.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ27.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><inline-formula id="IEq7"><alternatives><tex-math id="M67">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\widehat {\alpha }_{i}$\end{document}</tex-math><mml:math id="M68"><mml:msub><mml:mrow><mml:mover accent="false"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq7.gif"></inline-graphic></alternatives></inline-formula> is defined in Eq. (<xref rid="Equ24" ref-type="">24</xref>).</p></sec></sec><sec id="Sec6"><title>Results and discussion</title><p>In this section, we illustrate the performance of our proposed model using both simulated and real data.</p><sec id="Sec7"><title>Simulation study</title><p>In the simulation study, we examine the performance of our proposed model with respect to the accuracy of parameter estimation and the forward prediction of the case incidence. First, we generate data using various parameters by the following steps:
<list list-type="bullet"><list-item><p>Set <italic>m</italic>=4 (the number of sub-populations) and <italic>T</italic>=100 (the number of time slots). These two numbers are randomly selected.</p></list-item><list-item><p>Randomly draw <italic>α</italic> from [0.05,0.09], <italic>δ</italic><sub><italic>i</italic></sub> from [0.02,0.08], and <italic>σ</italic><sub><italic>i</italic></sub> from [0.02,0.08].</p></list-item><list-item><p>Initialize <italic>I</italic><sub><italic>i</italic></sub>(0),1≤<italic>i</italic>≤<italic>m</italic>.</p></list-item><list-item><p>Simulate <italic>I</italic><sub><italic>i</italic></sub>(<italic>k</italic>+1)=<italic>I</italic><sub><italic>i</italic></sub>(<italic>k</italic>)+△<italic>I</italic><sub><italic>i</italic></sub>(<italic>k</italic>),<italic>k</italic>=0,1,⋯,<italic>T</italic>−1 using Eq. (<xref rid="Equ7" ref-type="">7</xref>) and <inline-formula id="IEq8"><alternatives><tex-math id="M69">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\triangle I_{i}(k)|I_{i}(t_{k})\sim N\left ((\alpha +\delta _{i}I_{i}(t_{k}))\triangle t(k),{\sigma _{i}^{2}}\triangle t(k)\right)$\end{document}</tex-math><mml:math id="M70"><mml:mo>△</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>△</mml:mo><mml:mi>t</mml:mi><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfenced></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq8.gif"></inline-graphic></alternatives></inline-formula>.</p></list-item></list></p><p>Three parameters, <italic>α</italic>,<italic>δ</italic><sub><italic>i</italic></sub>, and <italic>σ</italic><sub><italic>i</italic></sub>, in Eq. (<xref rid="Equ2" ref-type="">2</xref>) will be estimated from the simulated data. We conduct 100 replicates by repeating Step 2–4 and compare the estimated ones, <inline-formula id="IEq9"><alternatives><tex-math id="M71">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\hat {\alpha }, \hat {\delta _{i}},$\end{document}</tex-math><mml:math id="M72"><mml:mover accent="true"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>^</mml:mo></mml:mover><mml:mo>,</mml:mo></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq9.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq10"><alternatives><tex-math id="M73">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\hat {\sigma _{i}}$\end{document}</tex-math><mml:math id="M74"><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq10.gif"></inline-graphic></alternatives></inline-formula>, with the ground truth values in terms of the mean absolute percentage error (MAPE) defined as:
<disp-formula id="Equ28"><label>28</label><alternatives><tex-math id="M75">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} E_{\alpha} = \frac{1}{100}\sum\limits_{j=1}^{100}\left|\frac{\hat{\alpha}_{j} - \alpha_{j}}{\alpha_{j}}\right|, \end{array} $$ \end{document}</tex-math><mml:math id="M76"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn></mml:mrow></mml:munderover><mml:mfenced close="|" open="|" separators=""><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ28.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ29"><label>29</label><alternatives><tex-math id="M77">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} E_{\delta} = \frac{1}{100*m}\sum\limits_{i=0}^{m-1}\sum\limits_{j=1}^{100}\left|\frac{\hat{\delta}_{ij} - \delta_{ij}}{\delta_{ij}}\right|, \end{array} $$ \end{document}</tex-math><mml:math id="M78"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>δ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn><mml:mo>∗</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn></mml:mrow></mml:munderover><mml:mfenced close="|" open="|" separators=""><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ29.gif" position="anchor"></graphic></alternatives></disp-formula></p><p><disp-formula id="Equ30"><label>30</label><alternatives><tex-math id="M79">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} E_{\sigma} = \frac{1}{100*m}\sum\limits_{i=0}^{m-1}\sum\limits_{j=1}^{100}\left|\frac{\hat{\sigma}_{ij} - \sigma_{ij}}{\sigma_{ij}}\right|. \end{array} $$ \end{document}</tex-math><mml:math id="M80"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn><mml:mo>∗</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn></mml:mrow></mml:munderover><mml:mfenced close="|" open="|" separators=""><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ30.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>The mean absolute percentage errors (MAPEs) for <italic>E</italic><sub><italic>α</italic></sub>, <italic>E</italic><sub><italic>δ</italic></sub>, and <italic>E</italic><sub><italic>σ</italic></sub> are 10.0 <italic>%</italic>, 6.0 <italic>%</italic>, and 10.0 <italic>%</italic>, respectively. We plot the distribution of the estimated errors for 100 replicates for <inline-formula id="IEq11"><alternatives><tex-math id="M81">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\hat {\alpha }, \hat {\delta _{i}},$\end{document}</tex-math><mml:math id="M82"><mml:mover accent="true"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>^</mml:mo></mml:mover><mml:mo>,</mml:mo></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq11.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq12"><alternatives><tex-math id="M83">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\hat {\sigma _{i}}$\end{document}</tex-math><mml:math id="M84"><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq12.gif"></inline-graphic></alternatives></inline-formula> in Fig. <xref rid="Fig1" ref-type="fig">1</xref>. From Fig. <xref rid="Fig1" ref-type="fig">1</xref>, we can see that both the estimates of <inline-formula id="IEq13"><alternatives><tex-math id="M85">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\hat {\alpha }$\end{document}</tex-math><mml:math id="M86"><mml:mover accent="true"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq13.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq14"><alternatives><tex-math id="M87">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\hat {\sigma _{i}}$\end{document}</tex-math><mml:math id="M88"><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq14.gif"></inline-graphic></alternatives></inline-formula> have small variations. The variation of the estimate of <inline-formula id="IEq15"><alternatives><tex-math id="M89">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\hat {\delta _{i}}$\end{document}</tex-math><mml:math id="M90"><mml:mover accent="true"><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="40249_2016_199_Article_IEq15.gif"></inline-graphic></alternatives></inline-formula> is slightly larger but is still acceptable and is due to the uncertainty embedded in the stochastic process. We also use the estimated values of the parameters to generate the data and compare it with the simulated data using the ground truth values of the parameters. The correlation between them is 0.96. We randomly select one replicate and show the comparison results in Fig. <xref rid="Fig2" ref-type="fig">2</xref>. Basically, we can use the estimated parameters to accurately recover the ground truth data.
<fig id="Fig1"><label>Fig. 1</label><caption><p>The performance of parameter estimation. The mean absolute percentage errors for <italic>α</italic>, <italic>δ</italic>, and <italic>σ</italic> are 10.0, 6.0, and 10.0 <italic>%</italic>, respectively. <italic>α</italic> measures the auto-recovery rate of one particular infectious disease. <italic>δ</italic> measures the disease transmissibility within the population. <italic>σ</italic> measures the disease transmissibility between populations</p></caption><graphic xlink:href="40249_2016_199_Fig1_HTML" id="MO1"></graphic></fig><fig id="Fig2"><label>Fig. 2</label><caption><p>The comparison of original data and estimated data for four regions. The x-axis represents the time in days. The y-axis represents the total number of confirmed cases. The The original data is generated using the ground truth values of parameters while the estimated data is generated with estimated values of parameters. The correlation between them is 0.96</p></caption><graphic xlink:href="40249_2016_199_Fig2_HTML" id="MO2"></graphic></fig></p><p>Next, we conduct an experiment to test the prediction accuracy of our model. Let us consider a sequence of data points <italic>I</italic><sub><italic>ij</italic></sub>(<italic>t</italic>) over a time interval [0,<italic>T</italic>] for the <italic>i</italic><sub><italic>th</italic></sub> subpopulation in the <italic>j</italic><sub><italic>th</italic></sub> replicate. We choose a time point <italic>s</italic> and use the data points <italic>I</italic><sub><italic>ij</italic></sub>[0],<italic>I</italic><sub><italic>ij</italic></sub>[1],⋯,<italic>I</italic><sub><italic>ij</italic></sub>[<italic>s</italic>] as the training data and predict the data points <italic>I</italic><sub><italic>ij</italic></sub>(<italic>t</italic>) for <italic>s</italic><<italic>t</italic>≤<italic>T</italic>. <italic>s</italic>=80 is chosen in this experiment. The MAPE of the prediction is defined as
<disp-formula id="Equ31"><label>31</label><alternatives><tex-math id="M91">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document} $$\begin{array}{@{}rcl@{}} E_{pre} = \frac{1}{100*20*m}\sum\limits_{j=1}^{100}\sum\limits_{i=0}^{m-1}\sum\limits_{s=81}^{100}\left|\frac{\hat{I}_{ij}[t] - I_{ij}[t]}{I_{ij}[t]}\right|. \end{array} $$ \end{document}</tex-math><mml:math id="M92"><mml:mtable class="eqnarray" columnalign="left center right"><mml:mtr><mml:mtd class="eqnarray-1"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">pre</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn><mml:mo>∗</mml:mo><mml:mn>20</mml:mn><mml:mo>∗</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn></mml:mrow></mml:munderover><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>81</mml:mn></mml:mrow><mml:mrow><mml:mn>100</mml:mn></mml:mrow></mml:munderover><mml:mfenced close="|" open="|" separators=""><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>Î</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub><mml:mo>[</mml:mo><mml:mi>t</mml:mi><mml:mo>]</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub><mml:mo>[</mml:mo><mml:mi>t</mml:mi><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mtext mathvariant="italic">ij</mml:mtext></mml:mrow></mml:msub><mml:mo>[</mml:mo><mml:mi>t</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mi>.</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math><graphic xlink:href="40249_2016_199_Article_Equ31.gif" position="anchor"></graphic></alternatives></disp-formula></p><p>The MAPE of the prediction is 17.3 <italic>%</italic>, which indicates that our model can achieve around 82.7 <italic>%</italic> accuracy in terms of the prediction. Again, we randomly select one replicate and show the prediction results in Fig. <xref rid="Fig3" ref-type="fig">3</xref>.
<fig id="Fig3"><label>Fig. 3</label><caption><p>The prediction performance of our method using simulation data for four regions.The x-axis represents the time in days. The y-axis represents the total number of confirmed cases. The data for the first 80 days are used for training. The data for the last 20 days are used for testing. The mean absolute percentage error in testing is 17.3 <italic>%</italic></p></caption><graphic xlink:href="40249_2016_199_Fig3_HTML" id="MO3"></graphic></fig></p></sec><sec id="Sec8"><title>Real application</title><p>In the case study, we apply our model to the surveillance data from the 2009 H1N1 pandemic in Hong Kong. We acquired the time series data of the daily number of confirmed H1N1 cases with symptom onset from May 1, 2009 to May 23, 2010. The database includes 36 547 confirmed cases with demographic information on location, age, and sex along with the laboratory confirmation dates. The epidemic curve of confirmed H1N1 cases (see Fig. <xref rid="Fig4" ref-type="fig">4</xref>) reaches its peak at the end of September 2009, after which the intervention procedure comes into effect and the curve goes down. We use the data up to September 30, 2009, which comprises 27 898 cases (more than 2/3 of all cases). Hong Kong is geographically divided by 18 political areas (districts). Each district is considered as one sub-population in our proposed model. The time interval △<italic>t</italic>(<italic>k</italic>)(<italic>k</italic>=0,1,⋯,<italic>n</italic>−1) of H1N1 is set as 1 day. The total number of days is 100.
<fig id="Fig4"><label>Fig. 4</label><caption><p>Daily H1N1 epidemic curve in Hong Kong from May 1, 2009 to May 23, 2010. The epidemic curve of confirmed H1N1 cases reaches its peak at the end of September, 2010</p></caption><graphic xlink:href="40249_2016_199_Fig4_HTML" id="MO4"></graphic></fig></p><p>Figure <xref rid="Fig5" ref-type="fig">5</xref> gives the effect of the different components for the 18 political areas in Hong Kong. From Fig. <xref rid="Fig5" ref-type="fig">5</xref>, we can find that the effect of <italic>δ</italic>, which measures the internal disease spread within each district, varies less than the effect of <italic>σ</italic>, which measures the external disease spread between districts. In general, the speed of internal disease spread is closely connected with the population density and the external disease spread pattern is associated with the pattern of people’s daily travel. It is well known that Hong Kong has the highest population density in the world, and most districts are densely populated. However, it possesses a heavy heterogeneous traffic pattern, and there is intensive traffic between districts every day. Therefore, the imported infections for each district account for a critical factor in the disease spread, whereas the internal effects only play a very small role in the progression of disease spread.
<fig id="Fig5"><label>Fig. 5</label><caption><p>The estimation of two factors in disease spread for 18 districts for 2009 H1N1 pandemic in Hong Kong. <italic>δ</italic> measures the transmissibility of disease spread within each district. <italic>σ</italic> measures the transmissibility of disease spread from the neighbors of each region. This figure shows that the imported infections for each district account for a critical factor in the disease spread while the internal effects only play a very small role in the progression of disease spread</p></caption><graphic xlink:href="40249_2016_199_Fig5_HTML" id="MO5"></graphic></fig></p><p>We also use the H1N1 data to test the prediction accuracy of our model. The MAPEs for all districts are shown in Fig. <xref rid="Fig6" ref-type="fig">6</xref> and Table <xref rid="Tab1" ref-type="table">1</xref>. The average prediction error is 20.0 <italic>%</italic>. We notice that the prediction error for the district “TSUEN WAN” is very high because the number of daily infections in this district changes suddenly during the epidemic period. Figure <xref rid="Fig7" ref-type="fig">7</xref> shows the epidemic curves of the three regions with the lowest incidence rate. We can observe that between time slot 34 and 42, there is a sudden rise for the “TSUEN WAN” district. Such a change significantly affects the parameter estimation and thereby the prediction accuracy for the district “TSUEN WAN”. Although the incidence rates of the other two districts also low, their epidemic curves are relatively smooth in comparison with that of “TSUEN WAN”, indicating that the prediction accuracies of these two districts are higher than that of the “TSUEN WAN” district.
<fig id="Fig6"><label>Fig. 6</label><caption><p>The prediction errors for 18 districts using the real H1N1 data. The mean absolute percentage error is 20.0 <italic>%</italic></p></caption><graphic xlink:href="40249_2016_199_Fig6_HTML" id="MO6"></graphic></fig><fig id="Fig7"><label>Fig. 7</label><caption><p>The epidemic curve of three districts with the lowest incidence rates. The x-axis represents the time staring from May 1, 2009 to May 23, 2010. Between time slot 34 and 42, there is a sudden rise for the “TSUEN WAN” district. Such a change significantly affects the parameter estimation and thereby the prediction accuracy for the district “TSUEN WAN”</p></caption><graphic xlink:href="40249_2016_199_Fig7_HTML" id="MO7"></graphic></fig><table-wrap id="Tab1"><label>Table 1</label><caption><p>Prediction error for real data</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left">District</th><th align="left">WAN CHAI</th><th align="left">KWUN TONG</th><th align="left">KOWLOON CITY</th><th align="left">SAI KUNG</th><th align="left">WONG TAI SIN</th><th align="left">SHA TIN</th></tr></thead><tbody><tr><td align="justify">X Error</td><td align="justify">0.12</td><td align="justify">0.02</td><td align="justify">0.15</td><td align="justify">0.31</td><td align="justify">0.13</td><td align="justify">0.14</td></tr><tr><td align="justify">District</td><td align="justify">TUEN MUN</td><td align="justify">CENTRAL & WEST</td><td align="justify">NORTH</td><td align="justify">KWAI TSING</td><td align="justify">EASTER</td><td align="justify">TSUEN WAN</td></tr><tr><td align="justify">Error</td><td align="justify">0.09</td><td align="justify">0.09</td><td align="justify">0.08</td><td align="justify">0.30</td><td align="justify">0.25</td><td align="justify">0.65</td></tr><tr><td align="justify">District</td><td align="justify">SOUTHERN</td><td align="justify">SHAM SHUI PO</td><td align="justify">TAI PO</td><td align="justify">YAU TSIM MONG</td><td align="justify">ISLANDS</td><td align="justify">YUEN LONG</td></tr><tr><td align="justify">Error</td><td align="justify">0.38</td><td align="justify">0.06</td><td align="justify">0.15</td><td align="justify">0.29</td><td align="justify">0.08</td><td align="justify">0.25</td></tr></tbody></table><table-wrap-foot><p>The average error of 18 districts is 0.20</p></table-wrap-foot></table-wrap></p></sec></sec><sec id="Sec9" sec-type="conclusion"><title>Conclusions</title><p>Epidemic modelling offers a practical means for policy makers to evaluate the potential effects of intervention strategies. To do so, the accuracy of epidemic modelling with respect to the real-world disease transmission dynamics is essential and remains a challenging task due to the inaccessibility of many factors that affect the spread patterns of infectious diseases. In particular, heterogeneity should be taken into consideration when modelling the disease spread in non-random mixing populations. Many methods have been proposed to deal with heterogeneity in the study of epidemic dynamics, mostly using network-based epidemic models in which nodes correspond to spatial locations with reported incidences over time, and the directional links indicate the probability of disease transmission from one node to another over time. However, it is very challenging to determine the network topology. Many studies have used a geographical topology whereas others have used a mobility network inferred from the public transportation network or other sources. How to verify the inferred network topology is another challenging issue because the true epidemic network topology is unknown, and it may vary for different types of infectious diseases for the same population. Furthermore, the neighborhood effect estimation is non-trivial; it involves many parameters (a polynomial of the number of nodes) and requires a large amount of data to avoid overfitting. Such data may not always be available for the inference of network topology. Therefore, in this work, we propose an alternate approach to investigate the spatial heterogeneity from temporal-spatial surveillance data without the inference of network topology.</p><p>Our proposed model possesses several merits over the previous works. First, it quantifies the role of the heterogeneity in the analysis of the spread dynamics of infectious diseases in heterogeneous populations. Second, parameter estimation can be computed very quickly. Therefore, the prediction and the corresponding intervention policies can be implemented without delay in an outbreak of infectious disease. We apply our model on both the simulated data and the real data from the 2009 H1N1 epidemic in Hong Kong and achieve acceptable prediction accuracy. Based on the study of disease diffusion, the model proposed in this work can be extended to study other propagation patterns such as the Internet and World Wide Web, through which individuals form multiple communities in which information can propagate in a manner similar to that of infectious disease. We believe that our work makes theoretical and empirical contributions in many areas.</p><p>There are some limitations in our proposed stochastic model. First, it does not consider the epidemic network topology. However, how to infer such networks is another challenging task. To the best of our knowledge, the best way to do so is to use the contact data among some infected patients to verify the results, but such data are not always available and can be difficult to collect due to many issues (e.g., privacy). This issue may be addressed by using other types of data, such as daily commute data extracted from social networks. Second, our proposed model achieves a prediction accuracy of only around 80 <italic>%</italic>. We need to further improve it to allow its full use in real applications. Third, the proposed model is only suitable for the situation in which the susceptible population (or sub-population) maintains a relatively constant size and structure in a region. However, if the number of infected people in an epidemic is large or asymptomatic infection plays a central role (e.g., the malaria epidemic in Africa), the population factor should be taken into consideration in the model. Moreover, for a highly spatially heterogeneous outbreak (e.g., the Ebola epidemic) in which cases may seem to disappear due to reduced transmission in one area while growth may continue or rise in new locales, our proposed model may have problems in capturing these opposite dynamics in different regions. Fourth, because the proposed model is based on the classic SIR model, it only works in the situation in which the number of infected people grows exponentially. We will investigate resolutions to these limitations in our future work.</p></sec><sec sec-type="supplementary-material"><title>Additional file</title><sec id="Sec10"><p><supplementary-material content-type="local-data" id="MOESM1"><media xlink:href="40249_2016_199_MOESM1_ESM.pdf"><label>Additional file 1</label><caption><p>Multilingual abstracts in the five official working languages of the United Nations. (PDF 598 kb)</p></caption></media></supplementary-material></p></sec></sec></body><back><fn-group><fn><p>An erratum to this article is available at <ext-link ext-link-type="uri" xlink:href="http://dx.doi.org/10.1186/s40249-017-0269-3">http://dx.doi.org/10.1186/s40249-017-0269-3</ext-link>.</p></fn></fn-group><ack><title>Acknowledgements</title><p>This work is supported by Hong Kong Baptist University Strategic Development Fund and Hong Kong General Research Grant HKBU12202114.</p><sec id="d29e6027"><title>Authors’ contributions</title><p>RXM, JML, and XW conceived and designed the experiments. RXM implemented the software. JML and XW analysed the data. All authors were involved in the manuscript preparation. All authors read and approved the final manuscript.</p></sec><sec id="d29e6032" sec-type="COI-statement"><title>Competing interests</title><p>The authors declare that they have no competing interests.</p></sec></ack><ref-list id="Bib1"><title>References</title><ref id="CR1"><label>1</label><mixed-citation publication-type="other">World Health Organization Pandemic (H1N1). 2009. <ext-link ext-link-type="uri" xlink:href="http://www.who.int/csr/don/2010_01_08/en/index.html">http://www.who.int/csr/don/2010_01_08/en/index.html</ext-link>. Accessed 1 Aug 2010.</mixed-citation></ref><ref id="CR2"><label>2</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Cohen</surname><given-names>J</given-names></name><name><surname>Enserink</surname><given-names>M</given-names></name></person-group><article-title>As swine flu circles globe, scientists grapple with basic questions</article-title><source>Science</source><year>2009</year><volume>324</volume><issue>5927</issue><fpage>572</fpage><lpage>3</lpage><pub-id pub-id-type="doi">10.1126/science.324_572</pub-id><pub-id pub-id-type="pmid">19407164</pub-id></element-citation></ref><ref id="CR3"><label>3</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Ferguson</surname><given-names>NM</given-names></name><name><surname>Cummings</surname><given-names>DA</given-names></name><name><surname>Fraser</surname><given-names>C</given-names></name><name><surname>Cajka</surname><given-names>JC</given-names></name><name><surname>Cooley</surname><given-names>PC</given-names></name><name><surname>Burke</surname><given-names>DS</given-names></name></person-group><article-title>Strategies for mitigating an influenza pandemic</article-title><source>Nature</source><year>2006</year><volume>442</volume><issue>7101</issue><fpage>448</fpage><lpage>52</lpage><pub-id pub-id-type="doi">10.1038/nature04795</pub-id><pub-id pub-id-type="pmid">16642006</pub-id></element-citation></ref><ref id="CR4"><label>4</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Germann</surname><given-names>TC</given-names></name><name><surname>Kadau</surname><given-names>K</given-names></name><name><surname>Longini</surname><given-names>IM</given-names></name><name><surname>Macken</surname><given-names>CA</given-names></name></person-group><article-title>Mitigation strategies for pandemic influenza in the United States</article-title><source>Proc Natl Acad Sci</source><year>2006</year><volume>103</volume><issue>15</issue><fpage>5935</fpage><lpage>940</lpage><pub-id pub-id-type="doi">10.1073/pnas.0601266103</pub-id><pub-id pub-id-type="pmid">16585506</pub-id></element-citation></ref><ref id="CR5"><label>5</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Bailey</surname><given-names>NTJ</given-names></name><etal></etal></person-group><source>The Mathematical Theory of Infectious Diseases and Its Applications</source><year>1975</year><publisher-loc>High Wycombe</publisher-loc><publisher-name>Charles Griffin and Company Ltd</publisher-name></element-citation></ref><ref id="CR6"><label>6</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Anderson</surname><given-names>RM</given-names></name><name><surname>May</surname><given-names>RM</given-names></name></person-group><source>Infectious Diseases of Humans vol. 1</source><year>1991</year><publisher-loc>Oxford</publisher-loc><publisher-name>Oxford University Press</publisher-name></element-citation></ref><ref id="CR7"><label>7</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Heesterbeek</surname><given-names>J</given-names></name></person-group><source>Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. vol 5</source><year>2000</year><publisher-loc>New York City</publisher-loc><publisher-name>Wiley</publisher-name></element-citation></ref><ref id="CR8"><label>8</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Li</surname><given-names>MY</given-names></name><name><surname>Muldowney</surname><given-names>JS</given-names></name></person-group><article-title>Global stability for the SEIR model in epidemiology</article-title><source>Math Biosci</source><year>1995</year><volume>125</volume><issue>2</issue><fpage>155</fpage><lpage>64</lpage><pub-id pub-id-type="doi">10.1016/0025-5564(95)92756-5</pub-id><pub-id pub-id-type="pmid">7881192</pub-id></element-citation></ref><ref id="CR9"><label>9</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Kuznetsov</surname><given-names>YA</given-names></name><name><surname>Piccardi</surname><given-names>C</given-names></name></person-group><article-title>Bifurcation analysis of periodic SEIR and SIR epidemic models</article-title><source>J Math Biol</source><year>1994</year><volume>32</volume><issue>2</issue><fpage>109</fpage><lpage>21</lpage><pub-id pub-id-type="doi">10.1007/BF00163027</pub-id><pub-id pub-id-type="pmid">8145028</pub-id></element-citation></ref><ref id="CR10"><label>10</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hethcote</surname><given-names>HW</given-names></name></person-group><article-title>Qualitative analyses of communicable disease models</article-title><source>Math Biosci</source><year>1976</year><volume>28</volume><issue>3</issue><fpage>335</fpage><lpage>56</lpage><pub-id pub-id-type="doi">10.1016/0025-5564(76)90132-2</pub-id></element-citation></ref><ref id="CR11"><label>11</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Becker</surname><given-names>NG</given-names></name><name><surname>Britton</surname><given-names>T</given-names></name></person-group><article-title>Statistical studies of infectious disease incidence</article-title><source>J R Stat Soc Series B Stat Methodol</source><year>2002</year><volume>61</volume><issue>2</issue><fpage>287</fpage><lpage>307</lpage><pub-id pub-id-type="doi">10.1111/1467-9868.00177</pub-id></element-citation></ref><ref id="CR12"><label>12</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Ferrari</surname><given-names>MJ</given-names></name><name><surname>Bjørnstad</surname><given-names>ON</given-names></name><name><surname>Dobson</surname><given-names>AP</given-names></name></person-group><article-title>Estimation and inference of R0 of an infectious pathogen by a removal method</article-title><source>Math Biosci</source><year>2005</year><volume>198</volume><issue>1</issue><fpage>14</fpage><lpage>26</lpage><pub-id pub-id-type="doi">10.1016/j.mbs.2005.08.002</pub-id><pub-id pub-id-type="pmid">16216286</pub-id></element-citation></ref><ref id="CR13"><label>13</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Allen</surname><given-names>L</given-names></name></person-group><article-title>An introduction to stochastic epidemic models</article-title><source>Math Epidemiol</source><year>2008</year><volume>1945</volume><fpage>81</fpage><lpage>130</lpage><pub-id pub-id-type="doi">10.1007/978-3-540-78911-6_3</pub-id></element-citation></ref><ref id="CR14"><label>14</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Cooper</surname><given-names>BS</given-names></name><name><surname>Pitman</surname><given-names>RJ</given-names></name><name><surname>Edmunds</surname><given-names>WJ</given-names></name><name><surname>Gay</surname><given-names>NJ</given-names></name><etal></etal></person-group><article-title>Delaying the international spread of pandemic influenza</article-title><source>PLoS Med</source><year>2006</year><volume>3</volume><issue>6</issue><fpage>212</fpage><pub-id pub-id-type="doi">10.1371/journal.pmed.0030212</pub-id></element-citation></ref><ref id="CR15"><label>15</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hufnagel</surname><given-names>L</given-names></name><name><surname>Brockmann</surname><given-names>D</given-names></name><name><surname>Geisel</surname><given-names>T</given-names></name></person-group><article-title>Forecast and control of epidemics in a globalized world</article-title><source>Proc Natl Acad Sci U S A</source><year>2004</year><volume>101</volume><issue>42</issue><fpage>15124</fpage><lpage>15129</lpage><pub-id pub-id-type="doi">10.1073/pnas.0308344101</pub-id><pub-id pub-id-type="pmid">15477600</pub-id></element-citation></ref><ref id="CR16"><label>16</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hollingsworth</surname><given-names>TD</given-names></name><name><surname>Ferguson</surname><given-names>NM</given-names></name><name><surname>Anderson</surname><given-names>RM</given-names></name></person-group><article-title>Will travel restrictions control the international spread of pandemic influenza?</article-title><source>Nat Med</source><year>2006</year><volume>12</volume><issue>5</issue><fpage>497</fpage><lpage>9</lpage><pub-id pub-id-type="doi">10.1038/nm0506-497</pub-id><pub-id pub-id-type="pmid">16675989</pub-id></element-citation></ref><ref id="CR17"><label>17</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Keeling</surname><given-names>MJ</given-names></name><name><surname>Woolhouse</surname><given-names>ME</given-names></name><name><surname>Shaw</surname><given-names>DJ</given-names></name><name><surname>Matthews</surname><given-names>L</given-names></name><name><surname>Chase-Topping</surname><given-names>M</given-names></name><name><surname>Haydon</surname><given-names>DT</given-names></name><name><surname>Cornell</surname><given-names>SJ</given-names></name><name><surname>Kappey</surname><given-names>J</given-names></name><name><surname>Wilesmith</surname><given-names>J</given-names></name><name><surname>Grenfell</surname><given-names>BT</given-names></name></person-group><article-title>Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape</article-title><source>Science</source><year>2001</year><volume>294</volume><issue>5543</issue><fpage>813</fpage><lpage>7</lpage><pub-id pub-id-type="doi">10.1126/science.1065973</pub-id><pub-id pub-id-type="pmid">11679661</pub-id></element-citation></ref><ref id="CR18"><label>18</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Ferguson</surname><given-names>NM</given-names></name><name><surname>Cummings</surname><given-names>DA</given-names></name><name><surname>Cauchemez</surname><given-names>S</given-names></name><name><surname>Fraser</surname><given-names>C</given-names></name><name><surname>Riley</surname><given-names>S</given-names></name><name><surname>Meeyai</surname><given-names>A</given-names></name><name><surname>Iamsirithaworn</surname><given-names>S</given-names></name><name><surname>Burke</surname><given-names>DS</given-names></name></person-group><article-title>Strategies for containing an emerging influenza pandemic in Southeast Asia</article-title><source>Nature</source><year>2005</year><volume>437</volume><issue>7056</issue><fpage>209</fpage><lpage>14</lpage><pub-id pub-id-type="doi">10.1038/nature04017</pub-id><pub-id pub-id-type="pmid">16079797</pub-id></element-citation></ref><ref id="CR19"><label>19</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Longini</surname><given-names>IM</given-names></name><name><surname>Nizam</surname><given-names>A</given-names></name><name><surname>Xu</surname><given-names>S</given-names></name><name><surname>Ungchusak</surname><given-names>K</given-names></name><name><surname>Hanshaoworakul</surname><given-names>W</given-names></name><name><surname>Cummings</surname><given-names>DA</given-names></name><name><surname>Halloran</surname><given-names>ME</given-names></name></person-group><article-title>Containing pandemic influenza at the source</article-title><source>Science</source><year>2005</year><volume>309</volume><issue>5737</issue><fpage>1083</fpage><lpage>1087</lpage><pub-id pub-id-type="doi">10.1126/science.1115717</pub-id><pub-id pub-id-type="pmid">16079251</pub-id></element-citation></ref><ref id="CR20"><label>20</label><element-citation publication-type="journal"><person-group person-group-type="author"><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>Epidemic dynamics and endemic states in complex networks</article-title><source>Phys Rev E</source><year>2001</year><volume>63</volume><issue>6</issue><fpage>066117</fpage><pub-id pub-id-type="doi">10.1103/PhysRevE.63.066117</pub-id></element-citation></ref><ref id="CR21"><label>21</label><mixed-citation publication-type="other">Pastor-Satorras R, Vespignani A. Epidemics and immunization in scale-free networks. 2002. arXiv preprint cond-mat/0205260.</mixed-citation></ref><ref id="CR22"><label>22</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Kuperman</surname><given-names>M</given-names></name><name><surname>Abramson</surname><given-names>G</given-names></name></person-group><article-title>Small world effect in an epidemiological model</article-title><source>Phys Rev Lett</source><year>2001</year><volume>86</volume><issue>13</issue><fpage>2909</fpage><lpage>912</lpage><pub-id pub-id-type="doi">10.1103/PhysRevLett.86.2909</pub-id><pub-id pub-id-type="pmid">11290070</pub-id></element-citation></ref><ref id="CR23"><label>23</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Newman</surname><given-names>ME</given-names></name><name><surname>Jensen</surname><given-names>I</given-names></name><name><surname>Ziff</surname><given-names>R</given-names></name></person-group><article-title>Percolation and epidemics in a two-dimensional small world</article-title><source>Phys Rev E</source><year>2002</year><volume>65</volume><issue>2</issue><fpage>021904</fpage><pub-id pub-id-type="doi">10.1103/PhysRevE.65.021904</pub-id></element-citation></ref><ref id="CR24"><label>24</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Boguná</surname><given-names>M</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>Absence of epidemic threshold in scale-free networks with degree correlations</article-title><source>Phys Rev Lett</source><year>2003</year><volume>90</volume><issue>2</issue><fpage>028701</fpage><pub-id pub-id-type="doi">10.1103/PhysRevLett.90.028701</pub-id><pub-id pub-id-type="pmid">12570587</pub-id></element-citation></ref><ref id="CR25"><label>25</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Grassberger</surname><given-names>P</given-names></name></person-group><article-title>On the critical behavior of the general epidemic process and dynamical percolation</article-title><source>Math Biosci</source><year>1983</year><volume>63</volume><issue>2</issue><fpage>157</fpage><lpage>72</lpage><pub-id pub-id-type="doi">10.1016/0025-5564(82)90036-0</pub-id></element-citation></ref><ref id="CR26"><label>26</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Watts</surname><given-names>DJ</given-names></name><name><surname>Strogatz</surname><given-names>SH</given-names></name></person-group><article-title>Collective dynamics of small-world networks</article-title><source>Nature</source><year>1998</year><volume>393</volume><issue>6684</issue><fpage>440</fpage><lpage>2</lpage><pub-id pub-id-type="doi">10.1038/30918</pub-id><pub-id pub-id-type="pmid">9623998</pub-id></element-citation></ref><ref id="CR27"><label>27</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Barabási</surname><given-names>AL</given-names></name><name><surname>Albert</surname><given-names>R</given-names></name></person-group><article-title>Emergence of scaling in random networks</article-title><source>Science</source><year>1999</year><volume>286</volume><issue>5439</issue><fpage>509</fpage><lpage>12</lpage><pub-id pub-id-type="doi">10.1126/science.286.5439.509</pub-id><pub-id pub-id-type="pmid">10521342</pub-id></element-citation></ref><ref id="CR28"><label>28</label><mixed-citation publication-type="other">Erd, 6s P, Rényi A. On the evolution of random graphs. Publ Math Inst Hungar Acad Sci. 1960; 5:17–61.</mixed-citation></ref><ref id="CR29"><label>29</label><element-citation publication-type="book"><person-group person-group-type="author"><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>Epidemics and immunization in scale-free networks</article-title><source>Handbook of graphs and networks: from the genome to the internet</source><year>2005</year><publisher-loc>Hoboken</publisher-loc><publisher-name>Wiley Online Library</publisher-name></element-citation></ref><ref id="CR30"><label>30</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Rogers</surname><given-names>EM</given-names></name></person-group><source>Diffusion of Innovations</source><year>2010</year><publisher-loc>New York</publisher-loc><publisher-name>Simon and Schuster</publisher-name></element-citation></ref><ref id="CR31"><label>31</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Leskovec</surname><given-names>J</given-names></name><name><surname>Adamic</surname><given-names>LA</given-names></name><name><surname>Huberman</surname><given-names>BA</given-names></name></person-group><article-title>The dynamics of viral marketing</article-title><source>ACM T Web</source><year>2007</year><volume>1</volume><issue>1</issue><fpage>5</fpage><pub-id pub-id-type="doi">10.1145/1232722.1232727</pub-id></element-citation></ref><ref id="CR32"><label>32</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Kumar</surname><given-names>R</given-names></name><name><surname>Novak</surname><given-names>J</given-names></name><name><surname>Raghavan</surname><given-names>P</given-names></name><name><surname>Tomkins</surname><given-names>A</given-names></name></person-group><article-title>On the bursty evolution of Blogspace</article-title><source>World Wide Web</source><year>2005</year><volume>8</volume><issue>2</issue><fpage>159</fpage><lpage>78</lpage><pub-id pub-id-type="doi">10.1007/s11280-004-4872-4</pub-id></element-citation></ref><ref id="CR33"><label>33</label><mixed-citation publication-type="other">Leskovec J, McGlohon M, Faloutsos C, Glance NS, Hurst M. Patterns of cascading behavior in large blog graphs. In: SDM, vol. 7. SIAM: 2007. p. 551–6.</mixed-citation></ref><ref id="CR34"><label>34</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Salathé</surname><given-names>M</given-names></name><name><surname>Kazandjieva</surname><given-names>M</given-names></name><name><surname>Lee</surname><given-names>JW</given-names></name><name><surname>Levis</surname><given-names>P</given-names></name><name><surname>Feldman</surname><given-names>MW</given-names></name><name><surname>Jones</surname><given-names>JH</given-names></name></person-group><article-title>A high-resolution human contact network for infectious disease transmission</article-title><source>Proc Natl Acad Sci</source><year>2010</year><volume>107</volume><issue>51</issue><fpage>22020</fpage><lpage>2025</lpage><pub-id pub-id-type="doi">10.1073/pnas.1009094108</pub-id><pub-id pub-id-type="pmid">21149721</pub-id></element-citation></ref><ref id="CR35"><label>35</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Stehlé</surname><given-names>J</given-names></name><name><surname>Voirin</surname><given-names>N</given-names></name><name><surname>Barrat</surname><given-names>A</given-names></name><name><surname>Cattuto</surname><given-names>C</given-names></name><name><surname>Isella</surname><given-names>L</given-names></name><name><surname>Pinton</surname><given-names>JF</given-names></name><name><surname>Quaggiotto</surname><given-names>M</given-names></name><name><surname>Van den Broeck</surname><given-names>W</given-names></name><name><surname>Régis</surname><given-names>C</given-names></name><name><surname>Lina</surname><given-names>B</given-names></name><etal></etal></person-group><article-title>High-resolution measurements of face-to-face contact patterns in a primary school</article-title><source>PloS ONE</source><year>2011</year><volume>6</volume><issue>8</issue><fpage>23176</fpage><pub-id pub-id-type="doi">10.1371/journal.pone.0023176</pub-id></element-citation></ref><ref id="CR36"><label>36</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Dickison</surname><given-names>M</given-names></name><name><surname>Havlin</surname><given-names>S</given-names></name><name><surname>Stanley</surname><given-names>HE</given-names></name></person-group><article-title>Epidemics on interconnected networks</article-title><source>Phys Rev E</source><year>2012</year><volume>85</volume><issue>6</issue><fpage>066109</fpage><pub-id pub-id-type="doi">10.1103/PhysRevE.85.066109</pub-id></element-citation></ref><ref id="CR37"><label>37</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hancock</surname><given-names>K</given-names></name><name><surname>Veguilla</surname><given-names>V</given-names></name><name><surname>Lu</surname><given-names>X</given-names></name><name><surname>Zhong</surname><given-names>W</given-names></name><name><surname>Butler</surname><given-names>EN</given-names></name><name><surname>Sun</surname><given-names>H</given-names></name><name><surname>Liu</surname><given-names>F</given-names></name><name><surname>Dong</surname><given-names>L</given-names></name><name><surname>DeVos</surname><given-names>JR</given-names></name><name><surname>Gargiullo</surname><given-names>PM</given-names></name><etal></etal></person-group><article-title>Cross-reactive antibody responses to the 2009 pandemic H1N1 influenza virus</article-title><source>N Engl J Med</source><year>2009</year><volume>361</volume><issue>20</issue><fpage>1945</fpage><lpage>1952</lpage><pub-id pub-id-type="doi">10.1056/NEJMoa0906453</pub-id><pub-id pub-id-type="pmid">19745214</pub-id></element-citation></ref><ref id="CR38"><label>38</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Riley</surname><given-names>S</given-names></name><name><surname>Fraser</surname><given-names>C</given-names></name><name><surname>Donnelly</surname><given-names>CA</given-names></name><name><surname>Ghani</surname><given-names>AC</given-names></name><name><surname>Abu-Raddad</surname><given-names>LJ</given-names></name><name><surname>Hedley</surname><given-names>AJ</given-names></name><name><surname>Leung</surname><given-names>GM</given-names></name><name><surname>Ho</surname><given-names>LM</given-names></name><name><surname>Lam</surname><given-names>TH</given-names></name><name><surname>Thach</surname><given-names>TQ</given-names></name><etal></etal></person-group><article-title>Transmission dynamics of the etiological agent of SARS in Hong Kong: impact of public health interventions</article-title><source>Science</source><year>2003</year><volume>300</volume><issue>5627</issue><fpage>1961</fpage><lpage>1966</lpage><pub-id pub-id-type="doi">10.1126/science.1086478</pub-id><pub-id pub-id-type="pmid">12766206</pub-id></element-citation></ref><ref id="CR39"><label>39</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Mills</surname><given-names>CE</given-names></name><name><surname>Robins</surname><given-names>JM</given-names></name><name><surname>Lipsitch</surname><given-names>M</given-names></name></person-group><article-title>Transmissibility of 1918 pandemic influenza</article-title><source>Nature</source><year>2004</year><volume>432</volume><issue>7019</issue><fpage>904</fpage><lpage>6</lpage><pub-id pub-id-type="doi">10.1038/nature03063</pub-id><pub-id pub-id-type="pmid">15602562</pub-id></element-citation></ref><ref id="CR40"><label>40</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Birrell</surname><given-names>PJ</given-names></name><name><surname>Ketsetzis</surname><given-names>G</given-names></name><name><surname>Gay</surname><given-names>NJ</given-names></name><name><surname>Cooper</surname><given-names>BS</given-names></name><name><surname>Presanis</surname><given-names>AM</given-names></name><name><surname>Harris</surname><given-names>RJ</given-names></name><name><surname>Charlett</surname><given-names>A</given-names></name><name><surname>Zhang</surname><given-names>XS</given-names></name><name><surname>White</surname><given-names>PJ</given-names></name><name><surname>Pebody</surname><given-names>RG</given-names></name><etal></etal></person-group><article-title>Bayesian modelling to unmask and predict influenza A/H1N1pdm dynamics in London</article-title><source>Proc Natl Acad Sci</source><year>2011</year><volume>108</volume><issue>45</issue><fpage>18238</fpage><lpage>18243</lpage><pub-id pub-id-type="doi">10.1073/pnas.1103002108</pub-id><pub-id pub-id-type="pmid">22042838</pub-id></element-citation></ref><ref id="CR41"><label>41</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Towers</surname><given-names>S</given-names></name><name><surname>Geisse</surname><given-names>KV</given-names></name><name><surname>Zheng</surname><given-names>Y</given-names></name><name><surname>Feng</surname><given-names>Z</given-names></name></person-group><article-title>Antiviral treatment for pandemic influenza: Assessing potential repercussions using a seasonally forced SIR model</article-title><source>J Theor Biol</source><year>2011</year><volume>289</volume><fpage>259</fpage><lpage>68</lpage><pub-id pub-id-type="doi">10.1016/j.jtbi.2011.08.011</pub-id><pub-id pub-id-type="pmid">21867715</pub-id></element-citation></ref><ref id="CR42"><label>42</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Cauchemez</surname><given-names>S</given-names></name><name><surname>Valleron</surname><given-names>AJ</given-names></name><name><surname>Boelle</surname><given-names>PY</given-names></name><name><surname>Flahault</surname><given-names>A</given-names></name><name><surname>Ferguson</surname><given-names>NM</given-names></name></person-group><article-title>Estimating the impact of school closure on influenza transmission from sentinel data</article-title><source>Nature</source><year>2008</year><volume>452</volume><issue>7188</issue><fpage>750</fpage><lpage>4</lpage><pub-id pub-id-type="doi">10.1038/nature06732</pub-id><pub-id pub-id-type="pmid">18401408</pub-id></element-citation></ref><ref id="CR43"><label>43</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Cauchemez</surname><given-names>S</given-names></name><name><surname>Donnelly</surname><given-names>CA</given-names></name><name><surname>Reed</surname><given-names>C</given-names></name><name><surname>Ghani</surname><given-names>AC</given-names></name><name><surname>Fraser</surname><given-names>C</given-names></name><name><surname>Kent</surname><given-names>CK</given-names></name><name><surname>Finelli</surname><given-names>L</given-names></name><name><surname>Ferguson</surname><given-names>NM</given-names></name></person-group><article-title>Household transmission of 2009 pandemic influenza A (H1N1) virus in the United States</article-title><source>N Engl J Med</source><year>2009</year><volume>361</volume><issue>27</issue><fpage>2619</fpage><lpage>627</lpage><pub-id pub-id-type="doi">10.1056/NEJMoa0905498</pub-id><pub-id pub-id-type="pmid">20042753</pub-id></element-citation></ref><ref id="CR44"><label>44</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Lessler</surname><given-names>J</given-names></name><name><surname>Reich</surname><given-names>NG</given-names></name><name><surname>Cummings</surname><given-names>DA</given-names></name></person-group><article-title>Outbreak of 2009 pandemic influenza a (h1n1) at a New York city school</article-title><source>N Engl J Med</source><year>2009</year><volume>361</volume><issue>27</issue><fpage>2628</fpage><lpage>636</lpage><pub-id pub-id-type="doi">10.1056/NEJMoa0906089</pub-id><pub-id pub-id-type="pmid">20042754</pub-id></element-citation></ref><ref id="CR45"><label>45</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Klick</surname><given-names>B</given-names></name><name><surname>Nishiura</surname><given-names>H</given-names></name><name><surname>Ng</surname><given-names>S</given-names></name><name><surname>Fang</surname><given-names>VJ</given-names></name><name><surname>Leung</surname><given-names>GM</given-names></name><name><surname>Peiris</surname><given-names>JM</given-names></name><name><surname>Cowling</surname><given-names>BJ</given-names></name></person-group><article-title>Transmissibility of seasonal and pandemic influenza in a cohort of households in Hong Kong in 2009</article-title><source>Epidemiology (Cambridge)</source><year>2011</year><volume>22</volume><issue>6</issue><fpage>793</fpage></element-citation></ref><ref id="CR46"><label>46</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Yang</surname><given-names>Y</given-names></name><name><surname>Sugimoto</surname><given-names>JD</given-names></name><name><surname>Halloran</surname><given-names>ME</given-names></name><name><surname>Basta</surname><given-names>NE</given-names></name><name><surname>Chao</surname><given-names>DL</given-names></name><name><surname>Matrajt</surname><given-names>L</given-names></name><name><surname>Potter</surname><given-names>G</given-names></name><name><surname>Kenah</surname><given-names>E</given-names></name><name><surname>Longini</surname><given-names>IM</given-names></name></person-group><article-title>The transmissibility and control of pandemic influenza a (H1N1) virus</article-title><source>Science</source><year>2009</year><volume>326</volume><issue>5953</issue><fpage>729</fpage><lpage>33</lpage><pub-id pub-id-type="doi">10.1126/science.1177373</pub-id><pub-id pub-id-type="pmid">19745114</pub-id></element-citation></ref><ref id="CR47"><label>47</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Jin</surname><given-names>Z</given-names></name><name><surname>Zhang</surname><given-names>J</given-names></name><name><surname>Song</surname><given-names>LP</given-names></name><name><surname>Sun</surname><given-names>GQ</given-names></name><name><surname>Kan</surname><given-names>J</given-names></name><name><surname>Zhu</surname><given-names>H</given-names></name></person-group><article-title>Modelling and analysis of influenza a (H1N1) on networks</article-title><source>BMC Public Health</source><year>2011</year><volume>11</volume><issue>Suppl 1</issue><fpage>9</fpage><pub-id pub-id-type="pmid">21208413</pub-id></element-citation></ref><ref id="CR48"><label>48</label><mixed-citation publication-type="other">World Health Organization. Ebola Situation Reports. 2015. Available at: <ext-link ext-link-type="uri" xlink:href="http://apps.who.int/ebola/en/current-situation/ebolasituation-report">http://apps.who.int/ebola/en/current-situation/ebolasituation-report</ext-link>. Accessed 29 June 2015.</mixed-citation></ref><ref id="CR49"><label>49</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Camacho</surname><given-names>A</given-names></name><name><surname>Kucharski</surname><given-names>A</given-names></name><name><surname>Aki-Sawyerr</surname><given-names>Y</given-names></name><name><surname>White</surname><given-names>MA</given-names></name><name><surname>Flasche</surname><given-names>S</given-names></name><name><surname>Baguelin</surname><given-names>M</given-names></name><name><surname>Pollington</surname><given-names>T</given-names></name><name><surname>Carney</surname><given-names>JR</given-names></name><name><surname>Glover</surname><given-names>R</given-names></name><name><surname>Smout</surname><given-names>E</given-names></name><etal></etal></person-group><article-title>Temporal changes in Ebola transmission in sierra leone and implications for control requirements: a real-time modelling study</article-title><source>PLoS Curr.</source><year>2015</year><volume>7</volume><fpage>PMC4339317</fpage></element-citation></ref><ref id="CR50"><label>50</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Chowell</surname><given-names>D</given-names></name><name><surname>Castillo-Chavez</surname><given-names>C</given-names></name><name><surname>Krishna</surname><given-names>S</given-names></name><name><surname>Qiu</surname><given-names>X</given-names></name><name><surname>Anderson</surname><given-names>KS</given-names></name></person-group><article-title>Modelling the effect of early detection of Ebola</article-title><source>Lancet Infect Dis</source><year>2015</year><volume>15</volume><issue>2</issue><fpage>148</fpage><lpage>9</lpage><pub-id pub-id-type="doi">10.1016/S1473-3099(14)71084-9</pub-id><pub-id pub-id-type="pmid">25749063</pub-id></element-citation></ref><ref id="CR51"><label>51</label><mixed-citation publication-type="other">Chowell G, Hengartner NW, Castillo-Chavez C, Fenimore PW, Hyman J. The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda. J Theor Biol. 2004; 229(1):119–26.</mixed-citation></ref><ref id="CR52"><label>52</label><mixed-citation publication-type="other">Chowell G, Viboud C, Hyman JM, Simonsen L. The Western Africa Ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates. PLoS Curr. 2015. doi:<ext-link ext-link-type="uri" xlink:href="http://dx.doi.org/10.1371/currents.outbreaks.8b55f4bad99ac5c5db3663e916803261">10.1371/currents.outbreaks.8b55f4bad99ac5c5db3663e916803261</ext-link>.</mixed-citation></ref><ref id="CR53"><label>53</label><mixed-citation publication-type="other">Fisman D, Khoo E, Tuite A. Early epidemic dynamics of the West African 2014 Ebola outbreak: estimates derived with a simple two-parameter model. PLoS Curr. 2014. doi:<ext-link ext-link-type="uri" xlink:href="http://dx.doi.org/10.1371/currents.outbreaks.89c0d3783f36958d96ebbae97348d571">10.1371/currents.outbreaks.89c0d3783f36958d96ebbae97348d571</ext-link>.</mixed-citation></ref><ref id="CR54"><label>54</label><mixed-citation publication-type="other">Gomes MF, y Piontti AP, Rossi L, Chao D, Longini I, Halloran ME, Vespignani A. Assessing the international spreading risk associated with the 2014 West African Ebola outbreak. PLoS Curr. 2014. doi:<ext-link ext-link-type="uri" xlink:href="http://dx.doi.org/10.1371/currents.outbreaks.cd818f63d40e24aef769dda7df9e0da5">10.1371/currents.outbreaks.cd818f63d40e24aef769dda7df9e0da5</ext-link>.</mixed-citation></ref><ref id="CR55"><label>55</label><mixed-citation publication-type="other">Althaus CL. Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa. PLoS Curr. 2014. doi:<ext-link ext-link-type="uri" xlink:href="http://dx.doi.org/10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288">10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288</ext-link>.</mixed-citation></ref><ref id="CR56"><label>56</label><mixed-citation publication-type="other">Webb G, Browne C, Huo X, Seydi O, Seydi M, Magal P. A model of the 2014 Ebola epidemic in West Africa with contact tracing. PLoS Curr. 2014. doi:<ext-link ext-link-type="uri" xlink:href="http://dx.doi.org/10.1371/currents.outbreaks.846b2a31ef37018b7d1126a9c8adf22a">10.1371/currents.outbreaks.846b2a31ef37018b7d1126a9c8adf22a</ext-link>.</mixed-citation></ref><ref id="CR57"><label>57</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Carroll</surname><given-names>MW</given-names></name><name><surname>Matthews</surname><given-names>DA</given-names></name><name><surname>Hiscox</surname><given-names>JA</given-names></name><name><surname>Elmore</surname><given-names>MJ</given-names></name><name><surname>Pollakis</surname><given-names>G</given-names></name><name><surname>Rambaut</surname><given-names>A</given-names></name><name><surname>Hewson</surname><given-names>R</given-names></name><name><surname>García-Dorival</surname><given-names>I</given-names></name><name><surname>Bore</surname><given-names>JA</given-names></name><name><surname>Koundouno</surname><given-names>R</given-names></name><etal></etal></person-group><article-title>Temporal and spatial analysis of the 2014-2015 Ebola virus outbreak in West Africa</article-title><source>Nature.</source><year>2015</year><volume>524</volume><issue>7563</issue><fpage>97</fpage><lpage>101</lpage><pub-id pub-id-type="doi">10.1038/nature14594</pub-id><pub-id pub-id-type="pmid">26083749</pub-id></element-citation></ref><ref id="CR58"><label>58</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Chowell</surname><given-names>G</given-names></name><name><surname>Nishiura</surname><given-names>H</given-names></name></person-group><article-title>Transmission dynamics and control of Ebola virus disease (EVD): a review</article-title><source>BMC Med</source><year>2014</year><volume>12</volume><issue>1</issue><fpage>196</fpage><pub-id pub-id-type="doi">10.1186/s12916-014-0196-0</pub-id><pub-id pub-id-type="pmid">25300956</pub-id></element-citation></ref><ref id="CR59"><label>59</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>van Dijk</surname><given-names>A</given-names></name><name><surname>Aramini</surname><given-names>J</given-names></name><name><surname>Edge</surname><given-names>G</given-names></name><name><surname>Moore</surname><given-names>KM</given-names></name></person-group><article-title>Real-time surveillance for respiratory disease outbreaks, Ontario, Canada</article-title><source>Emerg Infect Diseases</source><year>2009</year><volume>15</volume><issue>5</issue><fpage>799</fpage><pub-id pub-id-type="doi">10.3201/eid1505.081174</pub-id><pub-id pub-id-type="pmid">19402974</pub-id></element-citation></ref></ref-list></back></pmc></record>Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Sante/explor/SrasV1/Data/Pmc/Corpus
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000389 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Pmc/Corpus/biblio.hfd -nk 000389 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien
|wiki= Sante
|area= SrasV1
|flux= Pmc
|étape= Corpus
|type= RBID
|clé= PMC:5178099
|texte= Stochastic modelling of infectious diseases for heterogeneous populations
}}
Pour générer des pages wiki
HfdIndexSelect -h $EXPLOR_AREA/Data/Pmc/Corpus/RBID.i -Sk "pubmed:28003016" \
| HfdSelect -Kh $EXPLOR_AREA/Data/Pmc/Corpus/biblio.hfd \
| NlmPubMed2Wicri -a SrasV1
| This area was generated with Dilib version V0.6.33. |  |