Serveur d'exploration SRAS

Attention, ce site est en cours de développement !
Attention, site généré par des moyens informatiques à partir de corpus bruts.
Les informations ne sont donc pas validées.

Challenges, Opportunities and Theoretical Epidemiology

Identifieur interne : 000096 ( Pmc/Corpus ); précédent : 000095; suivant : 000097

Challenges, Opportunities and Theoretical Epidemiology

Auteurs : Fred Brauer ; Carlos Castillo-Chavez ; Zhilan Feng

Source :

RBID : PMC:7123038

Abstract

Lessons learned from the HIV pandemic, SARS in 2003, the 2009 H1N1 influenza pandemic, the 2014 Ebola outbreak in West Africa, and the ongoing Zika outbreaks in the Americas can be framed under a public health policy model that responds after the fact. Responses often come through reallocation of resources from one disease control effort to a new pressing need. The operating models of preparedness and response are ill-equipped to prevent or ameliorate disease emergence or reemergence at global scales. Epidemiological challenges that are a threat to the economic stability of many regions of the world, particularly those depending on travel and trade, remain at the forefront of the Global Commons. Consequently, efforts to quantify the impact of mobility and trade on disease dynamics have dominated the interests of theoreticians for some time. Our experience includes an H1N1 influenza pandemic crisscrossing the world during 2009 and 2010, the 2014 Ebola outbreaks, limited to regions of West Africa lacking appropriate medical facilities, health infrastructure, and sufficient levels of preparedness and education, and the expanding Zika outbreaks, moving expeditiously across habitats suitable for Aedes aegypti. These provide opportunities to quantify the impact of disease emergence or reemergence on the decisions that individuals take in response to real or perceived disease risks. The case of SARS 2003 in 2003, the efforts to reduce the burden of H1N1 influenza cases in 2009, and the challenges faced in reducing the number of Ebola cases in 2014 are the three recent scenarios that required a timely global response. Studies addressing the impact of centralized sources of information, the impact of information along social connections, or the role of past disease outbreak experiences on the risk-aversion decisions that individuals undertake may help identify and quantify the role of human responses to disease dynamics while recognizing the importance of assessing the timing of disease emergence and reemergence. The co-evolving human responses to disease dynamics are prototypical of the feedbacks that define complex adaptive systems. In short, we live in a socioepisphere being reshaped by ecoepidemiology in the “Era of Information.”


Url:
DOI: 10.1007/978-1-4939-9828-9_16
PubMed: NONE
PubMed Central: 7123038

Links to Exploration step

PMC:7123038

Le document en format XML

<record>
<TEI>
<teiHeader>
<fileDesc>
<titleStmt>
<title xml:lang="en">Challenges, Opportunities and Theoretical Epidemiology</title>
<author>
<name sortKey="Brauer, Fred" sort="Brauer, Fred" uniqKey="Brauer F" first="Fred" last="Brauer">Fred Brauer</name>
<affiliation>
<nlm:aff id="Aff19">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.17091.3e</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2288 9830</institution-id>
<institution>Department of Mathematics,</institution>
<institution>University of British Columbia,</institution>
</institution-wrap>
Vancouver, BC Canada</nlm:aff>
</affiliation>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" sort="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C" first="Carlos" last="Castillo-Chavez">Carlos Castillo-Chavez</name>
<affiliation>
<nlm:aff id="Aff20">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.215654.1</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2151 2636</institution-id>
<institution>Mathematical and Computational Modeling Center (MCMSC), Department of Mathematics and Statistics,</institution>
<institution>Arizona State University,</institution>
</institution-wrap>
Tempe, AZ USA</nlm:aff>
</affiliation>
</author>
<author>
<name sortKey="Feng, Zhilan" sort="Feng, Zhilan" uniqKey="Feng Z" first="Zhilan" last="Feng">Zhilan Feng</name>
<affiliation>
<nlm:aff id="Aff21">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.169077.e</institution-id>
<institution-id institution-id-type="ISNI">0000 0004 1937 2197</institution-id>
<institution>Department of Mathematics,</institution>
<institution>Purdue University,</institution>
</institution-wrap>
West Lafayette, IN USA</nlm:aff>
</affiliation>
</author>
</titleStmt>
<publicationStmt>
<idno type="wicri:source">PMC</idno>
<idno type="pmc">7123038</idno>
<idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7123038</idno>
<idno type="RBID">PMC:7123038</idno>
<idno type="doi">10.1007/978-1-4939-9828-9_16</idno>
<idno type="pmid">NONE</idno>
<date when="2019">2019</date>
<idno type="wicri:Area/Pmc/Corpus">000096</idno>
<idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000096</idno>
</publicationStmt>
<sourceDesc>
<biblStruct>
<analytic>
<title xml:lang="en" level="a" type="main">Challenges, Opportunities and Theoretical Epidemiology</title>
<author>
<name sortKey="Brauer, Fred" sort="Brauer, Fred" uniqKey="Brauer F" first="Fred" last="Brauer">Fred Brauer</name>
<affiliation>
<nlm:aff id="Aff19">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.17091.3e</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2288 9830</institution-id>
<institution>Department of Mathematics,</institution>
<institution>University of British Columbia,</institution>
</institution-wrap>
Vancouver, BC Canada</nlm:aff>
</affiliation>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" sort="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C" first="Carlos" last="Castillo-Chavez">Carlos Castillo-Chavez</name>
<affiliation>
<nlm:aff id="Aff20">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.215654.1</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2151 2636</institution-id>
<institution>Mathematical and Computational Modeling Center (MCMSC), Department of Mathematics and Statistics,</institution>
<institution>Arizona State University,</institution>
</institution-wrap>
Tempe, AZ USA</nlm:aff>
</affiliation>
</author>
<author>
<name sortKey="Feng, Zhilan" sort="Feng, Zhilan" uniqKey="Feng Z" first="Zhilan" last="Feng">Zhilan Feng</name>
<affiliation>
<nlm:aff id="Aff21">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.169077.e</institution-id>
<institution-id institution-id-type="ISNI">0000 0004 1937 2197</institution-id>
<institution>Department of Mathematics,</institution>
<institution>Purdue University,</institution>
</institution-wrap>
West Lafayette, IN USA</nlm:aff>
</affiliation>
</author>
</analytic>
<series>
<title level="j">Mathematical Models in Epidemiology</title>
<imprint>
<date when="2019">2019</date>
</imprint>
</series>
</biblStruct>
</sourceDesc>
</fileDesc>
<profileDesc>
<textClass></textClass>
</profileDesc>
</teiHeader>
<front>
<div type="abstract" xml:lang="en">
<p id="Par1">Lessons learned from the HIV pandemic, SARS in 2003, the 2009 H1N1 influenza pandemic, the 2014 Ebola outbreak in West Africa, and the ongoing Zika outbreaks in the Americas can be framed under a public health policy model that responds after the fact. Responses often come through reallocation of resources from one disease control effort to a new pressing need. The operating models of preparedness and response are ill-equipped to prevent or ameliorate disease emergence or reemergence at global scales. Epidemiological challenges that are a threat to the economic stability of many regions of the world, particularly those depending on travel and trade, remain at the forefront of the Global Commons. Consequently, efforts to quantify the impact of mobility and trade on disease dynamics have dominated the interests of theoreticians for some time. Our experience includes an H1N1 influenza pandemic crisscrossing the world during 2009 and 2010, the 2014 Ebola outbreaks, limited to regions of West Africa lacking appropriate medical facilities, health infrastructure, and sufficient levels of preparedness and education, and the expanding Zika outbreaks, moving expeditiously across habitats suitable for Aedes aegypti. These provide opportunities to quantify the impact of disease emergence or reemergence on the decisions that individuals take in response to real or perceived disease risks. The case of SARS 2003 in 2003, the efforts to reduce the burden of H1N1 influenza cases in 2009, and the challenges faced in reducing the number of Ebola cases in 2014 are the three recent scenarios that required a timely global response. Studies addressing the impact of centralized sources of information, the impact of information along social connections, or the role of past disease outbreak experiences on the risk-aversion decisions that individuals undertake may help identify and quantify the role of human responses to disease dynamics while recognizing the importance of assessing the timing of disease emergence and reemergence. The co-evolving human responses to disease dynamics are prototypical of the feedbacks that define complex adaptive systems. In short, we live in a socioepisphere being reshaped by ecoepidemiology in the “Era of Information.”</p>
</div>
</front>
<back>
<div1 type="bibliography">
<listBibl>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Andersson, Dan I" uniqKey="Andersson D">Dan I Andersson</name>
</author>
<author>
<name sortKey="Levin, Bruce R" uniqKey="Levin B">Bruce R Levin</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Andreasen, Viggo" uniqKey="Andreasen V">Viggo Andreasen</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Andreasen, Viggo" uniqKey="Andreasen V">Viggo Andreasen</name>
</author>
<author>
<name sortKey="Lin, Juan" uniqKey="Lin J">Juan Lin</name>
</author>
<author>
<name sortKey="Levin, Simon A" uniqKey="Levin S">Simon A. Levin</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Arino, Julien" uniqKey="Arino J">Julien Arino</name>
</author>
<author>
<name sortKey="Jordan, Richard" uniqKey="Jordan R">Richard Jordan</name>
</author>
<author>
<name sortKey="Van Den Driessche, P" uniqKey="Van Den Driessche P">P. van den Driessche</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Arino, Julien" uniqKey="Arino J">Julien Arino</name>
</author>
<author>
<name sortKey="Van Den Driessche, P" uniqKey="Van Den Driessche P">P. van den Driessche</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Auld, M Christopher" uniqKey="Auld M">M.Christopher Auld</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bajardi, Paolo" uniqKey="Bajardi P">Paolo Bajardi</name>
</author>
<author>
<name sortKey="Poletto, Chiara" uniqKey="Poletto C">Chiara Poletto</name>
</author>
<author>
<name sortKey="Ramasco, Jose J" uniqKey="Ramasco J">Jose J. Ramasco</name>
</author>
<author>
<name sortKey="Tizzoni, Michele" uniqKey="Tizzoni M">Michele Tizzoni</name>
</author>
<author>
<name sortKey="Colizza, Vittoria" uniqKey="Colizza V">Vittoria Colizza</name>
</author>
<author>
<name sortKey="Vespignani, Alessandro" uniqKey="Vespignani A">Alessandro Vespignani</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bansal, Shweta" uniqKey="Bansal S">Shweta Bansal</name>
</author>
<author>
<name sortKey="Grenfell, Bryan T" uniqKey="Grenfell B">Bryan T Grenfell</name>
</author>
<author>
<name sortKey="Meyers, Lauren Ancel" uniqKey="Meyers L">Lauren Ancel Meyers</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Baroyan, O V" uniqKey="Baroyan O">O. V. Baroyan</name>
</author>
<author>
<name sortKey="Rvachev, L A" uniqKey="Rvachev L">L. A. Rvachev</name>
</author>
<author>
<name sortKey="Basilevsky, U V" uniqKey="Basilevsky U">U. V. Basilevsky</name>
</author>
<author>
<name sortKey="Ermakov, V V" uniqKey="Ermakov V">V. V. Ermakov</name>
</author>
<author>
<name sortKey="Frank, K D" uniqKey="Frank K">K. D. Frank</name>
</author>
<author>
<name sortKey="Rvachev, M A" uniqKey="Rvachev M">M. A. Rvachev</name>
</author>
<author>
<name sortKey="Shashkov, V A" uniqKey="Shashkov V">V. A. Shashkov</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bellomo, Nicola" uniqKey="Bellomo N">NICOLA BELLOMO</name>
</author>
<author>
<name sortKey="Piccoli, Benedetto" uniqKey="Piccoli B">BENEDETTO PICCOLI</name>
</author>
<author>
<name sortKey="Tosin, Andrea" uniqKey="Tosin A">ANDREA TOSIN</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bichara, Derdei" uniqKey="Bichara D">Derdei Bichara</name>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bichara, Derdei" uniqKey="Bichara D">Derdei Bichara</name>
</author>
<author>
<name sortKey="Kang, Yun" uniqKey="Kang Y">Yun Kang</name>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
<author>
<name sortKey="Horan, Richard" uniqKey="Horan R">Richard Horan</name>
</author>
<author>
<name sortKey="Perrings, Charles" uniqKey="Perrings C">Charles Perrings</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Brauer, Fred" uniqKey="Brauer F">Fred Brauer</name>
</author>
<author>
<name sortKey="Van Den Driessche, Pauline" uniqKey="Van Den Driessche P">Pauline van den Driessche</name>
</author>
<author>
<name sortKey="Wu, Jianhong" uniqKey="Wu J">Jianhong Wu</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Castillo Chavez, C" uniqKey="Castillo Chavez C">C. Castillo-Chavez</name>
</author>
<author>
<name sortKey="Barley, K" uniqKey="Barley K">K. Barley</name>
</author>
<author>
<name sortKey="Bichara, D" uniqKey="Bichara D">D. Bichara</name>
</author>
<author>
<name sortKey="Chowell, D" uniqKey="Chowell D">D. Chowell</name>
</author>
<author>
<name sortKey="Diaz Herrera, E" uniqKey="Diaz Herrera E">E. Diaz Herrera</name>
</author>
<author>
<name sortKey="Espinoza, B" uniqKey="Espinoza B">B. Espinoza</name>
</author>
<author>
<name sortKey="Moreno, V" uniqKey="Moreno V">V. Moreno</name>
</author>
<author>
<name sortKey="Towers, S" uniqKey="Towers S">S. Towers</name>
</author>
<author>
<name sortKey="Yong, K E" uniqKey="Yong K">K. E. Yong</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
<author>
<name sortKey="Bichara, Derdei" uniqKey="Bichara D">Derdei Bichara</name>
</author>
<author>
<name sortKey="Morin, Benjamin R" uniqKey="Morin B">Benjamin R. Morin</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
<author>
<name sortKey="Curtiss, Roy" uniqKey="Curtiss R">Roy Curtiss</name>
</author>
<author>
<name sortKey="Daszak, Peter" uniqKey="Daszak P">Peter Daszak</name>
</author>
<author>
<name sortKey="Levin, Simon A" uniqKey="Levin S">Simon A Levin</name>
</author>
<author>
<name sortKey="Patterson Lomba, Oscar" uniqKey="Patterson Lomba O">Oscar Patterson-Lomba</name>
</author>
<author>
<name sortKey="Perrings, Charles" uniqKey="Perrings C">Charles Perrings</name>
</author>
<author>
<name sortKey="Poste, George" uniqKey="Poste G">George Poste</name>
</author>
<author>
<name sortKey="Towers, Sherry" uniqKey="Towers S">Sherry Towers</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Castillo Chavez, C" uniqKey="Castillo Chavez C">C. Castillo-Chavez</name>
</author>
<author>
<name sortKey="Hethcote, H W" uniqKey="Hethcote H">H. W. Hethcote</name>
</author>
<author>
<name sortKey="Andreasen, V" uniqKey="Andreasen V">V. Andreasen</name>
</author>
<author>
<name sortKey="Levin, S A" uniqKey="Levin S">S. A. Levin</name>
</author>
<author>
<name sortKey="Liu, W M" uniqKey="Liu W">W. M. Liu</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
<author>
<name sortKey="Song, Baojun" uniqKey="Song B">Baojun Song</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Cauchemez, S" uniqKey="Cauchemez S">S. Cauchemez</name>
</author>
<author>
<name sortKey="Bhattarai, A" uniqKey="Bhattarai A">A. Bhattarai</name>
</author>
<author>
<name sortKey="Marchbanks, T L" uniqKey="Marchbanks T">T. L. Marchbanks</name>
</author>
<author>
<name sortKey="Fagan, R P" uniqKey="Fagan R">R. P. Fagan</name>
</author>
<author>
<name sortKey="Ostroff, S" uniqKey="Ostroff S">S. Ostroff</name>
</author>
<author>
<name sortKey="Ferguson, N M" uniqKey="Ferguson N">N. M. Ferguson</name>
</author>
<author>
<name sortKey="Swerdlow, D" uniqKey="Swerdlow D">D. Swerdlow</name>
</author>
<author>
<name sortKey="Sodha, S V" uniqKey="Sodha S">S. V. Sodha</name>
</author>
<author>
<name sortKey="Moll, M E" uniqKey="Moll M">M. E. Moll</name>
</author>
<author>
<name sortKey="Angulo, F J" uniqKey="Angulo F">F. J. Angulo</name>
</author>
<author>
<name sortKey="Palekar, R" uniqKey="Palekar R">R. Palekar</name>
</author>
<author>
<name sortKey="Archer, W R" uniqKey="Archer W">W. R. Archer</name>
</author>
<author>
<name sortKey="Finelli, L" uniqKey="Finelli L">L. Finelli</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Chen, Frederick" uniqKey="Chen F">Frederick Chen</name>
</author>
<author>
<name sortKey="Jiang, Miaohua" uniqKey="Jiang M">Miaohua Jiang</name>
</author>
<author>
<name sortKey="Rabidoux, Scott" uniqKey="Rabidoux S">Scott Rabidoux</name>
</author>
<author>
<name sortKey="Robinson, Stephen" uniqKey="Robinson S">Stephen Robinson</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Chen, Frederick H" uniqKey="Chen F">Frederick H. Chen</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Chen, Frederick H" uniqKey="Chen F">Frederick H. Chen</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Chew, Cynthia" uniqKey="Chew C">Cynthia Chew</name>
</author>
<author>
<name sortKey="Eysenbach, Gunther" uniqKey="Eysenbach G">Gunther Eysenbach</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Chow, Karen" uniqKey="Chow K">Karen Chow</name>
</author>
<author>
<name sortKey="Wang, Xiaohong" uniqKey="Wang X">Xiaohong Wang</name>
</author>
<author>
<name sortKey="Curtiss, R" uniqKey="Curtiss R">R. Curtiss</name>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Chowell, G" uniqKey="Chowell G">G. Chowell</name>
</author>
<author>
<name sortKey="Fenimore, P W" uniqKey="Fenimore P">P.W. Fenimore</name>
</author>
<author>
<name sortKey="Castillo Garsow, M A" uniqKey="Castillo Garsow M">M.A. Castillo-Garsow</name>
</author>
<author>
<name sortKey="Castillo Chavez, C" uniqKey="Castillo Chavez C">C. Castillo-Chavez</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Chowell, Gerardo" uniqKey="Chowell G">Gerardo Chowell</name>
</author>
<author>
<name sortKey="Nishiura, Hiroshi" uniqKey="Nishiura H">Hiroshi Nishiura</name>
</author>
<author>
<name sortKey="Bettencourt, Luis M A" uniqKey="Bettencourt L">Luís M.A Bettencourt</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Chowell, Gerardo" uniqKey="Chowell G">Gerardo Chowell</name>
</author>
<author>
<name sortKey="Viboud, Cecile" uniqKey="Viboud C">Cécile Viboud</name>
</author>
<author>
<name sortKey="Wang, Xiaohong" uniqKey="Wang X">Xiaohong Wang</name>
</author>
<author>
<name sortKey="Bertozzi, Stefano M" uniqKey="Bertozzi S">Stefano M. Bertozzi</name>
</author>
<author>
<name sortKey="Miller, Mark A" uniqKey="Miller M">Mark A. Miller</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Clark, Colin W" uniqKey="Clark C">Colin W. Clark</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Clark, Colin W" uniqKey="Clark C">Colin W. Clark</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Clark, Colin W" uniqKey="Clark C">Colin W. Clark</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Cosner, C" uniqKey="Cosner C">C. Cosner</name>
</author>
<author>
<name sortKey="Beier, J C" uniqKey="Beier J">J.C. Beier</name>
</author>
<author>
<name sortKey="Cantrell, R S" uniqKey="Cantrell R">R.S. Cantrell</name>
</author>
<author>
<name sortKey="Impoinvil, D" uniqKey="Impoinvil D">D. Impoinvil</name>
</author>
<author>
<name sortKey="Kapitanski, L" uniqKey="Kapitanski L">L. Kapitanski</name>
</author>
<author>
<name sortKey="Potts, M D" uniqKey="Potts M">M.D. Potts</name>
</author>
<author>
<name sortKey="Troyo, A" uniqKey="Troyo A">A. Troyo</name>
</author>
<author>
<name sortKey="Ruan, S" uniqKey="Ruan S">S. Ruan</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Daszak, Peter" uniqKey="Daszak P">Peter Daszak</name>
</author>
<author>
<name sortKey="Plowright, R K" uniqKey="Plowright R">R. K. Plowright</name>
</author>
<author>
<name sortKey="Epstein, J H" uniqKey="Epstein J">J. H. Epstein</name>
</author>
<author>
<name sortKey="Pulliam, J" uniqKey="Pulliam J">J. Pulliam</name>
</author>
<author>
<name sortKey="Abdul Rahman, S" uniqKey="Abdul Rahman S">S. Abdul Rahman</name>
</author>
<author>
<name sortKey="Field, H E" uniqKey="Field H">H. E. Field</name>
</author>
<author>
<name sortKey="Jamaluddin, A" uniqKey="Jamaluddin A">A. Jamaluddin</name>
</author>
<author>
<name sortKey="Sharifah, S H" uniqKey="Sharifah S">S. H. Sharifah</name>
</author>
<author>
<name sortKey="Smith, C S" uniqKey="Smith C">C. S. Smith</name>
</author>
<author>
<name sortKey="Olival, K J" uniqKey="Olival K">K. J. Olival</name>
</author>
<author>
<name sortKey="Luby, S" uniqKey="Luby S">S. Luby</name>
</author>
<author>
<name sortKey="Halpin, K" uniqKey="Halpin K">K. Halpin</name>
</author>
<author>
<name sortKey="Hyatt, A D" uniqKey="Hyatt A">A. D. Hyatt</name>
</author>
<author>
<name sortKey="Cunningham, A A" uniqKey="Cunningham A">A. A. Cunningham</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Daszak, Peter" uniqKey="Daszak P">PETER DASZAK</name>
</author>
<author>
<name sortKey="Tabor, Gary M" uniqKey="Tabor G">GARY M. TABOR</name>
</author>
<author>
<name sortKey="Kilpatrick, A Marm" uniqKey="Kilpatrick A">A MARM KILPATRICK</name>
</author>
<author>
<name sortKey="Epstein, Jon" uniqKey="Epstein J">JON EPSTEIN</name>
</author>
<author>
<name sortKey="Plowright, Raina" uniqKey="Plowright R">RAINA PLOWRIGHT</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Domingo, E" uniqKey="Domingo E">E. Domingo</name>
</author>
<author>
<name sortKey="Holland, J J" uniqKey="Holland J">J. J. Holland</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Epstein, Joshua M" uniqKey="Epstein J">Joshua M. Epstein</name>
</author>
<author>
<name sortKey="Parker, Jon" uniqKey="Parker J">Jon Parker</name>
</author>
<author>
<name sortKey="Cummings, Derek" uniqKey="Cummings D">Derek Cummings</name>
</author>
<author>
<name sortKey="Hammond, Ross A" uniqKey="Hammond R">Ross A. Hammond</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Erdem, Mustafa" uniqKey="Erdem M">Mustafa Erdem</name>
</author>
<author>
<name sortKey="Safan, Muntaser" uniqKey="Safan M">Muntaser Safan</name>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Feng, Zhilan" uniqKey="Feng Z">Zhilan Feng</name>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
<author>
<name sortKey="Capurro, Angel F" uniqKey="Capurro A">Angel F. Capurro</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Fenichel, E P" uniqKey="Fenichel E">E. P. Fenichel</name>
</author>
<author>
<name sortKey="Castillo Chavez, C" uniqKey="Castillo Chavez C">C. Castillo-Chavez</name>
</author>
<author>
<name sortKey="Ceddia, M G" uniqKey="Ceddia M">M. G. Ceddia</name>
</author>
<author>
<name sortKey="Chowell, G" uniqKey="Chowell G">G. Chowell</name>
</author>
<author>
<name sortKey="Parra, P A G" uniqKey="Parra P">P. A. G. Parra</name>
</author>
<author>
<name sortKey="Hickling, G J" uniqKey="Hickling G">G. J. Hickling</name>
</author>
<author>
<name sortKey="Holloway, G" uniqKey="Holloway G">G. Holloway</name>
</author>
<author>
<name sortKey="Horan, R" uniqKey="Horan R">R. Horan</name>
</author>
<author>
<name sortKey="Morin, B" uniqKey="Morin B">B. Morin</name>
</author>
<author>
<name sortKey="Perrings, C" uniqKey="Perrings C">C. Perrings</name>
</author>
<author>
<name sortKey="Springborn, M" uniqKey="Springborn M">M. Springborn</name>
</author>
<author>
<name sortKey="Velazquez, L" uniqKey="Velazquez L">L. Velazquez</name>
</author>
<author>
<name sortKey="Villalobos, C" uniqKey="Villalobos C">C. Villalobos</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Fenichel, Eli P" uniqKey="Fenichel E">Eli P. Fenichel</name>
</author>
<author>
<name sortKey="Kuminoff, Nicolai V" uniqKey="Kuminoff N">Nicolai V. Kuminoff</name>
</author>
<author>
<name sortKey="Chowell, Gerardo" uniqKey="Chowell G">Gerardo Chowell</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Ferguson, Neil" uniqKey="Ferguson N">Neil Ferguson</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Funk, S" uniqKey="Funk S">S. Funk</name>
</author>
<author>
<name sortKey="Gilad, E" uniqKey="Gilad E">E. Gilad</name>
</author>
<author>
<name sortKey="Jansen, V A A" uniqKey="Jansen V">V.A.A. Jansen</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Funk, S" uniqKey="Funk S">S. Funk</name>
</author>
<author>
<name sortKey="Gilad, E" uniqKey="Gilad E">E. Gilad</name>
</author>
<author>
<name sortKey="Watkins, C" uniqKey="Watkins C">C. Watkins</name>
</author>
<author>
<name sortKey="Jansen, V A A" uniqKey="Jansen V">V. A. A. Jansen</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Gersovitz, Mark" uniqKey="Gersovitz M">Mark Gersovitz</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Hadeler, K P" uniqKey="Hadeler K">K.P. Hadeler</name>
</author>
<author>
<name sortKey="Castillo Chavez, C" uniqKey="Castillo Chavez C">C. Castillo-Chavez</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Hansen, Elsa" uniqKey="Hansen E">Elsa Hansen</name>
</author>
<author>
<name sortKey="Day, Troy" uniqKey="Day T">Troy Day</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Helbing, Dirk" uniqKey="Helbing D">Dirk Helbing</name>
</author>
<author>
<name sortKey="Farkas, Illes" uniqKey="Farkas I">Illés Farkas</name>
</author>
<author>
<name sortKey="Vicsek, Tamas" uniqKey="Vicsek T">Tamás Vicsek</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Helbing, Dirk" uniqKey="Helbing D">Dirk Helbing</name>
</author>
<author>
<name sortKey="Keltsch, Joachim" uniqKey="Keltsch J">Joachim Keltsch</name>
</author>
<author>
<name sortKey="Molnar, Peter" uniqKey="Molnar P">Péter Molnár</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Hethcote, Herbert W" uniqKey="Hethcote H">Herbert W. Hethcote</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Hethcote, H W" uniqKey="Hethcote H">H.W. Hethcote</name>
</author>
<author>
<name sortKey="Yi, Li" uniqKey="Yi L">Li Yi</name>
</author>
<author>
<name sortKey="Zhujun, Jing" uniqKey="Zhujun J">Jing Zhujun</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Hon Rio, Nildimar Alves" uniqKey="Hon Rio N">Nildimar Alves Honório</name>
</author>
<author>
<name sortKey="Silva, Wellington Da Costa" uniqKey="Silva W">Wellington da Costa Silva</name>
</author>
<author>
<name sortKey="Leite, Paulo Jose" uniqKey="Leite P">Paulo José Leite</name>
</author>
<author>
<name sortKey="Goncalves, Jaylei Monteiro" uniqKey="Goncalves J">Jaylei Monteiro Gonçalves</name>
</author>
<author>
<name sortKey="Lounibos, Leon Philip" uniqKey="Lounibos L">Leon Philip Lounibos</name>
</author>
<author>
<name sortKey="Lourenco De Oliveira, Ricardo" uniqKey="Lourenco De Oliveira R">Ricardo Lourenço-de-Oliveira</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Hsieh, Ying Hen" uniqKey="Hsieh Y">Ying-Hen Hsieh</name>
</author>
<author>
<name sortKey="Liu, Junli" uniqKey="Liu J">Junli Liu</name>
</author>
<author>
<name sortKey="Tzeng, Yun Huei" uniqKey="Tzeng Y">Yun-Huei Tzeng</name>
</author>
<author>
<name sortKey="Wu, Jianhong" uniqKey="Wu J">Jianhong Wu</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Iggidr, Aberrahman" uniqKey="Iggidr A">Aberrahman Iggidr</name>
</author>
<author>
<name sortKey="Sallet, Gauthier" uniqKey="Sallet G">Gauthier Sallet</name>
</author>
<author>
<name sortKey="Souza, Max O" uniqKey="Souza M">Max O. Souza</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Karesh, William B" uniqKey="Karesh W">William B. Karesh</name>
</author>
<author>
<name sortKey="Cook, Robert A" uniqKey="Cook R">Robert A. Cook</name>
</author>
<author>
<name sortKey="Bennett, Elizabeth L" uniqKey="Bennett E">Elizabeth L. Bennett</name>
</author>
<author>
<name sortKey="Newcomb, James" uniqKey="Newcomb J">James Newcomb</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Khan, Kamran" uniqKey="Khan K">Kamran Khan</name>
</author>
<author>
<name sortKey="Arino, Julien" uniqKey="Arino J">Julien Arino</name>
</author>
<author>
<name sortKey="Hu, Wei" uniqKey="Hu W">Wei Hu</name>
</author>
<author>
<name sortKey="Raposo, Paulo" uniqKey="Raposo P">Paulo Raposo</name>
</author>
<author>
<name sortKey="Sears, Jennifer" uniqKey="Sears J">Jennifer Sears</name>
</author>
<author>
<name sortKey="Calderon, Felipe" uniqKey="Calderon F">Felipe Calderon</name>
</author>
<author>
<name sortKey="Heidebrecht, Christine" uniqKey="Heidebrecht C">Christine Heidebrecht</name>
</author>
<author>
<name sortKey="Macdonald, Michael" uniqKey="Macdonald M">Michael Macdonald</name>
</author>
<author>
<name sortKey="Liauw, Jessica" uniqKey="Liauw J">Jessica Liauw</name>
</author>
<author>
<name sortKey="Chan, Angie" uniqKey="Chan A">Angie Chan</name>
</author>
<author>
<name sortKey="Gardam, Michael" uniqKey="Gardam M">Michael Gardam</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Kilpatrick, A M" uniqKey="Kilpatrick A">A. M. Kilpatrick</name>
</author>
<author>
<name sortKey="Chmura, A A" uniqKey="Chmura A">A. A. Chmura</name>
</author>
<author>
<name sortKey="Gibbons, D W" uniqKey="Gibbons D">D. W. Gibbons</name>
</author>
<author>
<name sortKey="Fleischer, R C" uniqKey="Fleischer R">R. C. Fleischer</name>
</author>
<author>
<name sortKey="Marra, P P" uniqKey="Marra P">P. P. Marra</name>
</author>
<author>
<name sortKey="Daszak, P" uniqKey="Daszak P">P. Daszak</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Klein, Eili" uniqKey="Klein E">EILI KLEIN</name>
</author>
<author>
<name sortKey="Laxminarayan, Ramanan" uniqKey="Laxminarayan R">RAMANAN LAXMINARAYAN</name>
</author>
<author>
<name sortKey="Smith, David L" uniqKey="Smith D">DAVID L. SMITH</name>
</author>
<author>
<name sortKey="Gilligan, Christopher A" uniqKey="Gilligan C">CHRISTOPHER A. GILLIGAN</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Koram, K A" uniqKey="Koram K">K.A. Koram</name>
</author>
<author>
<name sortKey="Bennett, S" uniqKey="Bennett S">S. Bennett</name>
</author>
<author>
<name sortKey="Adiamah, J H" uniqKey="Adiamah J">J.H. Adiamah</name>
</author>
<author>
<name sortKey="Greenwood, B M" uniqKey="Greenwood B">B.M. Greenwood</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Levin, Simon A" uniqKey="Levin S">Simon A. Levin</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Levin, S A" uniqKey="Levin S">S. A. Levin</name>
</author>
<author>
<name sortKey="Paine, R T" uniqKey="Paine R">R. T. Paine</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Levins, R" uniqKey="Levins R">R. Levins</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Levinthal, Daniel A" uniqKey="Levinthal D">Daniel A. Levinthal</name>
</author>
<author>
<name sortKey="March, James G" uniqKey="March J">James G. March</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Lipsitch, M" uniqKey="Lipsitch M">M. Lipsitch</name>
</author>
<author>
<name sortKey="Bergstrom, C T" uniqKey="Bergstrom C">C. T. Bergstrom</name>
</author>
<author>
<name sortKey="Levin, B R" uniqKey="Levin B">B. R. Levin</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Lipsitch, Marc" uniqKey="Lipsitch M">Marc Lipsitch</name>
</author>
<author>
<name sortKey="Cohen, Ted" uniqKey="Cohen T">Ted Cohen</name>
</author>
<author>
<name sortKey="Murray, Megan" uniqKey="Murray M">Megan Murray</name>
</author>
<author>
<name sortKey="Levin, Bruce R" uniqKey="Levin B">Bruce R Levin</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Moghadas, Seyed M" uniqKey="Moghadas S">Seyed M Moghadas</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Moghadas, Seyed M" uniqKey="Moghadas S">SEYED M. MOGHADAS</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Moghadas, Seyed M" uniqKey="Moghadas S">Seyed M. Moghadas</name>
</author>
<author>
<name sortKey="Bowman, Christopher S" uniqKey="Bowman C">Christopher S. Bowman</name>
</author>
<author>
<name sortKey="Rost, Gergely" uniqKey="Rost G">Gergely Röst</name>
</author>
<author>
<name sortKey="Wu, Jianhong" uniqKey="Wu J">Jianhong Wu</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Morin, Benjamin R" uniqKey="Morin B">BENJAMIN R. MORIN</name>
</author>
<author>
<name sortKey="Fenichel, Eli P" uniqKey="Fenichel E">ELI P. FENICHEL</name>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">CARLOS CASTILLO-CHAVEZ</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Morin, Benjamin R" uniqKey="Morin B">Benjamin R. Morin</name>
</author>
<author>
<name sortKey="Perrings, Charles" uniqKey="Perrings C">Charles Perrings</name>
</author>
<author>
<name sortKey="Kinzig, Ann" uniqKey="Kinzig A">Ann Kinzig</name>
</author>
<author>
<name sortKey="Levin, Simon" uniqKey="Levin S">Simon Levin</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Morin, Benjamin R" uniqKey="Morin B">Benjamin R. Morin</name>
</author>
<author>
<name sortKey="Perrings, Charles" uniqKey="Perrings C">Charles Perrings</name>
</author>
<author>
<name sortKey="Levin, Simon" uniqKey="Levin S">Simon Levin</name>
</author>
<author>
<name sortKey="Kinzig, Ann" uniqKey="Kinzig A">Ann Kinzig</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Naresh, R" uniqKey="Naresh R">R. Naresh</name>
</author>
<author>
<name sortKey="Tripathi, A" uniqKey="Tripathi A">A. Tripathi</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Newman, M E J" uniqKey="Newman M">M. E. J. Newman</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Nuno, M" uniqKey="Nuno M">M. Nuno</name>
</author>
<author>
<name sortKey="Reichert, T A" uniqKey="Reichert T">T. A. Reichert</name>
</author>
<author>
<name sortKey="Chowell, G" uniqKey="Chowell G">G. Chowell</name>
</author>
<author>
<name sortKey="Gumel, A B" uniqKey="Gumel A">A. B. Gumel</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Paine, R T" uniqKey="Paine R">R. T. Paine</name>
</author>
<author>
<name sortKey="Levin, Simon A" uniqKey="Levin S">Simon A. Levin</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Pavlin, Boris I" uniqKey="Pavlin B">Boris I. Pavlin</name>
</author>
<author>
<name sortKey="Schloegel, Lisa M" uniqKey="Schloegel L">Lisa M. Schloegel</name>
</author>
<author>
<name sortKey="Daszak, Peter" uniqKey="Daszak P">Peter Daszak</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Pennings, Joost M E" uniqKey="Pennings J">Joost M.E. Pennings</name>
</author>
<author>
<name sortKey="Wansink, Brian" uniqKey="Wansink B">Brian Wansink</name>
</author>
<author>
<name sortKey="Meulenberg, Matthew T G" uniqKey="Meulenberg M">Matthew T.G. Meulenberg</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Perra, Nicola" uniqKey="Perra N">Nicola Perra</name>
</author>
<author>
<name sortKey="Balcan, Duygu" uniqKey="Balcan D">Duygu Balcan</name>
</author>
<author>
<name sortKey="Goncalves, Bruno" uniqKey="Goncalves B">Bruno Gonçalves</name>
</author>
<author>
<name sortKey="Vespignani, Alessandro" uniqKey="Vespignani A">Alessandro Vespignani</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Perrings, Charles" uniqKey="Perrings C">Charles Perrings</name>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
<author>
<name sortKey="Chowell, Gerardo" uniqKey="Chowell G">Gerardo Chowell</name>
</author>
<author>
<name sortKey="Daszak, Peter" uniqKey="Daszak P">Peter Daszak</name>
</author>
<author>
<name sortKey="Fenichel, Eli P" uniqKey="Fenichel E">Eli P. Fenichel</name>
</author>
<author>
<name sortKey="Finnoff, David" uniqKey="Finnoff D">David Finnoff</name>
</author>
<author>
<name sortKey="Horan, Richard D" uniqKey="Horan R">Richard D. Horan</name>
</author>
<author>
<name sortKey="Kilpatrick, A Marm" uniqKey="Kilpatrick A">A. Marm Kilpatrick</name>
</author>
<author>
<name sortKey="Kinzig, Ann P" uniqKey="Kinzig A">Ann P. Kinzig</name>
</author>
<author>
<name sortKey="Kuminoff, Nicolai V" uniqKey="Kuminoff N">Nicolai V. Kuminoff</name>
</author>
<author>
<name sortKey="Levin, Simon" uniqKey="Levin S">Simon Levin</name>
</author>
<author>
<name sortKey="Morin, Benjamin" uniqKey="Morin B">Benjamin Morin</name>
</author>
<author>
<name sortKey="Smith, Katherine F" uniqKey="Smith K">Katherine F. Smith</name>
</author>
<author>
<name sortKey="Springborn, Michael" uniqKey="Springborn M">Michael Springborn</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Porco, Travis C" uniqKey="Porco T">Travis C. Porco</name>
</author>
<author>
<name sortKey="Small, Peter M" uniqKey="Small P">Peter M. Small</name>
</author>
<author>
<name sortKey="Blower, Sally M" uniqKey="Blower S">Sally M. Blower</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Preisser, Evan L" uniqKey="Preisser E">Evan L. Preisser</name>
</author>
<author>
<name sortKey="Bolnick, Daniel I" uniqKey="Bolnick D">Daniel I. Bolnick</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Raimundo, Silvia Martorano" uniqKey="Raimundo S">Silvia Martorano Raimundo</name>
</author>
<author>
<name sortKey="Engel, Alejandro B" uniqKey="Engel A">Alejandro B. Engel</name>
</author>
<author>
<name sortKey="Yang, Hyun Mo" uniqKey="Yang H">Hyun Mo Yang</name>
</author>
<author>
<name sortKey="Bassanezi, Rodney Carlos" uniqKey="Bassanezi R">Rodney Carlos Bassanezi</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Reluga, Timothy C" uniqKey="Reluga T">Timothy C. Reluga</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Rimmelzwaan, G F" uniqKey="Rimmelzwaan G">G. F. Rimmelzwaan</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Rodriguez, D" uniqKey="Rodriguez D">D RodrÍguez</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Ruktanonchai, Nick W" uniqKey="Ruktanonchai N">Nick W. Ruktanonchai</name>
</author>
<author>
<name sortKey="Smith, David L" uniqKey="Smith D">David L. Smith</name>
</author>
<author>
<name sortKey="De Leenheer, Patrick" uniqKey="De Leenheer P">Patrick De Leenheer</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Schulzer, Michael" uniqKey="Schulzer M">MICHAEL SCHULZER</name>
</author>
<author>
<name sortKey="Radhamani, M P" uniqKey="Radhamani M">M P RADHAMANI</name>
</author>
<author>
<name sortKey="Grzybowski, Stefan" uniqKey="Grzybowski S">STEFAN GRZYBOWSKI</name>
</author>
<author>
<name sortKey="Mak, Edwin" uniqKey="Mak E">EDWIN MAK</name>
</author>
<author>
<name sortKey="Fitzgerald, J Mark" uniqKey="Fitzgerald J">J MARK FITZGERALD</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Smith, David L" uniqKey="Smith D">David L Smith</name>
</author>
<author>
<name sortKey="Dushoff, Jonathan" uniqKey="Dushoff J">Jonathan Dushoff</name>
</author>
<author>
<name sortKey="Mckenzie, F Ellis" uniqKey="Mckenzie F">F. Ellis McKenzie</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Tatem, Andrew J" uniqKey="Tatem A">Andrew J. Tatem</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Tatem, A J" uniqKey="Tatem A">A. J. Tatem</name>
</author>
<author>
<name sortKey="Hay, S I" uniqKey="Hay S">S. I. Hay</name>
</author>
<author>
<name sortKey="Rogers, D J" uniqKey="Rogers D">D. J. Rogers</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Towers, Sherry" uniqKey="Towers S">Sherry Towers</name>
</author>
<author>
<name sortKey="Afzal, Shehzad" uniqKey="Afzal S">Shehzad Afzal</name>
</author>
<author>
<name sortKey="Bernal, Gilbert" uniqKey="Bernal G">Gilbert Bernal</name>
</author>
<author>
<name sortKey="Bliss, Nadya" uniqKey="Bliss N">Nadya Bliss</name>
</author>
<author>
<name sortKey="Brown, Shala" uniqKey="Brown S">Shala Brown</name>
</author>
<author>
<name sortKey="Espinoza, Baltazar" uniqKey="Espinoza B">Baltazar Espinoza</name>
</author>
<author>
<name sortKey="Jackson, Jasmine" uniqKey="Jackson J">Jasmine Jackson</name>
</author>
<author>
<name sortKey="Judson Garcia, Julia" uniqKey="Judson Garcia J">Julia Judson-Garcia</name>
</author>
<author>
<name sortKey="Khan, Maryam" uniqKey="Khan M">Maryam Khan</name>
</author>
<author>
<name sortKey="Lin, Michael" uniqKey="Lin M">Michael Lin</name>
</author>
<author>
<name sortKey="Mamada, Robert" uniqKey="Mamada R">Robert Mamada</name>
</author>
<author>
<name sortKey="Moreno, Victor M" uniqKey="Moreno V">Victor M. Moreno</name>
</author>
<author>
<name sortKey="Nazari, Fereshteh" uniqKey="Nazari F">Fereshteh Nazari</name>
</author>
<author>
<name sortKey="Okuneye, Kamaldeen" uniqKey="Okuneye K">Kamaldeen Okuneye</name>
</author>
<author>
<name sortKey="Ross, Mary L" uniqKey="Ross M">Mary L. Ross</name>
</author>
<author>
<name sortKey="Rodriguez, Claudia" uniqKey="Rodriguez C">Claudia Rodriguez</name>
</author>
<author>
<name sortKey="Medlock, Jan" uniqKey="Medlock J">Jan Medlock</name>
</author>
<author>
<name sortKey="Ebert, David" uniqKey="Ebert D">David Ebert</name>
</author>
<author>
<name sortKey="Castillo Chavez, Carlos" uniqKey="Castillo Chavez C">Carlos Castillo-Chavez</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Webb, Glenn" uniqKey="Webb G">Glenn Webb</name>
</author>
<author>
<name sortKey="Blaser, Martin" uniqKey="Blaser M">Martin Blaser</name>
</author>
<author>
<name sortKey="Zhu, Huaiping" uniqKey="Zhu H">Huaiping Zhu</name>
</author>
<author>
<name sortKey="Ardal, Sten" uniqKey="Ardal S">Sten Ardal</name>
</author>
<author>
<name sortKey="Wu, Jianhong" uniqKey="Wu J">Jianhong Wu</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Wesolowski, A" uniqKey="Wesolowski A">A. Wesolowski</name>
</author>
<author>
<name sortKey="Eagle, N" uniqKey="Eagle N">N. Eagle</name>
</author>
<author>
<name sortKey="Tatem, A J" uniqKey="Tatem A">A. J. Tatem</name>
</author>
<author>
<name sortKey="Smith, D L" uniqKey="Smith D">D. L. Smith</name>
</author>
<author>
<name sortKey="Noor, A M" uniqKey="Noor A">A. M. Noor</name>
</author>
<author>
<name sortKey="Snow, R W" uniqKey="Snow R">R. W. Snow</name>
</author>
<author>
<name sortKey="Buckee, C O" uniqKey="Buckee C">C. O. Buckee</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Wilson, Edward O" uniqKey="Wilson E">Edward O. Wilson</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Xiao, Yanyu" uniqKey="Xiao Y">Yanyu Xiao</name>
</author>
<author>
<name sortKey="Brauer, Fred" uniqKey="Brauer F">Fred Brauer</name>
</author>
<author>
<name sortKey="Moghadas, Seyed M" uniqKey="Moghadas S">Seyed M. Moghadas</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Yorke, James A" uniqKey="Yorke J">JAMES A. YORKE</name>
</author>
<author>
<name sortKey="Hethcote, Herbert W" uniqKey="Hethcote H">HERBERT W. HETHCOTE</name>
</author>
<author>
<name sortKey="Nold, Annett" uniqKey="Nold A">ANNETT NOLD</name>
</author>
</analytic>
</biblStruct>
</listBibl>
</div1>
</back>
</TEI>
<pmc article-type="chapter-article">
<pmc-dir>properties open_access</pmc-dir>
<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">978-1-4939-9828-9</journal-id>
<journal-id journal-id-type="doi">10.1007/978-1-4939-9828-9</journal-id>
<journal-id journal-id-type="nlm-ta">Mathematical Models in Epidemiology</journal-id>
<journal-title-group>
<journal-title>Mathematical Models in Epidemiology</journal-title>
</journal-title-group>
<isbn publication-format="print">978-1-4939-9826-5</isbn>
<isbn publication-format="electronic">978-1-4939-9828-9</isbn>
</journal-meta>
<article-meta>
<article-id pub-id-type="pmc">7123038</article-id>
<article-id pub-id-type="publisher-id">16</article-id>
<article-id pub-id-type="doi">10.1007/978-1-4939-9828-9_16</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Article</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>Challenges, Opportunities and Theoretical Epidemiology</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Brauer</surname>
<given-names>Fred</given-names>
</name>
<xref ref-type="aff" rid="Aff19">19</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
<xref ref-type="aff" rid="Aff20">20</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Feng</surname>
<given-names>Zhilan</given-names>
</name>
<xref ref-type="aff" rid="Aff21">21</xref>
</contrib>
<aff id="Aff19">
<label>19</label>
<institution-wrap>
<institution-id institution-id-type="GRID">grid.17091.3e</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2288 9830</institution-id>
<institution>Department of Mathematics,</institution>
<institution>University of British Columbia,</institution>
</institution-wrap>
Vancouver, BC Canada</aff>
<aff id="Aff20">
<label>20</label>
<institution-wrap>
<institution-id institution-id-type="GRID">grid.215654.1</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2151 2636</institution-id>
<institution>Mathematical and Computational Modeling Center (MCMSC), Department of Mathematics and Statistics,</institution>
<institution>Arizona State University,</institution>
</institution-wrap>
Tempe, AZ USA</aff>
<aff id="Aff21">
<label>21</label>
<institution-wrap>
<institution-id institution-id-type="GRID">grid.169077.e</institution-id>
<institution-id institution-id-type="ISNI">0000 0004 1937 2197</institution-id>
<institution>Department of Mathematics,</institution>
<institution>Purdue University,</institution>
</institution-wrap>
West Lafayette, IN USA</aff>
</contrib-group>
<pub-date pub-type="epub">
<day>25</day>
<month>06</month>
<year>2019</year>
</pub-date>
<pub-date pub-type="collection">
<year>2019</year>
</pub-date>
<volume>69</volume>
<fpage>507</fpage>
<lpage>531</lpage>
<permissions>
<copyright-statement>© Springer Science+Business Media, LLC, part of Springer Nature 2019</copyright-statement>
<license>
<license-p>This article is made available via the PMC Open Access Subset for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.</license-p>
</license>
</permissions>
<abstract id="Abs1">
<p id="Par1">Lessons learned from the HIV pandemic, SARS in 2003, the 2009 H1N1 influenza pandemic, the 2014 Ebola outbreak in West Africa, and the ongoing Zika outbreaks in the Americas can be framed under a public health policy model that responds after the fact. Responses often come through reallocation of resources from one disease control effort to a new pressing need. The operating models of preparedness and response are ill-equipped to prevent or ameliorate disease emergence or reemergence at global scales. Epidemiological challenges that are a threat to the economic stability of many regions of the world, particularly those depending on travel and trade, remain at the forefront of the Global Commons. Consequently, efforts to quantify the impact of mobility and trade on disease dynamics have dominated the interests of theoreticians for some time. Our experience includes an H1N1 influenza pandemic crisscrossing the world during 2009 and 2010, the 2014 Ebola outbreaks, limited to regions of West Africa lacking appropriate medical facilities, health infrastructure, and sufficient levels of preparedness and education, and the expanding Zika outbreaks, moving expeditiously across habitats suitable for Aedes aegypti. These provide opportunities to quantify the impact of disease emergence or reemergence on the decisions that individuals take in response to real or perceived disease risks. The case of SARS 2003 in 2003, the efforts to reduce the burden of H1N1 influenza cases in 2009, and the challenges faced in reducing the number of Ebola cases in 2014 are the three recent scenarios that required a timely global response. Studies addressing the impact of centralized sources of information, the impact of information along social connections, or the role of past disease outbreak experiences on the risk-aversion decisions that individuals undertake may help identify and quantify the role of human responses to disease dynamics while recognizing the importance of assessing the timing of disease emergence and reemergence. The co-evolving human responses to disease dynamics are prototypical of the feedbacks that define complex adaptive systems. In short, we live in a socioepisphere being reshaped by ecoepidemiology in the “Era of Information.”</p>
</abstract>
<custom-meta-group>
<custom-meta>
<meta-name>issue-copyright-statement</meta-name>
<meta-value>© Springer Science+Business Media, LLC, part of Springer Nature 2019</meta-value>
</custom-meta>
</custom-meta-group>
</article-meta>
</front>
<body>
<p id="Par2">Lessons learned from the HIV pandemic, SARS in 2003, the 2009 H1N1 influenza pandemic, the 2014 Ebola outbreak in West Africa, and the ongoing Zika outbreaks in the Americas can be framed under a public health policy model that responds after the fact. Responses often come through reallocation of resources from one disease control effort to a new pressing need. The operating models of preparedness and response are ill-equipped to prevent or ameliorate disease emergence or reemergence at global scales [
<xref ref-type="bibr" rid="CR27">27</xref>
]. Epidemiological challenges that are a threat to the economic stability of many regions of the world, particularly those depending on travel and trade [
<xref ref-type="bibr" rid="CR132">132</xref>
], remain at the forefront of the Global Commons. Consequently, efforts to quantify the impact of mobility and trade on disease dynamics have dominated the interests of theoreticians for some time [
<xref ref-type="bibr" rid="CR14">14</xref>
,
<xref ref-type="bibr" rid="CR143">143</xref>
]. Our experience includes an H1N1 influenza pandemic crisscrossing the world during 2009 and 2010, the 2014 Ebola outbreaks, limited to regions of West Africa lacking appropriate medical facilities, health infrastructure, and sufficient levels of preparedness and education, and the expanding Zika outbreaks, moving expeditiously across habitats suitable for Aedes aegypti. These provide opportunities to quantify the impact of disease emergence or reemergence on the decisions that individuals take in response to real or perceived disease risks [
<xref ref-type="bibr" rid="CR11">11</xref>
,
<xref ref-type="bibr" rid="CR62">62</xref>
,
<xref ref-type="bibr" rid="CR93">93</xref>
]. The case of SARS in 2003 [
<xref ref-type="bibr" rid="CR40">40</xref>
], the efforts to reduce the burden of H1N1 influenza cases in 2009 [
<xref ref-type="bibr" rid="CR33">33</xref>
,
<xref ref-type="bibr" rid="CR62">62</xref>
,
<xref ref-type="bibr" rid="CR80">80</xref>
,
<xref ref-type="bibr" rid="CR93">93</xref>
] and the challenges faced in reducing the number of Ebola cases in 2014 [
<xref ref-type="bibr" rid="CR24">24</xref>
,
<xref ref-type="bibr" rid="CR27">27</xref>
] are but three recent scenarios that required a timely global response. Studies addressing the impact of centralized sources of information [
<xref ref-type="bibr" rid="CR150">150</xref>
], the impact of information along social connections [
<xref ref-type="bibr" rid="CR33">33</xref>
,
<xref ref-type="bibr" rid="CR37">37</xref>
,
<xref ref-type="bibr" rid="CR42">42</xref>
], or the role of past disease outbreak experiences [
<xref ref-type="bibr" rid="CR105">105</xref>
,
<xref ref-type="bibr" rid="CR130">130</xref>
] on the risk-aversion decisions that individuals undertake may help identify and quantify the role of human responses to disease dynamics while recognizing the importance of assessing the timing of disease emergence and reemergence. The co-evolving human responses to disease dynamics are prototypical of the feedbacks that define complex adaptive systems. In short, we live in a socioepisphere being reshaped by ecoepidemiology in the “Era of Information”.</p>
<p id="Par3">What are the questions and modes of thinking that should be driving ongoing research on the dynamics, evolution, and control of epidemic diseases at the population level? The challenges of SARS, Ebola, Influenza, Zika, and other diseases are immense. While we may guess which emerging or re-emerging disease may lead to the next possible catastrophe, we cannot know. The contemporary philosopher Yogi Berra is rumored to have said, “Making predictions is hard, especially about the future”. There are some epidemiological topics that have already received some attention but are not yet fully developed. In the rest of this chapter we highlight some challenges, opportunities, and promising approaches in the study of disease dynamics at the population level.</p>
<sec id="Sec1">
<title>Disease and the Global Commons</title>
<p id="Par4">As has been noted, “The identification of a theoretical explanatory framework that accounts for the pattern regularity exhibited by a large number of host–parasite systems, including those sustained by host–vector epidemiological dynamics, is but one of the challenges facing the co-evolving fields of computational, evolutionary, and theoretical epidemiology” [
<xref ref-type="bibr" rid="CR25">25</xref>
]. Furthermore, “The emergence of new diseases, the persistence of recurring diseases and the re-emergence of old foes, the result of genetic changes or shifts in demographic, and environmental shifts have increased due to mobility, global connectedness, trade, bird migration, poverty and long-lasting violent conflicts. These diseases often present modeling challenges which may yield to existing analytic techniques but sometimes require new mathematics” [
<xref ref-type="bibr" rid="CR25">25</xref>
].</p>
<p id="Par5">The Global Commons are continuously reshaped by the ability of an increasing proportion of the human population to live, move, or trade nearly anywhere. Therefore, understanding the patterns of interactions between humans, or between humans and vectors, as well as their relationships to patterns of individual movement, particularly those of the highly mobile, is critical to public health responses that effectively ameliorate the ability of a disease to spread. In today’s world, hosts’ risk knowledge (good or bad information) when combined with the ability of public health officials to measure and properly communicate, in a timely manner, real or perceived information on disease risks, limit our ability to derail the spread of emergent and re-emergent diseases, at time scales that make a difference.</p>
<sec id="Sec2">
<title>Contagion and Tipping Points</title>
<p id="Par6">
<italic>Contagion</italic>
is believed to be the direct or indirect result of interactions between individuals experiencing radically different epidemiological, or immunological, or social states. Contagion tends to succeed within environments or communities that “facilitate” modes of infection among its members. Contagion is an “understood” or “believed” mode of disease transmission or of “socially transmitted” behaviors, popularized by Malcolm Gladwell, a journalist who made use of his general understanding of the concept of “contagion.” In his construction of reasonable or plausible explanations for the observed and documented dramatic reductions in car thefts and violent crimes in New York City in the 1990s, [
<xref ref-type="bibr" rid="CR71">71</xref>
] Gladwell expanded the use of the concept of contagion and tipping point in his development of a framework that captures—as the result of contagion—the spread of a multitude of social ills or virtues [
<xref ref-type="bibr" rid="CR72">72</xref>
]. Specifically, contagion is seen in [
<xref ref-type="bibr" rid="CR71">71</xref>
] as a force capable of starting and sustaining growth in criminal activity as long as a “critical mass” of individuals capable and willing to commit crimes is available. The growth in criminal activity in New York City is, according to Gladwell, the result of the “interactions” between a large enough pool of criminally active (infected) individuals and individuals susceptible to criminal contagion [
<xref ref-type="bibr" rid="CR71">71</xref>
]. Gladwell extends the perspective pioneered by Sir Ronald Ross [
<xref ref-type="bibr" rid="CR141">141</xref>
] and his “students” [
<xref ref-type="bibr" rid="CR90">90</xref>
<xref ref-type="bibr" rid="CR92">92</xref>
] to the field of social dynamics.</p>
<p id="Par7">Gladwell concludes as Ross did in 1911 that implementing control measures (crime contact-reduction measures) that bring the size of the population of criminals (the core) below a critical threshold (tipping point), are sufficient to explain the drastic reductions in criminal activity in NYC. Gladwell concludes, “There is probably no other place [NYC] in the country where violent crime has declined so far, so fast” [
<xref ref-type="bibr" rid="CR71">71</xref>
].</p>
</sec>
<sec id="Sec3">
<title>Geographic and Spatial Disease Spread</title>
<p id="Par8">The SARS epidemic of 2002–2003 emphasized the possibility of disease transmission over long distances through air travel, and this has led to metapopulation studies that account for long-distance transmission [
<xref ref-type="bibr" rid="CR5">5</xref>
<xref ref-type="bibr" rid="CR8">8</xref>
]. A metapopulation, in this context, is a population of populations linked by travel. A metapopulation model would have an associated, independent of travel, reproduction number as well as reproduction numbers that account for travel between patches, either temporary travel or permanent migration . This is an Eulerian perspective, describing migration between patches.</p>
<p id="Par9">An alternative approach to the modeling of the spatial spread of diseases is based on a Lagrangian perspective, which can be formulated, for example, in terms of residence times [
<xref ref-type="bibr" rid="CR18">18</xref>
,
<xref ref-type="bibr" rid="CR25">25</xref>
] . This approach has been introduced in Chap. 10.1007/978-1-4939-9828-9_15. In this structure, actual travel between patches is not described explicitly, and this makes the analysis less complicated. Calculation of the reproduction number and the final size relations is possible.</p>
<p id="Par10">Another aspect of the study of the spread of diseases is the spatial spread of diseases through diffusion. This has been introduced in Chap. 10.1007/978-1-4939-9828-9_14 of this book and has been examined in considerable detail in [
<xref ref-type="bibr" rid="CR136">136</xref>
], with particular emphasis on epidemic waves. </p>
</sec>
</sec>
<sec id="Sec4">
<title>Heterogeneity of Mixing, Cross-immunity, and Coinfection</title>
<p id="Par11">In epidemics, as in the rest of biology, the role of heterogeneity plays a fundamental role and a critical question arises: what is the level of heterogeneity that must be included to address a specific question properly? For example, first-order estimates of the fraction that must be vaccinated to eliminate a communicable disease can be handled with homogeneous mixing models while the elaboration of optimal vaccination strategies in real-life situations often require an age-structured model [
<xref ref-type="bibr" rid="CR28">28</xref>
,
<xref ref-type="bibr" rid="CR81">81</xref>
]. The study of nosocomial (in-hospital) infections provides an additional example of the role of heterogeneity in transmission or degree of susceptibility or resistance [
<xref ref-type="bibr" rid="CR38">38</xref>
,
<xref ref-type="bibr" rid="CR39">39</xref>
,
<xref ref-type="bibr" rid="CR106">106</xref>
]. The SARS epidemic provided a timely example of the criticality of heterogeneous mixing, in nosocomial transmission [
<xref ref-type="bibr" rid="CR85">85</xref>
,
<xref ref-type="bibr" rid="CR152">152</xref>
] . Since there was no treatment available during the SARS epidemic, the main management approach rested on the effectiveness of isolation of diagnosed infectives, quarantine of suspected infectives, and early diagnosis. Quarantine was decided by tracing of contacts made by infectives but in fact few quarantined individuals developed SARS symptoms. The role of early diagnosis and the effectiveness of isolation seemed to have been the key to SARS control with improvements in contact tracing also playing an important role in epidemic control.</p>
<p id="Par12">Another set of questions arises when one considers the immunological history of individuals or populations. There are many instances in which more than one strain of a disease is circulating within a population and the possibility of cross-immunity between strains becomes important [
<xref ref-type="bibr" rid="CR3">3</xref>
,
<xref ref-type="bibr" rid="CR4">4</xref>
]. Mathematically, co-strain co-circulation may lead to models that support a disease-free equilibrium (or non-uniform age distribution), equilibria in which only one strain persists, and an equilibrium in which two strains coexist. The role of cross-immunity in destabilizing disease dynamics (periodic solutions) has been studied extensively in the case of influenza models without age structure [
<xref ref-type="bibr" rid="CR57">57</xref>
,
<xref ref-type="bibr" rid="CR123">123</xref>
,
<xref ref-type="bibr" rid="CR124">124</xref>
,
<xref ref-type="bibr" rid="CR151">151</xref>
] and also in age-structured models [
<xref ref-type="bibr" rid="CR29">29</xref>
,
<xref ref-type="bibr" rid="CR30">30</xref>
]. Coinfections of more than one disease are also possible and their analysis requires more elaborate models. This is a real possibility with HIV and tuberculosis [
<xref ref-type="bibr" rid="CR96">96</xref>
,
<xref ref-type="bibr" rid="CR119">119</xref>
,
<xref ref-type="bibr" rid="CR133">133</xref>
,
<xref ref-type="bibr" rid="CR135">135</xref>
,
<xref ref-type="bibr" rid="CR140">140</xref>
,
<xref ref-type="bibr" rid="CR144">144</xref>
,
<xref ref-type="bibr" rid="CR154">154</xref>
].</p>
</sec>
<sec id="Sec5">
<title>Antibiotic Resistance</title>
<p id="Par13">In short-term disease outbreaks, antiviral treatment is one of the methods used to treat illness and also to decrease the basic reproduction number
<inline-formula id="IEq1">
<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} $$\mathcal {R}_0$$ \end{document}</tex-math>
<mml:math id="M2">
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq1.gif"></inline-graphic>
</alternatives>
</inline-formula>
and thus to lessen the number of cases of disease. However, many infectious pathogens can evolve and generate successor strains that confer drug resistance [
<xref ref-type="bibr" rid="CR55">55</xref>
]. The evolution of resistance is generally associated with impaired transmission fitness compared to the sensitive strains of the infectious pathogen [
<xref ref-type="bibr" rid="CR112">112</xref>
]. In the absence of treatment, resistant strains may be competitively disadvantaged compared to the sensitive strains and may go extinct. However, treatment prevents the growth and spread of sensitive strains, and therefore induces a selective pressure that favors the resistant strain to replicate and restores its fitness to a level suitable for successful transmission [
<xref ref-type="bibr" rid="CR2">2</xref>
]. This phenomenon has been observed in several infectious diseases, in particular for management of influenza infection using antiviral drugs [
<xref ref-type="bibr" rid="CR138">138</xref>
]. </p>
<p id="Par14">Previous models of influenza epidemics and pandemics have investigated strategies for antiviral treatment in order to reduce the epidemic final size (the total number of infections throughout the epidemic), while preventing widespread drug resistance in the population [
<xref ref-type="bibr" rid="CR77">77</xref>
,
<xref ref-type="bibr" rid="CR107">107</xref>
,
<xref ref-type="bibr" rid="CR111">111</xref>
,
<xref ref-type="bibr" rid="CR113">113</xref>
,
<xref ref-type="bibr" rid="CR114">114</xref>
]. Through computer simulations, these studies have shown that, when resistance is highly transmissible, there may be situations in which increasing the treatment rate may do more harm than good by causing a larger number of resistant cases than the decrease in cases produced by treatment of sensitive infections. A recent epidemic model [
<xref ref-type="bibr" rid="CR156">156</xref>
] has exhibited such behavior and suggested that there may be an optimal treatment rate for minimizing the final size [
<xref ref-type="bibr" rid="CR107">107</xref>
,
<xref ref-type="bibr" rid="CR111">111</xref>
,
<xref ref-type="bibr" rid="CR114">114</xref>
].</p>
<p id="Par15">In diseases such as tuberculosis, which operate on a very long time scale, the same problems arise but the modeling scenario is quite different. It is necessary to include demographic effects such as births and natural deaths in a model. This means that there may be an endemic equilibrium, and that the disease is always present in the population. Instead of studying the final size of an epidemic to measure the severity of a disease outbreak, it is more appropriate to consider the degree of prevalence of the disease in the population at endemic equilibrium as a measure of severity. For diseases such as tuberculosis, in which there are additional aspects such as reinfection, there may be additional difficulties caused by the possibility of backward bifurcations. The importance of understanding the dynamics of tuberculosis treatment suggests that this is a topic that should be pursued [
<xref ref-type="bibr" rid="CR60">60</xref>
].</p>
<p id="Par16">Antibiotic-resistant bacterial infections in hospitals are considered one of the biggest threat to public health. The British Chief Medical Officer, Dame Sally Davies, noted that “the problem of microbes becoming increasingly resistant to the most powerful drugs should be ranked alongside terrorism and climate change on the list of critical risks to the nation.” Yet while antibiotic use is rising, not least in agriculture for farmed animals and fish, resistance is steadily growing.</p>
<p id="Par17">The challenges posed by the persistence, evolution, and expansion of resistance to antimicrobials are critically important because the number of drugs is limited and no new ones have been created for three decades [
<xref ref-type="bibr" rid="CR2">2</xref>
,
<xref ref-type="bibr" rid="CR16">16</xref>
,
<xref ref-type="bibr" rid="CR38">38</xref>
,
<xref ref-type="bibr" rid="CR65">65</xref>
,
<xref ref-type="bibr" rid="CR99">99</xref>
]. We are facing a global crisis in antibiotics, the result of rapidly evolving resistance among microbes responsible for common infections that threaten to turn them into untreatable diseases. Every antibiotic ever developed is at risk of becoming useless. Antimicrobial resistance is on the rise in Europe, and elsewhere in the world.</p>
<p id="Par18">Dr. Margaret Chan, Director General of the World Health Organization, while addressing a meeting of infectious disease experts in Copenhagen, noted that “A post-antibiotic era means, in effect, an end to modern medicine as we know it. Things as common as strep throat or a child’s scratched knee could once again kill. For patients infected with some drug-resistant pathogens, mortality has been shown to increase by around 50 per cent.”
<xref ref-type="fn" rid="Fn1">1</xref>
</p>
<p id="Par20">Strategies suggested to curb the development of resistant hospital-acquired infections include antimicrobial cycling and mixing, that is, models of antibiotic use that make use of two distinct classes of antibiotics that may distributed over different schedules with the goal of slowing down the evolution of resistance. Cycling alternates both classes of drugs over pre-specified periods of time while mixing distributes both drugs simultaneously at random, that is, roughly half of the physicians would prescribe the first drug class while the other half would prescribe the second class. If the goal is to slow down single class drug resistance then “mixing” is the answer [
<xref ref-type="bibr" rid="CR16">16</xref>
] while if the goal is to minimize dual resistance (if such a possibility exists) then the best option is cycling [
<xref ref-type="bibr" rid="CR38">38</xref>
]. Of course, there are other factors that may accelerate resistance (physicians’ compliance) or slow down resistance (quarantine and isolation). All the above questions may be addressed via the use of contagion models [
<xref ref-type="bibr" rid="CR38">38</xref>
].</p>
</sec>
<sec id="Sec6">
<title>Mobility</title>
<p id="Par21">The Global Commons are continuously reshaped by the ability of an increasing proportion of the human population to live, move, or trade nearly anywhere. Therefore, understanding the patterns of interactions between humans, between humans and vectors, and the patterns of individuals’ movement, particularly those who are highly mobile, is critical in guiding public health responses to disease spread. In today’s world, hosts’ knowledge of information about risk, combined with the ability of public health officials to measure and properly communicate, in a timely manner, real or perceived information on disease risks, affects our ability to derail the spread of emergent and re-emergent diseases, at scales that make a difference.</p>
<p id="Par22">Simon Levin showed that understanding scale-dependent phenomena is intimately tied in to our understanding on how information at particular scales impact other scales. His four decade old seminal paper establishing the relationships between processes operating at different scales that highlighted how macroscopic features arise from microscopic processes open the door to the theoretical advances that have dominated the study of ecological and epidemiological systems [
<xref ref-type="bibr" rid="CR101">101</xref>
]. Specifically, the theory of metapopulations, common to the study of the models in this book [
<xref ref-type="bibr" rid="CR104">104</xref>
,
<xref ref-type="bibr" rid="CR155">155</xref>
], was used to establish the role that localized disturbances have had in maintaining biodiversity [
<xref ref-type="bibr" rid="CR103">103</xref>
,
<xref ref-type="bibr" rid="CR127">127</xref>
]. Kareiva et al. observe that there is a multitude of frameworks to study the role of disturbance, noting that, “Models that deal with dispersal and spatially distributed populations are extraordinarily varied, partly because they employ three distinct characterizations of space: as ‘islands’ (or ‘metapopulations’), as ‘stepping-stones’, or as a continuum” [
<xref ref-type="bibr" rid="CR88">88</xref>
]. We choose to deal with mobility in Chap. 10.1007/978-1-4939-9828-9_15 and this chapter uses a metapopulation approach [
<xref ref-type="bibr" rid="CR80">80</xref>
,
<xref ref-type="bibr" rid="CR93">93</xref>
,
<xref ref-type="bibr" rid="CR104">104</xref>
], with populations that exist on discrete “patches” defined by some characteristic(s) (i.e., location, disease risk, water availability, etc.). As is customary, patches are connected by their ability to transfer relevant information among themselves, which, in the context of disease dynamics, is modeled by the ability of individuals to move between patches. Patches may be constructed (defined) by species (human and mosquito) with movement explicitly modeled via patch-specific residence times and under a framework that sees disease dynamics as the result of location-dependent interactions and location characteristic average risks of infection [
<xref ref-type="bibr" rid="CR17">17</xref>
,
<xref ref-type="bibr" rid="CR18">18</xref>
].</p>
<p id="Par23">We observed that “It is therefore important to identify and quantify the processes responsible for observed epidemiological macroscopic patterns: the result of individual interactions in changing social and ecological landscapes” [
<xref ref-type="bibr" rid="CR25">25</xref>
]. In the rest of this chapter, we touch on some of the issues calling for the identification of an encompassing theoretical explanatory framework or frameworks. We do this by identifying some of the limitations of existing theory, in the context of particular epidemiological systems. The goal is fostering and re-energizing research that aims at disentangling the role of epidemiological and socioeconomic forces on disease dynamics. In short, epidemic models on social landscapes are better formulated as complex adaptive systems. Now the question becomes, “How does such a perspective help our understanding of epidemics and our ability to make informed adaptive decisions?” These are huge complex questions whose answers have engaged a large number of interdisciplinary and trans-disciplinary teams of researchers. What may be promising directions? In what follows, we discuss some of the modeling used to address some of the challenges and opportunities that we believe must be considered in the field of theoretical epidemiology.</p>
<sec id="Sec7">
<title>A Lagrangian Approach to Modeling Mobility and Infectious Disease Dynamics</title>
<p id="Par24">The deleterious impact of the use of cordons sanitaires [
<xref ref-type="bibr" rid="CR58">58</xref>
,
<xref ref-type="bibr" rid="CR100">100</xref>
] to limit the spread of Ebola in West Africa points to the importance of developing and implementing novel approaches that may ameliorate the impact of disease outbreaks in areas where timely response to the emergence of novel pathogens is not possible at this time.</p>
<p id="Par25">Disease risk is a function of the scale and the level of heterogeneity considered. Risk varies by countries and within a country by areas of localized poverty, or as a function of the availability and quality of sanitary/phytosanitary conditions, or as a result of access and the quality of health care, or variability on the levels of individual education, or as a result of engrained cultural practices and norms. Travel and trade, easily bypassing in today’s world the natural or cultural boundaries defined by many of factors just outlined, are now seen as engines that drive the spread of pests and pathogens across regional and global scales. Hence, the identification of explanatory frameworks that help to disentangle the role of epidemiological, socioeconomic, and cultural perspectives on disease dynamics becomes evident and necessary in the Global Commons. Further, since the work of Sir Ronald Ross over a century ago [
<xref ref-type="bibr" rid="CR141">141</xref>
], efforts to develop a mathematical framework that allow us to tease out the role of various mechanisms on disease spread while enhancing our understanding of what may be the most effective measures to manage or eliminate a disease, the fields of mathematical and theoretical epidemiology have developed into rich and useful fields of their own. Their role in the development of public health policy and the study of disease evolution within hosts (immunology) and between populations and its relationship to the study of host–pathogen interactions within ecology or community ecology are now integral components of the education and training of theoreticians and practitioners alike [
<xref ref-type="bibr" rid="CR1">1</xref>
,
<xref ref-type="bibr" rid="CR12">12</xref>
,
<xref ref-type="bibr" rid="CR19">19</xref>
<xref ref-type="bibr" rid="CR23">23</xref>
,
<xref ref-type="bibr" rid="CR26">26</xref>
,
<xref ref-type="bibr" rid="CR41">41</xref>
,
<xref ref-type="bibr" rid="CR53">53</xref>
,
<xref ref-type="bibr" rid="CR54">54</xref>
,
<xref ref-type="bibr" rid="CR59">59</xref>
,
<xref ref-type="bibr" rid="CR74">74</xref>
,
<xref ref-type="bibr" rid="CR76">76</xref>
,
<xref ref-type="bibr" rid="CR82">82</xref>
,
<xref ref-type="bibr" rid="CR102">102</xref>
,
<xref ref-type="bibr" rid="CR122">122</xref>
,
<xref ref-type="bibr" rid="CR157">157</xref>
].</p>
<p id="Par26">The use of (per capita) contact or activity rates in modeling the interactions between individuals, that is, who mixes with whom or who interacts with whom, has been the natural social dynamics currency used to model human-to-human or vector-to-human interactions in the context of the transmission dynamics of communicable diseases. The “physics or chemistry traditions” are used to model disease transmission as the result of the “collisions” between individuals (with different energy or activity levels) in different epidemiological states. Further, movement, typically modeled using a metapopulation approach, is seen as the relocation between patches of non-identifiable individuals. The scholarly and extensive review in [
<xref ref-type="bibr" rid="CR83">83</xref>
] addresses this perspective within homogeneous and heterogeneous mixing (age-structured) population models (see also [
<xref ref-type="bibr" rid="CR30">30</xref>
]). Weakening the assumption of homogeneous mixing via contacts in epidemiology has been addressed using network-based analyses that identify host contact patterns and clusters [
<xref ref-type="bibr" rid="CR13">13</xref>
,
<xref ref-type="bibr" rid="CR120">120</xref>
,
<xref ref-type="bibr" rid="CR121">121</xref>
,
<xref ref-type="bibr" rid="CR128">128</xref>
] (and references therein with [
<xref ref-type="bibr" rid="CR121">121</xref>
] offering an extensive review). Focusing on how each individual is connected within the population has been used to address the effects of host behavioral response on disease prevalence (see [
<xref ref-type="bibr" rid="CR67">67</xref>
,
<xref ref-type="bibr" rid="CR68">68</xref>
,
<xref ref-type="bibr" rid="CR110">110</xref>
] for a review). Other approaches have included the effects of behavioral changes triggered by “fear” and/or awareness of disease [
<xref ref-type="bibr" rid="CR56">56</xref>
,
<xref ref-type="bibr" rid="CR66">66</xref>
,
<xref ref-type="bibr" rid="CR131">131</xref>
,
<xref ref-type="bibr" rid="CR134">134</xref>
]. Although this stress-induced behavior may benefit public health efforts in some cases, it can also cause somewhat unpredictable outcomes [
<xref ref-type="bibr" rid="CR75">75</xref>
].</p>
<p id="Par27">However, the fact remains that our ability to determine (and hence define) what an effective contact is in the context of communicable diseases, that is, our ability to measure the average number of contacts that a typical patch resident has per unit of time and where, has been hampered by high levels of uncertainty. Therefore, when we ask, what is the average rate of contacts that an individual has while riding a packed subway in Japan or Mexico City, or what is the average rate of contacts that an individual has at a religious event involving hundreds of thousands of people, including pilgrimages, one quickly arrives at the conclusion that different observers are extremely likely to arrive at very distinct understandings and quantifications of the frequency, intensity, and levels of heterogeneity involved. In short, this perspective puts emphasis on the use of a different currency (residence times) because measuring contacts at the places where the risk of infection is the highest, pilgrimages, massive religious ceremonies, “Woodstock time events”, packed subways, and other forms of mass gathering or transportation have not been done to the satisfaction of most researchers. The risk of acquiring an infectious disease within a flight can be measured at least in principle as a function of the time that each individual of x-type spends flying, the number of passengers, and the likelihood that an infectious individual is on board. For example, measuring the risk of acquiring tuberculosis, an airborne disease that may spread by air circulation in a flight, may be more a function of the duration of the flight and the seating arrangement than the average rate of contacts per passenger within the flight (see [
<xref ref-type="bibr" rid="CR31">31</xref>
] and references therein). Furthermore, replication studies that measure risk of infection in a given environment may indeed be possible under a residence time model. In short, the risks of acquiring an infection can be quantified as a function of the time spent (residence time) within each particular environment. The Lagrangian modeling approach builds (epidemiological) models by tracking individuals’ patch-residence times and estimating their contacts according to the time spent in each environment [
<xref ref-type="bibr" rid="CR32">32</xref>
]. The value of these models increases when we have the ability to assess risk as a patch-specific characteristic. In short, the use of a Lagrangian modeling perspective rather than the use of contacts is tied to the difficulties that must be faced when the goal is to measure the average rate of contacts per type-
<italic>x</italic>
individual in the environments that facilitate transmission the most.</p>
<p id="Par28">The Lagrangian approach is highlighted here via the formulation of a disease model involving the joint dynamics of an
<italic>n</italic>
-patch geographically structured population with individuals moving back and forth from their place of residence to other patches. Each of these patches (or environments) is defined by its associated risk of residence-time infection. Patch risk measurements account for environmental, health, and socioeconomic conditions. The Lagrangian approach [
<xref ref-type="bibr" rid="CR73">73</xref>
,
<xref ref-type="bibr" rid="CR125">125</xref>
,
<xref ref-type="bibr" rid="CR126">126</xref>
] keeps track of the identity of hosts regardless of their geographical/spatial position. The use of Lagrangian modeling in living systems was, to the best of our knowledge, pioneered and popularized by Okubo and Levin [
<xref ref-type="bibr" rid="CR125">125</xref>
,
<xref ref-type="bibr" rid="CR126">126</xref>
] in the context of animal aggregation. Recently, Lagrangian approaches have also been used to model human crowd movement and behavior [
<xref ref-type="bibr" rid="CR15">15</xref>
,
<xref ref-type="bibr" rid="CR49">49</xref>
,
<xref ref-type="bibr" rid="CR78">78</xref>
,
<xref ref-type="bibr" rid="CR79">79</xref>
] and in the context of bioterrorism [
<xref ref-type="bibr" rid="CR31">31</xref>
].</p>
<p id="Par29">Here, host-residence status and mobility across patches are monitored with the help of a residence times matrix
<inline-formula id="IEq2">
<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} $$\mathbb {P}=(p_{ij})_{1\le i,j\le n}$$ \end{document}</tex-math>
<mml:math id="M4">
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo></mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo></mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq2.gif"></inline-graphic>
</alternatives>
</inline-formula>
, where
<italic>p</italic>
<sub>
<italic>ij</italic>
</sub>
is the proportion of time residents of Patch
<italic>i</italic>
spend in Patch
<italic>j</italic>
. Letting
<italic>N</italic>
<sub>
<italic>i</italic>
</sub>
denote the population of Patch
<italic>i</italic>
predispersal, that is, when patches are isolated, we conclude that effective population size in Patch
<italic>i</italic>
, at time
<italic>t</italic>
, is given by
<inline-formula id="IEq3">
<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} $$\sum _{j=1}^np_{ji}N_j$$ \end{document}</tex-math>
<mml:math id="M6">
<mml:msubsup>
<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:mi>n</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq3.gif"></inline-graphic>
</alternatives>
</inline-formula>
. That is, the effective population within each patch must account for the residents and visitors to Patch
<italic>i</italic>
at time
<italic>t</italic>
. A typical SIS model captures this Lagrangian approach in an
<italic>n</italic>
- patch setting via the system of nonlinear differential equations:
<disp-formula id="Equ1">
<label>16.1</label>
<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} $$\displaystyle \begin{aligned} \begin{aligned} \dot{S}_i&=b_i-d_iS_i+\gamma_iI_i-\sum_{j=1}^n(S_i \text{ infected in Patch } j)\\ \dot{I}_i&=\sum_{j=1}^n(S_i \text{ infected in Patch } j)-\gamma_iI_i-d_iI_i, \end{aligned} \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M8" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mrow>
<mml:mover>
<mml:mrow>
<mml:mi>S</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>
<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:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>S</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>γ</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:munderover accentunder="false">
<mml:mrow>
<mml:mo mathsize="big"></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:mi>n</mml:mi>
</mml:mrow>
</mml:munderover>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mspace width="0.3em"></mml:mspace>
<mml:mtext>infected in Patch</mml:mtext>
<mml:mspace width="0.3em"></mml:mspace>
<mml:mi>j</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mrow>
<mml:mi>İ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
<mml:mtd>
<mml:mo>=</mml:mo>
<mml:munderover accentunder="false">
<mml:mrow>
<mml:mo mathsize="big"></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:mi>n</mml:mi>
</mml:mrow>
</mml:munderover>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mspace width="0.3em"></mml:mspace>
<mml:mtext>infected in Patch</mml:mtext>
<mml:mspace width="0.3em"></mml:mspace>
<mml:mi>j</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo></mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>γ</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>d</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:mtd>
</mml:mtr>
</mml:mtable>
</mml:mtd>
<mml:mtd class="align-2"></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equ1.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
where
<italic>b</italic>
<sub>
<italic>i</italic>
</sub>
,
<italic>d</italic>
<sub>
<italic>i</italic>
</sub>
, and
<italic>γ</italic>
<sub>
<italic>i</italic>
</sub>
denote the constant recruitment, the per capita natural death, and recovery rates, respectively, in Patch
<italic>i</italic>
. The effective population
<inline-formula id="IEq4">
<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} $$\sum _{j=1}^np_{ij}N_j$$ \end{document}</tex-math>
<mml:math id="M10">
<mml:msubsup>
<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:mi>n</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq4.gif"></inline-graphic>
</alternatives>
</inline-formula>
in each Patch
<italic>i</italic>
,
<italic>i</italic>
 = 1, …,
<italic>n</italic>
includes
<inline-formula id="IEq5">
<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} $$\sum _{j=1}^np_{ij}I_j$$ \end{document}</tex-math>
<mml:math id="M12">
<mml:msubsup>
<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:mi>n</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq5.gif"></inline-graphic>
</alternatives>
</inline-formula>
infected individuals. Therefore, the infection term is modeled as follows:
<disp-formula id="Equa">
<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} $$\displaystyle \begin{aligned}S_i \text{ infected in patch }j=\beta_j\times p_{ij}S_i\times \frac{\sum_{k=1}^np_{kj}I_k}{\sum_{k=1}^np_{kj}N_k}.\end{aligned}$$ \end{document}</tex-math>
<mml:math id="M14" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:msub>
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mspace width="0.3em"></mml:mspace>
<mml:mtext>infected in patch</mml:mtext>
<mml:mspace width="0.3em"></mml:mspace>
<mml:mi>j</mml:mi>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>×</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>S</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>×</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mo></mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mo></mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mn>.</mml:mn>
</mml:mtd>
<mml:mtd class="align-2"></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equa.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
</p>
<p id="Par30">The likelihood of infection in each patch is tied to the environmental risks, defined by the “transmission/risk” vector
<inline-formula id="IEq6">
<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} $$\mathcal {B}=(\beta _1,\beta _2,\dots ,\beta _n)^t$$ \end{document}</tex-math>
<mml:math id="M16">
<mml:mi mathvariant="normal">B</mml:mi>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq6.gif"></inline-graphic>
</alternatives>
</inline-formula>
and the proportion of time spent in particular area. Letting
<italic>I</italic>
 = (
<italic>I</italic>
<sub>1</sub>
,
<italic>I</italic>
<sub>2</sub>
, …,
<italic>I</italic>
<sub>
<italic>n</italic>
</sub>
)
<sup>
<italic>t</italic>
</sup>
,
<inline-formula id="IEq7">
<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} $$\bar {N}=(\frac {b_1}{d_1},\frac {b_2}{d_2},\dots ,\frac {b_n}{d_n})^t$$ \end{document}</tex-math>
<mml:math id="M18" displaystyle="true">
<mml:mover>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq7.gif"></inline-graphic>
</alternatives>
</inline-formula>
,
<inline-formula id="IEq8">
<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} $$\tilde {N}=\mathbb {P}t\bar {N}$$ \end{document}</tex-math>
<mml:math id="M20" displaystyle="true">
<mml:mi>Ñ</mml:mi>
<mml:mo>=</mml:mo>
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mi>t</mml:mi>
<mml:mover>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq8.gif"></inline-graphic>
</alternatives>
</inline-formula>
,
<italic>d</italic>
 = (
<italic>d</italic>
<sub>1</sub>
,
<italic>d</italic>
<sub>2</sub>
, …,
<italic>d</italic>
<sub>
<italic>n</italic>
</sub>
)
<sup>
<italic>t</italic>
</sup>
, and
<italic>γ</italic>
 = (
<italic>γ</italic>
<sub>1</sub>
,
<italic>γ</italic>
<sub>2</sub>
, …,
<italic>γ</italic>
<sub>
<italic>n</italic>
</sub>
)
<sup>
<italic>t</italic>
</sup>
allows to rewrite System
<xref rid="Equ1" ref-type="">16.1</xref>
in the following single vectorial form
<disp-formula id="Equ2">
<label>16.2</label>
<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} $$\displaystyle \begin{aligned} \dot{I}&=diag(\bar{N}-I)\mathbb{P}diag(\mathcal{B})diag(\tilde{N})^{-1}\mathbb{P}^tI-diag(d+\gamma)I. \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M22" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:mi>İ</mml:mi>
</mml:mtd>
<mml:mtd class="align-2">
<mml:mo>=</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mover>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
<mml:mo></mml:mo>
<mml:mi>I</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi mathvariant="normal">B</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>Ñ</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mi>I</mml:mi>
<mml:mo></mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>d</mml:mi>
<mml:mo>+</mml:mo>
<mml:mi>γ</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>I</mml:mi>
<mml:mn>.</mml:mn>
</mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equ2.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
</p>
<p id="Par31">The dynamics of the disease in all of the patches depends on the patch connectivity structure. Therefore, if the residence-time matrix
<inline-formula id="IEq9">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M24">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq9.gif"></inline-graphic>
</alternatives>
</inline-formula>
is irreducible, patches are strongly connected, then system 2 supports a sharp threshold property. That is, the disease dies out or persists (in all patches) whenever the basic reproduction number
<inline-formula id="IEq10">
<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} $$\mathcal {R}_0$$ \end{document}</tex-math>
<mml:math id="M26">
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq10.gif"></inline-graphic>
</alternatives>
</inline-formula>
is less than or greater than unity [
<xref ref-type="bibr" rid="CR18">18</xref>
].
<inline-formula id="IEq11">
<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} $$\mathcal {R}_0$$ \end{document}</tex-math>
<mml:math id="M28">
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq11.gif"></inline-graphic>
</alternatives>
</inline-formula>
is given by
<disp-formula id="Equb">
<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} $$\displaystyle \begin{aligned}\mathcal{R}_0=\rho(diag(\bar{N})\mathbb{P}diag(\mathcal{B})diag(\tilde{N})^{-1}\mathbb{P}^tV^{-1}),\end{aligned}$$ \end{document}</tex-math>
<mml:math id="M30" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mi>ρ</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mover>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi mathvariant="normal">B</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>Ñ</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:msup>
<mml:mrow>
<mml:mi>V</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>,</mml:mo>
</mml:mtd>
<mml:mtd class="align-2"></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equb.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
where
<italic>ρ</italic>
denotes the spectral radius and
<italic>V</italic>
 = −
<italic>diag</italic>
(
<italic>d</italic>
 + 
<italic>γ</italic>
). The dynamics of the system when the matrix
<inline-formula id="IEq12">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M32">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq12.gif"></inline-graphic>
</alternatives>
</inline-formula>
is not irreducible can be characterized using the following patch-specific basic reproduction numbers:
<disp-formula id="Equc">
<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} $$\displaystyle \begin{aligned}\mathcal{R}_0^i(\mathbb{P})=\frac{\beta_i}{\gamma_i+d_i}\times\sum_{j=1}^n\left(\frac{\beta_j}{\beta_i}\right)p_{ij}\left(\frac{p_{ij}\left(\frac{b_i}{d_i}\right)}{\sum_{k=1}^np_{kj}b_kd_k}\right).\end{aligned}$$ \end{document}</tex-math>
<mml:math id="M34" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>γ</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>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>×</mml:mo>
<mml:munderover accentunder="false">
<mml:mrow>
<mml:mo mathsize="big"></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:mi>n</mml:mi>
</mml:mrow>
</mml:munderover>
<mml:mfenced close=")" open="(" separators="">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>β</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>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mfenced close=")" open="(" separators="">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mfenced close=")" open="(" separators="">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mo></mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
<mml:mn>.</mml:mn>
</mml:mtd>
<mml:mtd class="align-2"></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equc.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
</p>
<p id="Par32">The disease persists in Patch i whenever
<inline-formula id="IEq13">
<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} $$\mathcal {R}_0^i(\mathbb {P})>1$$ \end{document}</tex-math>
<mml:math id="M36">
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>></mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq13.gif"></inline-graphic>
</alternatives>
</inline-formula>
, whereas the disease dies out in Patch
<italic>i</italic>
if
<italic>p</italic>
<sub>
<italic>kj</italic>
</sub>
 = 0 for all
<italic>k</italic>
 = 1, …,
<italic>n</italic>
, and
<italic>k</italic>
<italic>i</italic>
,provided
<italic>p</italic>
<sub>
<italic>ij</italic>
</sub>
 > 0 and
<inline-formula id="IEq14">
<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} $$\mathcal {R}_0^i(\mathbb {P})<1.$$ \end{document}</tex-math>
<mml:math id="M38">
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo><</mml:mo>
<mml:mn>1.</mml:mn>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq14.gif"></inline-graphic>
</alternatives>
</inline-formula>
</p>
<p id="Par33">Patch-specific disease persistence can be established using the average Lyapunov theorem [
<xref ref-type="bibr" rid="CR86">86</xref>
] (see [
<xref ref-type="bibr" rid="CR18">18</xref>
] for more details).</p>
<p id="Par34">In Model
<xref rid="Equ2" ref-type="">16.2</xref>
, human behavior is crudely incorporated through the use of a constant mobility matrix
<inline-formula id="IEq15">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M40">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq15.gif"></inline-graphic>
</alternatives>
</inline-formula>
. The role that adaptive human behavior may play in response to disease dynamics is captured, also rather crudely, via a phenomenological approach that assumes that individuals avoid or spend less time in areas of high prevalence. This effect is captured by placing natural restrictions on the entries of
<inline-formula id="IEq16">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M42">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq16.gif"></inline-graphic>
</alternatives>
</inline-formula>
. The inequalities
<inline-formula id="IEq17">
<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} $$\frac {p_{ij}(I_i,I_j)}{\partial I_j}\le 0$$ \end{document}</tex-math>
<mml:math id="M44">
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</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>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi></mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo></mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq17.gif"></inline-graphic>
</alternatives>
</inline-formula>
and
<inline-formula id="IEq18">
<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} $$\frac {p_{ij}(I_i,I_j)}{\partial I_i}\ge 0$$ \end{document}</tex-math>
<mml:math id="M46">
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</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>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi></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:mrow>
</mml:mfrac>
<mml:mo></mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq18.gif"></inline-graphic>
</alternatives>
</inline-formula>
, for (
<italic>i</italic>
,
<italic>j</italic>
) ∈ 1, 2, guarantee the expected behavioral response. An example of such dependency could be captured by the following functions:
<inline-formula id="IEq19">
<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} $$p_{ii}(I_i,I_j)=\frac {\sigma _{ii}+\sigma _{ii}I_i+I_j}{1+I_i+I_j}$$ \end{document}</tex-math>
<mml:math id="M48">
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</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>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>σ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>σ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<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>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<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>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq19.gif"></inline-graphic>
</alternatives>
</inline-formula>
and
<inline-formula id="IEq20">
<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} $$p_{ij}(I_1,I_j)=\sigma _{ij}\frac {1+I_i}{1+I_i+I_j}$$ \end{document}</tex-math>
<mml:math id="M50">
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>σ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
<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:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<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>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq20.gif"></inline-graphic>
</alternatives>
</inline-formula>
, for (
<italic>i</italic>
,
<italic>j</italic>
) ∈ 1, 2 and
<italic>σ</italic>
<sub>
<italic>ij</italic>
</sub>
 = 
<italic>p</italic>
<sub>
<italic>ij</italic>
</sub>
(0, 0), are such that
<inline-formula id="IEq21">
<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} $$\sum _{j=1}^2 \sigma _{ij}=1$$ \end{document}</tex-math>
<mml:math id="M52">
<mml:msubsup>
<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>2</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow>
<mml:mi>σ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq21.gif"></inline-graphic>
</alternatives>
</inline-formula>
. The simulation below shows how a crude, density-dependent modeling mobility approach can alter the expected disease dynamics from those generated under constant
<inline-formula id="IEq22">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M54">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq22.gif"></inline-graphic>
</alternatives>
</inline-formula>
(Figs.
<xref rid="Fig1" ref-type="fig">16.1</xref>
and
<xref rid="Fig2" ref-type="fig">16.2</xref>
). In the special case, where there is no movement between patches (
<italic>p</italic>
<sub>12</sub>
 = 
<italic>p</italic>
<sub>21</sub>
 = 
<italic>σ</italic>
<sub>12</sub>
 = 
<italic>σ</italic>
21 = 0), that is, there is no behavioral change, the two populations support, as expected, the same dynamics (see the blue curves in Figs.
<xref rid="Fig1" ref-type="fig">16.1</xref>
and
<xref rid="Fig2" ref-type="fig">16.2</xref>
).
<fig id="Fig1">
<label>Fig. 16.1</label>
<caption>
<p>Dynamics of the disease in Patch 1 for three special cases. The symmetric residence times (
<italic>p</italic>
<sub>12</sub>
 = 
<italic>p</italic>
<sub>21</sub>
 = 
<italic>σ</italic>
<sub>12</sub>
 = 
<italic>σ</italic>
<sub>21</sub>
 = 0.5) are described by the solid and dashed black curves. The blue curves represent the case where there is no movement between patches, that is,
<italic>p</italic>
<sub>12</sub>
 = 
<italic>p</italic>
<sub>21</sub>
 = 
<italic>σ</italic>
<sub>12</sub>
 = 
<italic>σ</italic>
<sub>21</sub>
 = 0. The red curves represent the high-mobility case for which
<italic>p</italic>
<sub>12</sub>
 = 
<italic>p</italic>
<sub>21</sub>
 = 
<italic>σ</italic>
<sub>12</sub>
 = 
<italic>σ</italic>
<sub>21</sub>
 = 1. If there is no movement between the patches (blue curves), the disease dies out in the low risk Patch 1 in both approaches with
<inline-formula id="IEq23">
<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} $$\mathcal {R}^1_0= \frac {\beta _1}{d_1+\gamma _1}=0.7636$$ \end{document}</tex-math>
<mml:math id="M56">
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>γ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>=</mml:mo>
<mml:mn>0.7636</mml:mn>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq23.gif"></inline-graphic>
</alternatives>
</inline-formula>
. The vertical axis represents the prevalence of the disease in Patch 1. Figure courtesy of Ref. [
<xref ref-type="bibr" rid="CR18">18</xref>
]</p>
</caption>
<graphic xlink:href="211675_1_En_16_Fig1_HTML" id="MO1"></graphic>
</fig>
<fig id="Fig2">
<label>Fig. 16.2</label>
<caption>
<p>Dynamics of the disease in Patch 2. In the high-mobility case
<italic>p</italic>
<sub>12</sub>
 = 
<italic>p</italic>
<sub>21</sub>
 = 
<italic>σ</italic>
<sub>12</sub>
 = 
<italic>σ</italic>
<sub>21</sub>
 = 1 (and then
<italic>p</italic>
<sub>11</sub>
 = 
<italic>p</italic>
<sub>22</sub>
 = 
<italic>σ</italic>
<sub>11</sub>
 = 
<italic>σ</italic>
<sub>22</sub>
 = 0), the disease dies out (solid red curve) for
<inline-formula id="IEq24">
<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} $$ \mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M58">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq24.gif"></inline-graphic>
</alternatives>
</inline-formula>
constant, with
<inline-formula id="IEq25">
<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} $$\tilde {\mathcal {R}^2_0}= \frac {\beta _1}{\gamma _2+d_2}=0.8571$$ \end{document}</tex-math>
<mml:math id="M60" displaystyle="true">
<mml:mover>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
<mml:mo>~</mml:mo>
</mml:mover>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>γ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>=</mml:mo>
<mml:mn>0.8571</mml:mn>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq25.gif"></inline-graphic>
</alternatives>
</inline-formula>
. For the constant residence-time matrix, the system is strangely decoupled because individuals of Patch 1 spend all their time in Patch 2, whereas individuals of Patch 2 spend all their time in Patch 1. Hence, Patch 2 individuals (
<italic>d</italic>
<sub>2</sub>
and
<italic>μ</italic>
<sub>2</sub>
) are subject exclusively to the environmental conditions that define Patch 1 (
<italic>β</italic>
<sub>1</sub>
), and so the basic reproduction of the “isolated” Patch 1 is
<inline-formula id="IEq26">
<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} $$\tilde {\mathcal {R}^2_0}= \frac {\beta _1}{\gamma _2+d_2}$$ \end{document}</tex-math>
<mml:math id="M62" displaystyle="true">
<mml:mover>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
<mml:mo>~</mml:mo>
</mml:mover>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>γ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>d</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq26.gif"></inline-graphic>
</alternatives>
</inline-formula>
and the disease dies out because
<inline-formula id="IEq27">
<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} $$\tilde {\mathcal {R}^2_0}=0.8571$$ \end{document}</tex-math>
<mml:math id="M64" displaystyle="true">
<mml:mover>
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
<mml:mo>~</mml:mo>
</mml:mover>
<mml:mo>=</mml:mo>
<mml:mn>0.8571</mml:mn>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq27.gif"></inline-graphic>
</alternatives>
</inline-formula>
. The disease persists if
<inline-formula id="IEq28">
<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} $$ \mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M66">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq28.gif"></inline-graphic>
</alternatives>
</inline-formula>
state-dependent (dashed red curve) as
<inline-formula id="IEq29">
<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} $$p_{12}(I_1,I_2)= \frac {1+I_1}{1+I_1+I_2},p_{21}(I_1,I_2)= \frac {1+I_2}{1+I_1+I_2}$$ \end{document}</tex-math>
<mml:math id="M68">
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>12</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>21</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq29.gif"></inline-graphic>
</alternatives>
</inline-formula>
,
<inline-formula id="IEq30">
<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} $$p_{11}(I_1,I_2)= \frac {I_2}{1+I_1+I_2}$$ \end{document}</tex-math>
<mml:math id="M70">
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>11</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq30.gif"></inline-graphic>
</alternatives>
</inline-formula>
, and
<inline-formula id="IEq31">
<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} $$p_{22}(I_1,I_2)= \frac {I_1}{1+I_1+I_2}$$ \end{document}</tex-math>
<mml:math id="M72">
<mml:msub>
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>22</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq31.gif"></inline-graphic>
</alternatives>
</inline-formula>
. Figure courtesy of Ref. [
<xref ref-type="bibr" rid="CR18">18</xref>
]</p>
</caption>
<graphic xlink:href="211675_1_En_16_Fig2_HTML" id="MO2"></graphic>
</fig>
</p>
<p id="Par37">The speed at which the vector-borne Zika virus disease spread throughout Latin America, Central America, and the Caribbean (then hitting Mexico and the United States) was strongly linked to human mobility patterns. Travelers transport the disease and infect native mosquitoes. Here, it is assumed that vector mobility is negligible and the assumptions proceed to incorporate the life history and epidemiology of mosquitoes [
<xref ref-type="bibr" rid="CR10">10</xref>
,
<xref ref-type="bibr" rid="CR84">84</xref>
,
<xref ref-type="bibr" rid="CR98">98</xref>
,
<xref ref-type="bibr" rid="CR108">108</xref>
,
<xref ref-type="bibr" rid="CR109">109</xref>
,
<xref ref-type="bibr" rid="CR141">141</xref>
], which can be effectively captured by decoupling host and vector mobility [
<xref ref-type="bibr" rid="CR98">98</xref>
,
<xref ref-type="bibr" rid="CR145">145</xref>
]. Figure
<xref rid="Fig3" ref-type="fig">16.3</xref>
and System
<xref rid="Equ3" ref-type="">16.3</xref>
illustrate the approach. A Lagrangian model based on residence times has been proposed recently for vector-borne diseases like dengue, malaria, and Zika [
<xref ref-type="bibr" rid="CR17">17</xref>
]. The appropriateness of the Lagrangian approach for the study of the dynamics of vector-borne diseases lies also in its assessment of the life-history specifics of the vector involved [
<xref ref-type="bibr" rid="CR145">145</xref>
].
<disp-formula id="Equ3">
<label>16.3</label>
<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} $$\displaystyle \begin{aligned} \begin{aligned} \dot{I}_h&=\beta_{vh}diag(N_h-I_h)\mathbb{P}diag(a)diag(\mathbb{P}^tN_h)^{-1}I_v-diag(\mu+\gamma)I_h\\ \dot{I}_v&=\beta_{hv}diag(a)diag(N_v-I_v)diag(\mathbb{P}^tN_h)^{-1}\mathbb{P}^tI_h-diag(\mu_v+\delta)I_v. \end{aligned}{} \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M74" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mrow>
<mml:mi>İ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
<mml:mtd>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</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>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo></mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>μ</mml:mi>
<mml:mo>+</mml:mo>
<mml:mi>γ</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mrow>
<mml:mi>İ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
<mml:mtd>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</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>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo></mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>μ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:mi>δ</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mn>.</mml:mn>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mtd>
<mml:mtd class="align-2"></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equ3.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
<fig id="Fig3">
<label>Fig. 16.3</label>
<caption>
<p>Flow diagram of a Lagrangian model in which the host structure is decoupled from the vectors’ structure. Figure courtesy of Ref. [
<xref ref-type="bibr" rid="CR17">17</xref>
]</p>
</caption>
<graphic xlink:href="211675_1_En_16_Fig3_HTML" id="MO3"></graphic>
</fig>
</p>
<p id="Par39">Lagrangian approaches have been used to model vector-borne diseases (see [
<xref ref-type="bibr" rid="CR48">48</xref>
,
<xref ref-type="bibr" rid="CR87">87</xref>
,
<xref ref-type="bibr" rid="CR139">139</xref>
,
<xref ref-type="bibr" rid="CR142">142</xref>
,
<xref ref-type="bibr" rid="CR153">153</xref>
] and other references contained therein), although these researchers have not considered the impact that the residence-time matrix
<inline-formula id="IEq32">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M76">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq32.gif"></inline-graphic>
</alternatives>
</inline-formula>
may have on patch effective population size. Specifically, in [
<xref ref-type="bibr" rid="CR48">48</xref>
,
<xref ref-type="bibr" rid="CR142">142</xref>
], the effects of movement on patch population size at time
<italic>t</italic>
are ignored, namely, the population size in each Patch
<italic>j</italic>
is fixed at
<italic>N</italic>
<sub>
<italic>j</italic>
</sub>
. In [
<xref ref-type="bibr" rid="CR139">139</xref>
], it is assumed that human mobility across patches does not produce any “net” change on the patch population size. On the other hand, in Model
<xref rid="Equ3" ref-type="">16.3</xref>
the relationship between each patch population and mobility is dynamic and explicitly formulated. Moreover, the limited (vector mobility is ignored) Lagrangian approach used here to model the dynamics of vector-borne diseases captures some unique features because the “spatial” structure of mosquitoes is not the same as that of humans. Mosquitoes are stratified into
<italic>m</italic>
patches (that may represent, for example, oviposition or breeding sites or forests) with infection taking place still within each Patch
<italic>j</italic>
, characterized by its particular risk
<italic>β</italic>
<sub>
<italic>vh</italic>
</sub>
<italic>a</italic>
<sub>
<italic>j</italic>
</sub>
for
<italic>j</italic>
 = 1, …,
<italic>m</italic>
. Here,
<italic>β</italic>
<sub>
<italic>vh</italic>
</sub>
represents the infectiousness of human to mosquitoes bite with
<italic>a</italic>
<sub>
<italic>j</italic>
</sub>
denoting the per capita biting rate in Patch
<italic>j</italic>
. Hosts, on the other hand, are structured by social groups or age classes (
<italic>n</italic>
). This residence habitat division can be particularly useful in the study of the impact of target control strategies.</p>
<p id="Par40">The model in [
<xref ref-type="bibr" rid="CR17">17</xref>
] describes the interactions of
<italic>n</italic>
host groups in
<italic>m</italic>
patches via System
<xref rid="Equ3" ref-type="">16.3</xref>
, where
<disp-formula id="Equd">
<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} $$\displaystyle \begin{aligned} \begin{array}{rcl} I_h &\displaystyle =&\displaystyle [I_{h,1},I_{h,2},\dots,I_{h,n}]^t, I_v=[I_{v,1},I_{v,2},\dots,I_{v,m}]^t \\ N_h &\displaystyle =&\displaystyle [N_{h,1},N_{h,2},\dots,N_{h,n}]^t, \bar{N}_v=[\bar{N}_{v,1},\bar{N}_{v,2},\dots,\bar{N}_{v,m}]^t \\ \delta &\displaystyle =&\displaystyle [\delta_1,\delta_2,\dots,\delta_m]^t, a=[a_1,a_2,\dots,a_m]^t, and \mu=[\mu_1,\mu_2,\dots,\mu_n]^t. \end{array} \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M78" displaystyle="true">
<mml:mtable columnalign="right center left" displaystyle="true">
<mml:mtr>
<mml:mtd class="eqnarray-1">
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</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:msup>
<mml:mrow>
<mml:mo stretchy="false">[</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">]</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">[</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">]</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd class="eqnarray-1">
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</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:msup>
<mml:mrow>
<mml:mo stretchy="false">[</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">]</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mover>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">[</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mover>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mover>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mover>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">]</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd class="eqnarray-1">
<mml:mi>δ</mml:mi>
</mml:mtd>
<mml:mtd class="eqnarray-2">
<mml:mo>=</mml:mo>
</mml:mtd>
<mml:mtd class="eqnarray-3">
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">[</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>δ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>δ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>δ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">]</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">[</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">]</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>d</mml:mi>
<mml:mi>μ</mml:mi>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">[</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>μ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>μ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mstyle>
<mml:mo></mml:mo>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>μ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">]</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mn>.</mml:mn>
</mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equd.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
The infected host population is denoted by the vector
<italic>I</italic>
<sub>
<italic>h</italic>
</sub>
and the host population by
<italic>N</italic>
<sub>
<italic>h</italic>
</sub>
. The infected vector population is denoted by
<italic>I</italic>
<sub>
<italic>v</italic>
</sub>
and the mosquito population by
<italic>N</italic>
<sub>
<italic>v</italic>
</sub>
. The parameters
<italic>a</italic>
<sub>
<italic>i</italic>
</sub>
,
<italic>δ</italic>
<sub>
<italic>i</italic>
</sub>
, and
<italic>μ</italic>
<sub>
<italic>v</italic>
</sub>
denote the biting, death rate of control, and natural death rate of mosquitoes in Patch
<italic>j</italic>
, for
<italic>j</italic>
 = 1, …,
<italic>m</italic>
. The infectiousness of human to mosquitoes is
<italic>β</italic>
<sub>
<italic>vh</italic>
</sub>
, whereas the infectiousness of mosquitoes to humans is given by
<italic>β</italic>
<sub>
<italic>h</italic>
</sub>
<italic>v</italic>
. The host recovery and natural mortality rates are given, respectively, by
<italic>γ</italic>
and
<italic>μ</italic>
. Finally, the matrix
<inline-formula id="IEq33">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M80">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq33.gif"></inline-graphic>
</alternatives>
</inline-formula>
represents the proportion of time host of group
<italic>i</italic>
,
<italic>i</italic>
 = 1, …,
<italic>n</italic>
, spend in Patch
<italic>j</italic>
,
<italic>j</italic>
 = 1, …,
<italic>m</italic>
. The basic reproduction number of Model 3, with
<italic>m</italic>
patches and
<italic>n</italic>
groups, is given by
<inline-formula id="IEq34">
<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} $$\mathcal {R}_{0}^2(m,n)=\rho (M_{vh}M_{hv})$$ \end{document}</tex-math>
<mml:math id="M82">
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>m</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mi>ρ</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>M</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>M</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq34.gif"></inline-graphic>
</alternatives>
</inline-formula>
, where
<disp-formula id="Eque">
<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} $$\displaystyle \begin{aligned} M_{hv}&=\beta_{hv}diag(a)diag(\mathbb{P}^tN_h)^{-1}diag(N_v)\mathbb{P}^tdiag(\mu+\gamma)^{-1} \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M84" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:msub>
<mml:mrow>
<mml:mi>M</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
<mml:mtd class="align-2">
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>μ</mml:mi>
<mml:mo>+</mml:mo>
<mml:mi>γ</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mtd>
<mml:mtd></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Eque.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
and
<disp-formula id="Equf">
<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} $$\displaystyle \begin{aligned} M_{vh}&=\beta_{vh}diag(Nh)\mathbb{P}diag(\mathbb{P}^tN_h)^{-1}diag(a)diag(\mu_v+\delta)^{-1}. \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M86" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:msub>
<mml:mrow>
<mml:mi>M</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
<mml:mtd class="align-2">
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>β</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>N</mml:mi>
<mml:mi>h</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi mathvariant="double-struck"></mml:mi>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
<mml:msub>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>g</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>μ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:mi>δ</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo></mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mn>.</mml:mn>
</mml:mtd>
<mml:mtd></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equf.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
</p>
<p id="Par41">If the host–vector network configuration is irreducible, then System
<xref rid="Equ3" ref-type="">16.3</xref>
is cooperative and strongly concave with an irreducible Jacobian, hence the theory of monotone systems, particularly Smith’s results [
<xref ref-type="bibr" rid="CR146">146</xref>
], guarantee the existence of a sharp threshold. That is, the disease-free equilibrium is globally asymptotically stable if
<inline-formula id="IEq35">
<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} $$\mathcal {R}^2_0(m,n)$$ \end{document}</tex-math>
<mml:math id="M88">
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>m</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq35.gif"></inline-graphic>
</alternatives>
</inline-formula>
is less than unity and a unique globally asymptotic stable interior endemic equilibrium exists otherwise. The effects of various forms of heterogeneity on the basic reproduction number have been explored in [
<xref ref-type="bibr" rid="CR17">17</xref>
], and we have found, for example, that the irreducibility of the residence-time matrix
<inline-formula id="IEq36">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M90">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq36.gif"></inline-graphic>
</alternatives>
</inline-formula>
is no longer sufficient to ensure a sharp threshold property, although the irreducibility of the host–vector network configuration is necessary for such property [
<xref ref-type="bibr" rid="CR17">17</xref>
].</p>
<p id="Par42">The Lagrangian approach to disease modeling can use contacts [
<xref ref-type="bibr" rid="CR32">32</xref>
] or times or both as its currency. Here, we choose time-spatial-dependent risk, that is, we choose to handle social heterogeneity by keeping track of individuals’ social or geographical membership. In this context, it is possible to include adaptive responses, for example, via the inclusion of prevalence-dependent dispersal coefficients. In this setting, the underlying hypothesis is that host behavioral responses to disease are automatic: either constant or following a predefined function. The average residence time
<inline-formula id="IEq37">
<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} $$\mathbb {P}$$ \end{document}</tex-math>
<mml:math id="M92">
<mml:mi mathvariant="double-struck"></mml:mi>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq37.gif"></inline-graphic>
</alternatives>
</inline-formula>
incorporates the average behavior of all hosts in each patch. This assumption is rather crude because it implicitly assumes that hosts have accurate information on health status and patch prevalence and respond to risk of infection accordingly. The incorporation of the role that human decisions, as a function of what individuals value and the cost that individuals place on these choices and trade-offs, within systems that account for the overall population disease dynamics has been addressed recently [
<xref ref-type="bibr" rid="CR61">61</xref>
,
<xref ref-type="bibr" rid="CR132">132</xref>
] and discussed in economic epidemiology.</p>
</sec>
</sec>
<sec id="Sec8">
<title>Behavior, Economic Epidemiology, and Mobility</title>
<p id="Par43">The movement/behavior of individuals within and between patches may be driven by real or perceived personal economic risk and associated social dynamics. Embedding behavioral-driven decisions in epidemiological models has shed new perspectives on the modeling of disease dynamics [
<xref ref-type="bibr" rid="CR61">61</xref>
], expanding available option to manage infectious diseases [
<xref ref-type="bibr" rid="CR44">44</xref>
,
<xref ref-type="bibr" rid="CR62">62</xref>
]. Economic epidemiological modeling (EEM) has a history of addressing the role of individuals’ behavior when facing the risk of disease. However, it has often failed to incorporate within host-pathogen feedback mechanisms [
<xref ref-type="bibr" rid="CR34">34</xref>
<xref ref-type="bibr" rid="CR36">36</xref>
,
<xref ref-type="bibr" rid="CR52">52</xref>
,
<xref ref-type="bibr" rid="CR70">70</xref>
,
<xref ref-type="bibr" rid="CR97">97</xref>
,
<xref ref-type="bibr" rid="CR137">137</xref>
]. EEMs that account for host-pathogen feedback mechanisms has propelled the study of the ways that contact decisions impact disease emergence or alter infectious disease-transmission dynamics. Decisions involved may include the determination to engage in trade on particular routes [
<xref ref-type="bibr" rid="CR89">89</xref>
,
<xref ref-type="bibr" rid="CR94">94</xref>
,
<xref ref-type="bibr" rid="CR95">95</xref>
,
<xref ref-type="bibr" rid="CR129">129</xref>
], or to travel to specific places [
<xref ref-type="bibr" rid="CR62">62</xref>
,
<xref ref-type="bibr" rid="CR147">147</xref>
<xref ref-type="bibr" rid="CR149">149</xref>
], or to make contact with or to avoid particular types of people [
<xref ref-type="bibr" rid="CR61">61</xref>
,
<xref ref-type="bibr" rid="CR63">63</xref>
,
<xref ref-type="bibr" rid="CR116">116</xref>
]. EEMs advance the view that the emergence of novel zoonotic diseases, such as SARS or the Nipah virus, depend on the choices that bring people into contact with other species [
<xref ref-type="bibr" rid="CR50">50</xref>
,
<xref ref-type="bibr" rid="CR51">51</xref>
]. EEMs are usually built under the assumption that associated disease risks are among the factors that individuals must consider when making decisions. Individual decision-making processes, within epidemic outbreaks, must incorporate the humans’ cost–benefit-driven adaptive responses to risk.</p>
<sec id="Sec9">
<title>Economic Epidemiology</title>
<p id="Par44">Simple EEMs are, by mathematical necessity, initially built on classical compartmental epidemiological models that account for the orderly transition of individuals facing a communicable disease, through the susceptible, infected, and recovered disease stages: the result of social and environmental interactions. EEMs assume that the amount of activity one participates in, with whom, and where may all be envisioned as the solutions to an individual decision problem. It is assumed that individual decision problems are generated by rational-value formulations based on (driven by) personal, real or perceived, cost of disease, and disease avoidance: decisions constrained by underlying population-level disease dynamics. Thus, finding effective ways of modeling rational value connections to individualized cost–benefit analyses of disease risk is fundamental to the building of useful EEMs. It is a quite challenging enterprise.</p>
<p id="Par45">EEM approaches have precursors in the epidemiological literature [
<xref ref-type="bibr" rid="CR9">9</xref>
,
<xref ref-type="bibr" rid="CR64">64</xref>
,
<xref ref-type="bibr" rid="CR69">69</xref>
]. EEM construction has been strongly influenced by past and ongoing work on the exploitation of species [
<xref ref-type="bibr" rid="CR45">45</xref>
<xref ref-type="bibr" rid="CR47">47</xref>
], a literature that addresses optimal harvesting questions in the context of wild species, or the control of invasive pests, or the management of forestry system. The methodology for modeling behavior within an EEM rests on a proper specification of behavioral costs and a description of the payoffs linked to such behaviors; the stipulation of an appropriate objective function, congruent with the decision-makers’ goals; the coupling to the dynamics of the natural resource and/or infectious human capital; and the mechanisms available for a decision-maker to alter his or her behavior and the behaviors of those around him or her. Although not all motivations for mitigation against infection are monetary in nature, we choose to refer to them as economic.</p>
<p id="Par46">Modeling whether or not an individual undertakes infection-causing behavior provides a natural starting point since it is connected to the rate of generation of secondary cases of infection per unit of time, the so-called incidence rate. A simple incidence function that captures the instantaneous expectation of the rate of new infections at a given time is therefore given by
<disp-formula id="Equg">
<alternatives>
<tex-math id="M93">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned} S(t)cP_{SI}(t)\rho, \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M94" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:mi>S</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>c</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>P</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mi>I</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mi>ρ</mml:mi>
<mml:mo>,</mml:mo>
</mml:mtd>
<mml:mtd class="align-2"></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equg.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
where
<italic>S</italic>
(
<italic>t</italic>
) is the number of individuals susceptible to the disease,
<italic>c</italic>
is the average amount of activity they engage in,
<italic>P</italic>
<sub>
<italic>SI</italic>
</sub>
(
<italic>t</italic>
) is the probability that a unit of such activity takes the susceptible individual in contact with infectious individuals/material, and
<italic>ρ</italic>
is the probability that such contact successfully infects.</p>
<p id="Par47">A decision to reduce the volume of activity one engages in (lowering
<italic>c</italic>
) has been shown in many cases to be phenomenologically identical to reducing one’s chances of coming in contact with infection (lowering
<italic>P</italic>
<sub>
<italic>SI</italic>
</sub>
(
<italic>t</italic>
)) by altering where the activity takes place and with whom one engages or by substituting a particular behavior for a riskier one [
<xref ref-type="bibr" rid="CR62">62</xref>
,
<xref ref-type="bibr" rid="CR118">118</xref>
]. The modeling assumes that individuals derive benefits from making contacts but may incur costs associated with an infection. Hence, the modeling assumes that activity volume or contacts are chosen to maximize expected utility (rudimentarily, benefit less cost), balancing the marginal value of a contact against the increased risk of infection. The utility function is assumed to depend on the health status of the individual and the contacts that they make, that is, the utility of a representative individual of health status
<italic>h</italic>
is given, for example, by the function
<disp-formula id="Equ4">
<label>16.4</label>
<alternatives>
<tex-math id="M95">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned} U_h=U(h,C^h). {} \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M96" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:msub>
<mml:mrow>
<mml:mi>U</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mi>U</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>h</mml:mi>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
<mml:mn>.</mml:mn>
</mml:mtd>
<mml:mtd class="align-2"></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equ4.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
</p>
<p id="Par48">The utility function is assumed to be concave, decreasing in illness and increasing in contacts. If the probability of transitioning from susceptible to infected health status depends on the rate of contacts, the optimal choice of contacts is the solution to a dynamic programming problem:
<disp-formula id="Equ5">
<label>16.5</label>
<alternatives>
<tex-math id="M97">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned} V_t(h)=\underset{C_s}{max}\left\{U_t(h_t,C_t^h)+r\sum_j\rho^{hj}V_{t+1}(j)\right\}, {} \end{aligned} $$ \end{document}</tex-math>
<mml:math id="M98" displaystyle="true">
<mml:mtable columnalign="right left">
<mml:mtr>
<mml:mtd class="align-1">
<mml:msub>
<mml:mrow>
<mml:mi>V</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>h</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:munder>
<mml:mrow>
<mml:mi>m</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:munder>
<mml:mfenced close="}" open="{" separators="">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>U</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>r</mml:mi>
<mml:munder accentunder="false">
<mml:mrow>
<mml:mo mathsize="big"></mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:munder>
<mml:msup>
<mml:mrow>
<mml:mi>ρ</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msup>
<mml:msub>
<mml:mrow>
<mml:mi>V</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mfenced>
<mml:mo>,</mml:mo>
</mml:mtd>
<mml:mtd class="align-2"></mml:mtd>
<mml:mtd></mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:math>
<graphic xlink:href="211675_1_En_16_Chapter_Equ5.gif" position="anchor"></graphic>
</alternatives>
</disp-formula>
where
<italic>r</italic>
is the discount rate and
<italic>ρ</italic>
<sup>
<italic>hj</italic>
</sup>
is the probability of transition from health state
<italic>h</italic>
to health state
<italic>j</italic>
. This probability depends on the current state of the system, {
<italic>S</italic>
(
<italic>t</italic>
),
<italic>I</italic>
(
<italic>t</italic>
),
<italic>R</italic>
(
<italic>t</italic>
)}, the behavior of individuals in other health classes,
<italic>C</italic>
<sup>
<italic>h</italic>
</sup>
, and the behavior of individuals in the decision-makers’ own health class,
<inline-formula id="IEq38">
<alternatives>
<tex-math id="M99">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar {C}^{h}$$ \end{document}</tex-math>
<mml:math id="M100" displaystyle="true">
<mml:msup>
<mml:mrow>
<mml:mover>
<mml:mrow>
<mml:mi>C</mml:mi>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msup>
</mml:math>
<inline-graphic xlink:href="211675_1_En_16_Chapter_IEq38.gif"></inline-graphic>
</alternatives>
</inline-formula>
. In short, we have a complex adaptive system where individuals within the model, in this example, impact disease outcomes (through changes in the incidence). Eqs. (
<xref rid="Equ4" ref-type="">16.4</xref>
) and (
<xref rid="Equ5" ref-type="">16.5</xref>
) are both optimized from an individual perspective. Within this individual context, EEMs have shown that individual distancing, conditional on health status, plays an important role in the spread of infectious disease. However, it has also been shown that the provision of incentives for infectious individuals to self- quarantine is likely to be welfare-enhancing [
<xref ref-type="bibr" rid="CR43">43</xref>
,
<xref ref-type="bibr" rid="CR44">44</xref>
,
<xref ref-type="bibr" rid="CR61">61</xref>
,
<xref ref-type="bibr" rid="CR115">115</xref>
]. Thus, understanding how the individual responds to relative costs of disease and disease prevention is critical to the design of public policy that affects those costs. Indeed, the role of recovered individuals in protecting susceptible individuals has been generally overlooked in public health interventions, and yet it is known that their behavior is, in fact, critical to disease management due to the positive externality the individuals’ contacts generate once in an immune, non-disease-transmitting state [
<xref ref-type="bibr" rid="CR63">63</xref>
]. The benefits of herd immunity include the positive externality associated with acquired immunity but may, in turn, be nullified by nontargeted social-distancing policies that induce such immune individuals to reduce contacts. By incentivizing the maintenance of contacts by recovered individuals policy may lower the probability of susceptible individuals contacting infected individuals and/or allow susceptible and infected individuals to individually increase contacts without changing the probability of infection.</p>
</sec>
<sec id="Sec10">
<title>Lagrangian and Economic Epidemiology Models</title>
<p id="Par49">Theoretical epidemiology aims to disentangle the role of epidemiological and socioeconomic forces on disease dynamics. However, the role of behavior and individual decisions in response to a changing epidemic landscape has not been tackled systematically. In this chapter, we highlight alternative ways for modeling disease transmission that can use contacts as its currency or residence times or both. It seems evident that the use of contacts, in the context of influenza, Ebola, tuberculosis, or other communicable diseases (as opposed to sexually transmitted diseases), while intellectually satisfying, fails to recognized the fact that contacts cannot be measured effectively in settings where the risk of acquiring such infections is the highest. In fact, when contact-based models are fitted to data, it has become clear that contact rates play primarily the role of fitting parameters; in other words, if the goal is connecting models to data that include transmission mechanisms, then the use of contacts has serious shortcomings. Therefore, in order to advance the role of theory, we need models that are informed by data. Hence, the need to invest on efforts that bring forth alternative modes of modeling. While Lagrangian approaches are not a panacea, their use extends the possibilities because they depend on parameters like residence times and average time to infection for a given environment (risk), that is, parameters that can be measured. Frameworks should be explored and compared and their analyses contrasted. We have revisited recent work that equates behavior with cost–benefit decisions, which, in turn, are linked, within our framework, to health status and population-level dynamics, the components of a complex adaptive system. Connecting the Lagrangian movement-modeling approach with EEMs seems promising, albeit computationally and mathematically challenging. However, as discussed in [
<xref ref-type="bibr" rid="CR117">117</xref>
], the perception that the benefits of disease control are limited by the capacity of the weakest link in the chain to respond effectively is not a basic result of EEM models, which actually show that it may not be in within the ability of an individual in a poor community/country to do more risk mitigation. The need for richer communities or nations to find ways to incentivize greater levels of disease-risk mitigation in poor countries may be the best approach.</p>
<p id="Par50">Simon Levin, in his address as the 2004 recipient of the Heineken award, placed our narrow perspective in a broader powerful context:
<disp-quote>
<p id="Par51">A great challenge before us is thus to understand the dynamics of social norms, how they arise, how they spread, how they are sustained and how they change. Models of these dynamics have many of the same features as models of epidemic spread, no great surprise, since many aspects of culture have the characteristics of being social diseases. 1998 Heineken award winner Paul Ehrlich and I have been directing our collective energies to this problem, convinced that it is as important to understand the dynamics of the social systems in which we live as it is to understand the ecological systems themselves. Understanding the links between individual behavior and societal consequences, and characterizing the networks of interaction and influence, create the potential to change the reward structures so that the social costs of individual actions are brought down to the level of individual payoffs. It is a daunting task, both because of the amount we still must learn, and because of the ethical dilemmas that are implicit in any form of social engineering. But it is a task from which we cannot shrink, lest we squander the last of our diminishing resources.</p>
</disp-quote>
</p>
</sec>
</sec>
</body>
<back>
<fn-group>
<fn id="Fn1">
<label>1</label>
<p id="Par19">The Independent, Friday 16 March 2012.</p>
</fn>
</fn-group>
<ref-list id="Bib1">
<title>References</title>
<ref id="CR1">
<label>1.</label>
<mixed-citation publication-type="other">Anderson, R.M., R. M. May, and B. Anderson (1992) Infectious Diseases of Humans: Dynamics and Control, volume 28. Wiley Online Library.</mixed-citation>
</ref>
<ref id="CR2">
<label>2.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Andersson</surname>
<given-names>Dan I</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>Bruce R</given-names>
</name>
</person-group>
<article-title>The biological cost of antibiotic resistance</article-title>
<source>Current Opinion in Microbiology</source>
<year>1999</year>
<volume>2</volume>
<issue>5</issue>
<fpage>489</fpage>
<lpage>493</lpage>
<pub-id pub-id-type="pmid">10508723</pub-id>
</element-citation>
</ref>
<ref id="CR3">
<label>3.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Andreasen</surname>
<given-names>Viggo</given-names>
</name>
</person-group>
<article-title>Dynamics of annual influenza A epidemics with immuno-selection</article-title>
<source>Journal of Mathematical Biology</source>
<year>2003</year>
<volume>46</volume>
<issue>6</issue>
<fpage>504</fpage>
<lpage>536</lpage>
<pub-id pub-id-type="pmid">12783180</pub-id>
</element-citation>
</ref>
<ref id="CR4">
<label>4.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Andreasen</surname>
<given-names>Viggo</given-names>
</name>
<name>
<surname>Lin</surname>
<given-names>Juan</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>Simon A.</given-names>
</name>
</person-group>
<article-title>The dynamics of cocirculating influenza strains conferring partial cross-immunity</article-title>
<source>Journal of Mathematical Biology</source>
<year>1997</year>
<volume>35</volume>
<issue>7</issue>
<fpage>825</fpage>
<lpage>842</lpage>
<pub-id pub-id-type="pmid">9269738</pub-id>
</element-citation>
</ref>
<ref id="CR5">
<label>5.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Arino</surname>
<given-names>Julien</given-names>
</name>
<name>
<surname>Jordan</surname>
<given-names>Richard</given-names>
</name>
<name>
<surname>van den Driessche</surname>
<given-names>P.</given-names>
</name>
</person-group>
<article-title>Quarantine in a multi-species epidemic model with spatial dynamics</article-title>
<source>Mathematical Biosciences</source>
<year>2007</year>
<volume>206</volume>
<issue>1</issue>
<fpage>46</fpage>
<lpage>60</lpage>
<pub-id pub-id-type="pmid">16343557</pub-id>
</element-citation>
</ref>
<ref id="CR6">
<label>6.</label>
<mixed-citation publication-type="other">Arino, J. and P. Van Den Driessche (2003) The basic reproduction number in a multi-city compartmental epidemic model, In Positive Systems, pages 135–142. Springer</mixed-citation>
</ref>
<ref id="CR7">
<label>7.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Arino</surname>
<given-names>Julien</given-names>
</name>
<name>
<surname>van den Driessche</surname>
<given-names>P.</given-names>
</name>
</person-group>
<article-title>A multi-city epidemic model</article-title>
<source>Mathematical Population Studies</source>
<year>2003</year>
<volume>10</volume>
<issue>3</issue>
<fpage>175</fpage>
<lpage>193</lpage>
</element-citation>
</ref>
<ref id="CR8">
<label>8.</label>
<mixed-citation publication-type="other">Arino, J. and P. van den Driessche (2006) Metapopulation epidemic models, a survey, Fields Institute Communications
<bold>48</bold>
: 1–13.</mixed-citation>
</ref>
<ref id="CR9">
<label>9.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Auld</surname>
<given-names>M.Christopher</given-names>
</name>
</person-group>
<article-title>Choices, beliefs, and infectious disease dynamics</article-title>
<source>Journal of Health Economics</source>
<year>2003</year>
<volume>22</volume>
<issue>3</issue>
<fpage>361</fpage>
<lpage>377</lpage>
<pub-id pub-id-type="pmid">12683957</pub-id>
</element-citation>
</ref>
<ref id="CR10">
<label>10.</label>
<mixed-citation publication-type="other">Bailey, N.T., et al, (1982) The biomathematics of malaria: the biomathematics of diseases: 1, The biomathematics of malaria. The Biomathematics of Diseases
<bold>1</bold>
.</mixed-citation>
</ref>
<ref id="CR11">
<label>11.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bajardi</surname>
<given-names>Paolo</given-names>
</name>
<name>
<surname>Poletto</surname>
<given-names>Chiara</given-names>
</name>
<name>
<surname>Ramasco</surname>
<given-names>Jose J.</given-names>
</name>
<name>
<surname>Tizzoni</surname>
<given-names>Michele</given-names>
</name>
<name>
<surname>Colizza</surname>
<given-names>Vittoria</given-names>
</name>
<name>
<surname>Vespignani</surname>
<given-names>Alessandro</given-names>
</name>
</person-group>
<article-title>Human Mobility Networks, Travel Restrictions, and the Global Spread of 2009 H1N1 Pandemic</article-title>
<source>PLoS ONE</source>
<year>2011</year>
<volume>6</volume>
<issue>1</issue>
<fpage>e16591</fpage>
<pub-id pub-id-type="pmid">21304943</pub-id>
</element-citation>
</ref>
<ref id="CR12">
<label>12.</label>
<mixed-citation publication-type="other">Banks, H.T. and C. Castillo-Chavez (2003) Bioterrorism: Mathematical Modeling Applications in Homeland Security. SIAM.</mixed-citation>
</ref>
<ref id="CR13">
<label>13.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bansal</surname>
<given-names>Shweta</given-names>
</name>
<name>
<surname>Grenfell</surname>
<given-names>Bryan T</given-names>
</name>
<name>
<surname>Meyers</surname>
<given-names>Lauren Ancel</given-names>
</name>
</person-group>
<article-title>When individual behaviour matters: homogeneous and network models in epidemiology</article-title>
<source>Journal of The Royal Society Interface</source>
<year>2007</year>
<volume>4</volume>
<issue>16</issue>
<fpage>879</fpage>
<lpage>891</lpage>
</element-citation>
</ref>
<ref id="CR14">
<label>14.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Baroyan</surname>
<given-names>O. V.</given-names>
</name>
<name>
<surname>Rvachev</surname>
<given-names>L. A.</given-names>
</name>
<name>
<surname>Basilevsky</surname>
<given-names>U. V.</given-names>
</name>
<name>
<surname>Ermakov</surname>
<given-names>V. V.</given-names>
</name>
<name>
<surname>Frank</surname>
<given-names>K. D.</given-names>
</name>
<name>
<surname>Rvachev</surname>
<given-names>M. A.</given-names>
</name>
<name>
<surname>Shashkov</surname>
<given-names>V. A.</given-names>
</name>
</person-group>
<article-title>Computer modelling of influenza epidemics for the whole country (USSR)</article-title>
<source>Advances in Applied Probability</source>
<year>1971</year>
<volume>3</volume>
<issue>02</issue>
<fpage>224</fpage>
<lpage>226</lpage>
</element-citation>
</ref>
<ref id="CR15">
<label>15.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>BELLOMO</surname>
<given-names>NICOLA</given-names>
</name>
<name>
<surname>PICCOLI</surname>
<given-names>BENEDETTO</given-names>
</name>
<name>
<surname>TOSIN</surname>
<given-names>ANDREA</given-names>
</name>
</person-group>
<article-title>MODELING CROWD DYNAMICS FROM A COMPLEX SYSTEM VIEWPOINT</article-title>
<source>Mathematical Models and Methods in Applied Sciences</source>
<year>2012</year>
<volume>22</volume>
<issue>supp02</issue>
<fpage>1230004</fpage>
</element-citation>
</ref>
<ref id="CR16">
<label>16.</label>
<mixed-citation publication-type="other">Bergstrom, C. T., Lo, M.,
<italic>&</italic>
Lipsitch, M. (2004) Ecological theory suggests that antimicrobial cycling will not reduce antimicrobial resistance, Proc Natl Acad Sci.
<bold>101</bold>
: 13285–13290.</mixed-citation>
</ref>
<ref id="CR17">
<label>17.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bichara</surname>
<given-names>Derdei</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
</person-group>
<article-title>Vector-borne diseases models with residence times – A Lagrangian perspective</article-title>
<source>Mathematical Biosciences</source>
<year>2016</year>
<volume>281</volume>
<fpage>128</fpage>
<lpage>138</lpage>
<pub-id pub-id-type="pmid">27622812</pub-id>
</element-citation>
</ref>
<ref id="CR18">
<label>18.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bichara</surname>
<given-names>Derdei</given-names>
</name>
<name>
<surname>Kang</surname>
<given-names>Yun</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
<name>
<surname>Horan</surname>
<given-names>Richard</given-names>
</name>
<name>
<surname>Perrings</surname>
<given-names>Charles</given-names>
</name>
</person-group>
<article-title>SIS and SIR Epidemic Models Under Virtual Dispersal</article-title>
<source>Bulletin of Mathematical Biology</source>
<year>2015</year>
<volume>77</volume>
<issue>11</issue>
<fpage>2004</fpage>
<lpage>2034</lpage>
<pub-id pub-id-type="pmid">26489419</pub-id>
</element-citation>
</ref>
<ref id="CR19">
<label>19.</label>
<mixed-citation publication-type="other">Brauer F., and C. Castillo-Chávez (2013) Mathematical models for communicable diseases. No. 84 in CBMS-NSF regional conference series in applied mathematics Philadelphia: Society for Industrial and Applied Mathematics.</mixed-citation>
</ref>
<ref id="CR20">
<label>20.</label>
<element-citation publication-type="book">
<person-group person-group-type="editor">
<name>
<surname>Brauer</surname>
<given-names>Fred</given-names>
</name>
<name>
<surname>van den Driessche</surname>
<given-names>Pauline</given-names>
</name>
<name>
<surname>Wu</surname>
<given-names>Jianhong</given-names>
</name>
</person-group>
<source>Mathematical Epidemiology</source>
<year>2008</year>
<publisher-loc>Berlin, Heidelberg</publisher-loc>
<publisher-name>Springer Berlin Heidelberg</publisher-name>
</element-citation>
</ref>
<ref id="CR21">
<label>21.</label>
<mixed-citation publication-type="other">Castillo-Chavez, C. (2002) Mathematical Approaches for Emerging and Reemerging Infectious Diseases: an Introduction, Volume 1. Springer Science & Business Media.</mixed-citation>
</ref>
<ref id="CR22">
<label>22.</label>
<mixed-citation publication-type="other">Castillo Chavez, C. (2002) Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, volume 126. Springer, 2002.</mixed-citation>
</ref>
<ref id="CR23">
<label>23.</label>
<mixed-citation publication-type="other">Castillo-Chavez, C. (2013) Mathematical and Statistical Approaches to AIDS Epidemiology, volume 83. Springer Science & Business Media.</mixed-citation>
</ref>
<ref id="CR24">
<label>24.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Castillo-Chavez</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Barley</surname>
<given-names>K.</given-names>
</name>
<name>
<surname>Bichara</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Chowell</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Diaz Herrera</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Espinoza</surname>
<given-names>B.</given-names>
</name>
<name>
<surname>Moreno</surname>
<given-names>V.</given-names>
</name>
<name>
<surname>Towers</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Yong</surname>
<given-names>K. E.</given-names>
</name>
</person-group>
<article-title>Modeling Ebola at the Mathematical and Theoretical Biology Institute (MTBI)</article-title>
<source>Notices of the American Mathematical Society</source>
<year>2016</year>
<volume>63</volume>
<issue>04</issue>
<fpage>366</fpage>
<lpage>371</lpage>
</element-citation>
</ref>
<ref id="CR25">
<label>25.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
<name>
<surname>Bichara</surname>
<given-names>Derdei</given-names>
</name>
<name>
<surname>Morin</surname>
<given-names>Benjamin R.</given-names>
</name>
</person-group>
<article-title>Perspectives on the role of mobility, behavior, and time scales in the spread of diseases</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>2016</year>
<volume>113</volume>
<issue>51</issue>
<fpage>14582</fpage>
<lpage>14588</lpage>
</element-citation>
</ref>
<ref id="CR26">
<label>26.</label>
<mixed-citation publication-type="other">Castillo-Chavez, C. and G. Chowell (2011) Preface: Mathematical models, challenges, and lessons learned Special volume on influenza dynamics, volume 8. Math. Biosc. Eng.
<bold>8</bold>
:1–6.</mixed-citation>
</ref>
<ref id="CR27">
<label>27.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
<name>
<surname>Curtiss</surname>
<given-names>Roy</given-names>
</name>
<name>
<surname>Daszak</surname>
<given-names>Peter</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>Simon A</given-names>
</name>
<name>
<surname>Patterson-Lomba</surname>
<given-names>Oscar</given-names>
</name>
<name>
<surname>Perrings</surname>
<given-names>Charles</given-names>
</name>
<name>
<surname>Poste</surname>
<given-names>George</given-names>
</name>
<name>
<surname>Towers</surname>
<given-names>Sherry</given-names>
</name>
</person-group>
<article-title>Beyond Ebola: lessons to mitigate future pandemics</article-title>
<source>The Lancet Global Health</source>
<year>2015</year>
<volume>3</volume>
<issue>7</issue>
<fpage>e354</fpage>
<lpage>e355</lpage>
<pub-id pub-id-type="pmid">26087978</pub-id>
</element-citation>
</ref>
<ref id="CR28">
<label>28.</label>
<mixed-citation publication-type="other">Castillo-Chavez,C., Z. Feng,
<italic>et al</italic>
(1996) Optimal vaccination strategies for TB in age-structure populations, J. Math. Biol.
<bold>35</bold>
: 629–656.</mixed-citation>
</ref>
<ref id="CR29">
<label>29.</label>
<mixed-citation publication-type="other">Castillo-Chavez, C., H. W. Hethcote, V. Andreasen, S. A. Levin, and W. M. Liu (1988) Cross-immunity in the dynamics of homogeneous and heterogeneous populations, Proceedings of the Autumn Course Research Seminars Mathematical Ecology, pages 303–316.</mixed-citation>
</ref>
<ref id="CR30">
<label>30.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Castillo-Chavez</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Hethcote</surname>
<given-names>H. W.</given-names>
</name>
<name>
<surname>Andreasen</surname>
<given-names>V.</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>S. A.</given-names>
</name>
<name>
<surname>Liu</surname>
<given-names>W. M.</given-names>
</name>
</person-group>
<article-title>Epidemiological models with age structure, proportionate mixing, and cross-immunity</article-title>
<source>Journal of Mathematical Biology</source>
<year>1989</year>
<volume>27</volume>
<issue>3</issue>
<fpage>233</fpage>
<lpage>258</lpage>
<pub-id pub-id-type="pmid">2746140</pub-id>
</element-citation>
</ref>
<ref id="CR31">
<label>31.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
<name>
<surname>Song</surname>
<given-names>Baojun</given-names>
</name>
</person-group>
<article-title>Dynamical Models of Tuberculosis and Their Applications</article-title>
<source>Mathematical Biosciences and Engineering</source>
<year>2004</year>
<volume>1</volume>
<issue>2</issue>
<fpage>361</fpage>
<lpage>404</lpage>
<pub-id pub-id-type="pmid">20369977</pub-id>
</element-citation>
</ref>
<ref id="CR32">
<label>32.</label>
<mixed-citation publication-type="other">Castillo-Chavez, C., B. Song, and J. Zhang (2003) An epidemic model with virtual mass transportation: the case of smallpox in a large city, In Bioterrorism: Mathematical Modeling Applications in Homeland Security, pages 173–197. SIAM.</mixed-citation>
</ref>
<ref id="CR33">
<label>33.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Cauchemez</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Bhattarai</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Marchbanks</surname>
<given-names>T. L.</given-names>
</name>
<name>
<surname>Fagan</surname>
<given-names>R. P.</given-names>
</name>
<name>
<surname>Ostroff</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Ferguson</surname>
<given-names>N. M.</given-names>
</name>
<name>
<surname>Swerdlow</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Sodha</surname>
<given-names>S. V.</given-names>
</name>
<name>
<surname>Moll</surname>
<given-names>M. E.</given-names>
</name>
<name>
<surname>Angulo</surname>
<given-names>F. J.</given-names>
</name>
<name>
<surname>Palekar</surname>
<given-names>R.</given-names>
</name>
<name>
<surname>Archer</surname>
<given-names>W. R.</given-names>
</name>
<name>
<surname>Finelli</surname>
<given-names>L.</given-names>
</name>
</person-group>
<article-title>Role of social networks in shaping disease transmission during a community outbreak of 2009 H1N1 pandemic influenza</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>2011</year>
<volume>108</volume>
<issue>7</issue>
<fpage>2825</fpage>
<lpage>2830</lpage>
</element-citation>
</ref>
<ref id="CR34">
<label>34.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chen</surname>
<given-names>Frederick</given-names>
</name>
<name>
<surname>Jiang</surname>
<given-names>Miaohua</given-names>
</name>
<name>
<surname>Rabidoux</surname>
<given-names>Scott</given-names>
</name>
<name>
<surname>Robinson</surname>
<given-names>Stephen</given-names>
</name>
</person-group>
<article-title>Public avoidance and epidemics: Insights from an economic model</article-title>
<source>Journal of Theoretical Biology</source>
<year>2011</year>
<volume>278</volume>
<issue>1</issue>
<fpage>107</fpage>
<lpage>119</lpage>
<pub-id pub-id-type="pmid">21419135</pub-id>
</element-citation>
</ref>
<ref id="CR35">
<label>35.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chen</surname>
<given-names>Frederick H.</given-names>
</name>
</person-group>
<article-title>Rational behavioral response and the transmission of STDs</article-title>
<source>Theoretical Population Biology</source>
<year>2004</year>
<volume>66</volume>
<issue>4</issue>
<fpage>307</fpage>
<lpage>316</lpage>
<pub-id pub-id-type="pmid">15560909</pub-id>
</element-citation>
</ref>
<ref id="CR36">
<label>36.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chen</surname>
<given-names>Frederick H.</given-names>
</name>
</person-group>
<article-title>Modeling the effect of information quality on risk behavior change and the transmission of infectious diseases</article-title>
<source>Mathematical Biosciences</source>
<year>2009</year>
<volume>217</volume>
<issue>2</issue>
<fpage>125</fpage>
<lpage>133</lpage>
<pub-id pub-id-type="pmid">19059272</pub-id>
</element-citation>
</ref>
<ref id="CR37">
<label>37.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chew</surname>
<given-names>Cynthia</given-names>
</name>
<name>
<surname>Eysenbach</surname>
<given-names>Gunther</given-names>
</name>
</person-group>
<article-title>Pandemics in the Age of Twitter: Content Analysis of Tweets during the 2009 H1N1 Outbreak</article-title>
<source>PLoS ONE</source>
<year>2010</year>
<volume>5</volume>
<issue>11</issue>
<fpage>e14118</fpage>
<pub-id pub-id-type="pmid">21124761</pub-id>
</element-citation>
</ref>
<ref id="CR38">
<label>38.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chow</surname>
<given-names>Karen</given-names>
</name>
<name>
<surname>Wang</surname>
<given-names>Xiaohong</given-names>
</name>
<name>
<surname>Curtiss</surname>
<given-names>R.</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
</person-group>
<article-title>Evaluating the efficacy of antimicrobial cycling programmes and patient isolation on dual resistance in hospitals</article-title>
<source>Journal of Biological Dynamics</source>
<year>2011</year>
<volume>5</volume>
<issue>1</issue>
<fpage>27</fpage>
<lpage>43</lpage>
<pub-id pub-id-type="pmid">22877228</pub-id>
</element-citation>
</ref>
<ref id="CR39">
<label>39.</label>
<mixed-citation publication-type="other">Chow, K.C., X. Wang, and C. Castillo-Chávez (2007) A mathematical model of nosocomial infection and antibiotic resistance: evaluating the efficacy of antimicrobial cycling programs and patient isolation on dual resistance Mathematical and Theoretical Biology Institute archive, 2007.</mixed-citation>
</ref>
<ref id="CR40">
<label>40.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chowell</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Fenimore</surname>
<given-names>P.W.</given-names>
</name>
<name>
<surname>Castillo-Garsow</surname>
<given-names>M.A.</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>C.</given-names>
</name>
</person-group>
<article-title>SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism</article-title>
<source>Journal of Theoretical Biology</source>
<year>2003</year>
<volume>224</volume>
<issue>1</issue>
<fpage>1</fpage>
<lpage>8</lpage>
<pub-id pub-id-type="pmid">12900200</pub-id>
</element-citation>
</ref>
<ref id="CR41">
<label>41.</label>
<mixed-citation publication-type="other">Chowell, G., J. M. Hyman, L. M. Bettencourt, and C. Castillo-Chavez (2009) Mathematical and Statistical Estimation Approaches in Epidemiology, Springer.</mixed-citation>
</ref>
<ref id="CR42">
<label>42.</label>
<mixed-citation publication-type="other">Chowell, G., J. M. Hyman, S. Eubank, and C. Castillo-Chavez (2003) Scaling laws for the movement of people between locations in a large city, Phys. Rev. E
<bold>68</bold>
: 066102, 2003.</mixed-citation>
</ref>
<ref id="CR43">
<label>43.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chowell</surname>
<given-names>Gerardo</given-names>
</name>
<name>
<surname>Nishiura</surname>
<given-names>Hiroshi</given-names>
</name>
<name>
<surname>Bettencourt</surname>
<given-names>Luís M.A</given-names>
</name>
</person-group>
<article-title>Comparative estimation of the reproduction number for pandemic influenza from daily case notification data</article-title>
<source>Journal of The Royal Society Interface</source>
<year>2006</year>
<volume>4</volume>
<issue>12</issue>
<fpage>155</fpage>
<lpage>166</lpage>
</element-citation>
</ref>
<ref id="CR44">
<label>44.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Chowell</surname>
<given-names>Gerardo</given-names>
</name>
<name>
<surname>Viboud</surname>
<given-names>Cécile</given-names>
</name>
<name>
<surname>Wang</surname>
<given-names>Xiaohong</given-names>
</name>
<name>
<surname>Bertozzi</surname>
<given-names>Stefano M.</given-names>
</name>
<name>
<surname>Miller</surname>
<given-names>Mark A.</given-names>
</name>
</person-group>
<article-title>Adaptive Vaccination Strategies to Mitigate Pandemic Influenza: Mexico as a Case Study</article-title>
<source>PLoS ONE</source>
<year>2009</year>
<volume>4</volume>
<issue>12</issue>
<fpage>e8164</fpage>
<pub-id pub-id-type="pmid">19997603</pub-id>
</element-citation>
</ref>
<ref id="CR45">
<label>45.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Clark</surname>
<given-names>Colin W.</given-names>
</name>
</person-group>
<article-title>Profit Maximization and the Extinction of Animal Species</article-title>
<source>Journal of Political Economy</source>
<year>1973</year>
<volume>81</volume>
<issue>4</issue>
<fpage>950</fpage>
<lpage>961</lpage>
</element-citation>
</ref>
<ref id="CR46">
<label>46.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Clark</surname>
<given-names>Colin W.</given-names>
</name>
</person-group>
<article-title>A delayed-recruitment model of population dynamics, with an application to baleen whale populations</article-title>
<source>Journal of Mathematical Biology</source>
<year>1976</year>
<volume>3</volume>
<issue>3-4</issue>
<fpage>381</fpage>
<lpage>391</lpage>
<pub-id pub-id-type="pmid">1022837</pub-id>
</element-citation>
</ref>
<ref id="CR47">
<label>47.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Clark</surname>
<given-names>Colin W.</given-names>
</name>
</person-group>
<article-title>Mathematical Models in the Economics of Renewable Resources</article-title>
<source>SIAM Review</source>
<year>1979</year>
<volume>21</volume>
<issue>1</issue>
<fpage>81</fpage>
<lpage>99</lpage>
</element-citation>
</ref>
<ref id="CR48">
<label>48.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Cosner</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Beier</surname>
<given-names>J.C.</given-names>
</name>
<name>
<surname>Cantrell</surname>
<given-names>R.S.</given-names>
</name>
<name>
<surname>Impoinvil</surname>
<given-names>D.</given-names>
</name>
<name>
<surname>Kapitanski</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Potts</surname>
<given-names>M.D.</given-names>
</name>
<name>
<surname>Troyo</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Ruan</surname>
<given-names>S.</given-names>
</name>
</person-group>
<article-title>The effects of human movement on the persistence of vector-borne diseases</article-title>
<source>Journal of Theoretical Biology</source>
<year>2009</year>
<volume>258</volume>
<issue>4</issue>
<fpage>550</fpage>
<lpage>560</lpage>
<pub-id pub-id-type="pmid">19265711</pub-id>
</element-citation>
</ref>
<ref id="CR49">
<label>49.</label>
<mixed-citation publication-type="other">Cristiani, E., B. Piccoli, and A. Tosin (2014) Multiscale Modeling of Pedestrian Dynamics, volume 12. Springer.</mixed-citation>
</ref>
<ref id="CR50">
<label>50.</label>
<element-citation publication-type="book">
<person-group person-group-type="author">
<name>
<surname>Daszak</surname>
<given-names>Peter</given-names>
</name>
<name>
<surname>Plowright</surname>
<given-names>R. K.</given-names>
</name>
<name>
<surname>Epstein</surname>
<given-names>J. H.</given-names>
</name>
<name>
<surname>Pulliam</surname>
<given-names>J.</given-names>
</name>
<name>
<surname>Abdul Rahman</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Field</surname>
<given-names>H. E.</given-names>
</name>
<name>
<surname>Jamaluddin</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Sharifah</surname>
<given-names>S. H.</given-names>
</name>
<name>
<surname>Smith</surname>
<given-names>C. S.</given-names>
</name>
<name>
<surname>Olival</surname>
<given-names>K. J.</given-names>
</name>
<name>
<surname>Luby</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Halpin</surname>
<given-names>K.</given-names>
</name>
<name>
<surname>Hyatt</surname>
<given-names>A. D.</given-names>
</name>
<name>
<surname>Cunningham</surname>
<given-names>A. A.</given-names>
</name>
</person-group>
<article-title>The emergence of Nipah and Hendra virus: pathogen dynamics across a wildlife-livestock-human continuum</article-title>
<source>Disease Ecology</source>
<year>2006</year>
<fpage>186</fpage>
<lpage>201</lpage>
</element-citation>
</ref>
<ref id="CR51">
<label>51.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>DASZAK</surname>
<given-names>PETER</given-names>
</name>
<name>
<surname>TABOR</surname>
<given-names>GARY M.</given-names>
</name>
<name>
<surname>KILPATRICK</surname>
<given-names>A MARM</given-names>
</name>
<name>
<surname>EPSTEIN</surname>
<given-names>JON</given-names>
</name>
<name>
<surname>PLOWRIGHT</surname>
<given-names>RAINA</given-names>
</name>
</person-group>
<article-title>Conservation Medicine and a New Agenda for Emerging Diseases</article-title>
<source>Annals of the New York Academy of Sciences</source>
<year>2004</year>
<volume>1026</volume>
<issue>1</issue>
<fpage>1</fpage>
<lpage>11</lpage>
<pub-id pub-id-type="pmid">15604464</pub-id>
</element-citation>
</ref>
<ref id="CR52">
<label>52.</label>
<mixed-citation publication-type="other">Del Valle,S., H. Hethcote, J. M. Hyman, and C. Castillo-Chavez (2005) Effects of behavioral changes in a smallpox attack model. Math. Biosc.
<bold>195:</bold>
228–251.</mixed-citation>
</ref>
<ref id="CR53">
<label>53.</label>
<mixed-citation publication-type="other">Diekmann, O., H. Heesterbeek, and T. Britton (2012) Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press.</mixed-citation>
</ref>
<ref id="CR54">
<label>54.</label>
<mixed-citation publication-type="other">Diekmann, O. and J. A. P. Heesterbeek (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Volume 5. John Wiley & Sons.</mixed-citation>
</ref>
<ref id="CR55">
<label>55.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Domingo</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Holland</surname>
<given-names>J. J.</given-names>
</name>
</person-group>
<article-title>RNA VIRUS MUTATIONS AND FITNESS FOR SURVIVAL</article-title>
<source>Annual Review of Microbiology</source>
<year>1997</year>
<volume>51</volume>
<issue>1</issue>
<fpage>151</fpage>
<lpage>178</lpage>
</element-citation>
</ref>
<ref id="CR56">
<label>56.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Epstein</surname>
<given-names>Joshua M.</given-names>
</name>
<name>
<surname>Parker</surname>
<given-names>Jon</given-names>
</name>
<name>
<surname>Cummings</surname>
<given-names>Derek</given-names>
</name>
<name>
<surname>Hammond</surname>
<given-names>Ross A.</given-names>
</name>
</person-group>
<article-title>Coupled Contagion Dynamics of Fear and Disease: Mathematical and Computational Explorations</article-title>
<source>PLoS ONE</source>
<year>2008</year>
<volume>3</volume>
<issue>12</issue>
<fpage>e3955</fpage>
<pub-id pub-id-type="pmid">19079607</pub-id>
</element-citation>
</ref>
<ref id="CR57">
<label>57.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Erdem</surname>
<given-names>Mustafa</given-names>
</name>
<name>
<surname>Safan</surname>
<given-names>Muntaser</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
</person-group>
<article-title>Mathematical Analysis of an SIQR Influenza Model with Imperfect Quarantine</article-title>
<source>Bulletin of Mathematical Biology</source>
<year>2017</year>
<volume>79</volume>
<issue>7</issue>
<fpage>1612</fpage>
<lpage>1636</lpage>
<pub-id pub-id-type="pmid">28608046</pub-id>
</element-citation>
</ref>
<ref id="CR58">
<label>58.</label>
<mixed-citation publication-type="other">Espinoza, B., V. Moreno, D. Bichara, and C. Castillo-Chavez (2016) Assessing the efficiency of movement restriction as a control strategy of Ebola, In Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases, pages 123–145. Springer.</mixed-citation>
</ref>
<ref id="CR59">
<label>59.</label>
<mixed-citation publication-type="other">Feng, Z. (2014) Applications of Epidemiological Models to Public Health Policymaking: the Role of Heterogeneity in Model Predictions. World Scientific.</mixed-citation>
</ref>
<ref id="CR60">
<label>60.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Feng</surname>
<given-names>Zhilan</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
<name>
<surname>Capurro</surname>
<given-names>Angel F.</given-names>
</name>
</person-group>
<article-title>A Model for Tuberculosis with Exogenous Reinfection</article-title>
<source>Theoretical Population Biology</source>
<year>2000</year>
<volume>57</volume>
<issue>3</issue>
<fpage>235</fpage>
<lpage>247</lpage>
<pub-id pub-id-type="pmid">10828216</pub-id>
</element-citation>
</ref>
<ref id="CR61">
<label>61.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Fenichel</surname>
<given-names>E. P.</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Ceddia</surname>
<given-names>M. G.</given-names>
</name>
<name>
<surname>Chowell</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Parra</surname>
<given-names>P. A. G.</given-names>
</name>
<name>
<surname>Hickling</surname>
<given-names>G. J.</given-names>
</name>
<name>
<surname>Holloway</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Horan</surname>
<given-names>R.</given-names>
</name>
<name>
<surname>Morin</surname>
<given-names>B.</given-names>
</name>
<name>
<surname>Perrings</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Springborn</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Velazquez</surname>
<given-names>L.</given-names>
</name>
<name>
<surname>Villalobos</surname>
<given-names>C.</given-names>
</name>
</person-group>
<article-title>Adaptive human behavior in epidemiological models</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>2011</year>
<volume>108</volume>
<issue>15</issue>
<fpage>6306</fpage>
<lpage>6311</lpage>
</element-citation>
</ref>
<ref id="CR62">
<label>62.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Fenichel</surname>
<given-names>Eli P.</given-names>
</name>
<name>
<surname>Kuminoff</surname>
<given-names>Nicolai V.</given-names>
</name>
<name>
<surname>Chowell</surname>
<given-names>Gerardo</given-names>
</name>
</person-group>
<article-title>Skip the Trip: Air Travelers' Behavioral Responses to Pandemic Influenza</article-title>
<source>PLoS ONE</source>
<year>2013</year>
<volume>8</volume>
<issue>3</issue>
<fpage>e58249</fpage>
<pub-id pub-id-type="pmid">23526970</pub-id>
</element-citation>
</ref>
<ref id="CR63">
<label>63.</label>
<mixed-citation publication-type="other">Fenichel, E.P. and X. Wang (2013) The mechanism and phenomena of adaptive human behavior during an epidemic and the role of information, In Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, pages 153–168. Springer, 2013.</mixed-citation>
</ref>
<ref id="CR64">
<label>64.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Ferguson</surname>
<given-names>Neil</given-names>
</name>
</person-group>
<article-title>Capturing human behaviour</article-title>
<source>Nature</source>
<year>2007</year>
<volume>446</volume>
<issue>7137</issue>
<fpage>733</fpage>
<lpage>733</lpage>
<pub-id pub-id-type="pmid">17429381</pub-id>
</element-citation>
</ref>
<ref id="CR65">
<label>65.</label>
<mixed-citation publication-type="other">Fraser, C., S. Riley, R.M. Anderson, & N.M. Ferguson (2004) Factors that make an infectious disease outbreak controllable, Proc. Nat. Acad Sci.
<bold>101</bold>
: 6146–6151.</mixed-citation>
</ref>
<ref id="CR66">
<label>66.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Funk</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Gilad</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Jansen</surname>
<given-names>V.A.A.</given-names>
</name>
</person-group>
<article-title>Endemic disease, awareness, and local behavioural response</article-title>
<source>Journal of Theoretical Biology</source>
<year>2010</year>
<volume>264</volume>
<issue>2</issue>
<fpage>501</fpage>
<lpage>509</lpage>
<pub-id pub-id-type="pmid">20184901</pub-id>
</element-citation>
</ref>
<ref id="CR67">
<label>67.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Funk</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Gilad</surname>
<given-names>E.</given-names>
</name>
<name>
<surname>Watkins</surname>
<given-names>C.</given-names>
</name>
<name>
<surname>Jansen</surname>
<given-names>V. A. A.</given-names>
</name>
</person-group>
<article-title>The spread of awareness and its impact on epidemic outbreaks</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>2009</year>
<volume>106</volume>
<issue>16</issue>
<fpage>6872</fpage>
<lpage>6877</lpage>
</element-citation>
</ref>
<ref id="CR68">
<label>68.</label>
<mixed-citation publication-type="other">Funk, S., M. Salathé, and V. A. Jansen (2010) Modelling the influence of human behaviour on the spread of infectious diseases: a review, J. Roy. Soc. Interface page rsif20100142.</mixed-citation>
</ref>
<ref id="CR69">
<label>69.</label>
<mixed-citation publication-type="other">Geoffard, P.Y. and T. Philipson (1996) Rational epidemics and their public control, International economic review: 603–624.</mixed-citation>
</ref>
<ref id="CR70">
<label>70.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Gersovitz</surname>
<given-names>Mark</given-names>
</name>
</person-group>
<article-title>The Economics of Infection Control</article-title>
<source>Annual Review of Resource Economics</source>
<year>2011</year>
<volume>3</volume>
<issue>1</issue>
<fpage>277</fpage>
<lpage>296</lpage>
</element-citation>
</ref>
<ref id="CR71">
<label>71.</label>
<mixed-citation publication-type="other">Gladwell, M. (1996) The Tipping Point. New Yorker, June 3, 1996.</mixed-citation>
</ref>
<ref id="CR72">
<label>72.</label>
<mixed-citation publication-type="other">Gladwell, M. (2002) The Tipping Point: How Little Things Can Make a Big Difference. Back Bay Books/LittleLittle, Brown and Company, Time Warner Book Group.</mixed-citation>
</ref>
<ref id="CR73">
<label>73.</label>
<mixed-citation publication-type="other">Grünbaum, D. (1994) Translating stochastic density-dependent individual behavior with sensory constraints to an Eulerian model of animal swarming, J. Math. Biol.
<bold>33</bold>
: 139–161.</mixed-citation>
</ref>
<ref id="CR74">
<label>74.</label>
<mixed-citation publication-type="other">Gumel, A.B., C. Castillo-Chávez, R. E. Mickens, and D. P. Clemence (2006) Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges: AMS-IMS-SIAM Joint Summer Research Conference on Modeling the Dynamics of Human Diseases: Emerging Paradigms and Challenges, July 17–21, 2005, Snowbird, Utah, volume 410. American Mathematical Soc., 2006.</mixed-citation>
</ref>
<ref id="CR75">
<label>75.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hadeler</surname>
<given-names>K.P.</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>C.</given-names>
</name>
</person-group>
<article-title>A core group model for disease transmission</article-title>
<source>Mathematical Biosciences</source>
<year>1995</year>
<volume>128</volume>
<issue>1-2</issue>
<fpage>41</fpage>
<lpage>55</lpage>
<pub-id pub-id-type="pmid">7606144</pub-id>
</element-citation>
</ref>
<ref id="CR76">
<label>76.</label>
<mixed-citation publication-type="other">Hadeler, K.P. and J. Müller (2017) Cellular Automata: Analysis and Applications. Springer, 2017.</mixed-citation>
</ref>
<ref id="CR77">
<label>77.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hansen</surname>
<given-names>Elsa</given-names>
</name>
<name>
<surname>Day</surname>
<given-names>Troy</given-names>
</name>
</person-group>
<article-title>Optimal antiviral treatment strategies and the effects of resistance</article-title>
<source>Proceedings of the Royal Society B: Biological Sciences</source>
<year>2010</year>
<volume>278</volume>
<issue>1708</issue>
<fpage>1082</fpage>
<lpage>1089</lpage>
</element-citation>
</ref>
<ref id="CR78">
<label>78.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Helbing</surname>
<given-names>Dirk</given-names>
</name>
<name>
<surname>Farkas</surname>
<given-names>Illés</given-names>
</name>
<name>
<surname>Vicsek</surname>
<given-names>Tamás</given-names>
</name>
</person-group>
<article-title>Simulating dynamical features of escape panic</article-title>
<source>Nature</source>
<year>2000</year>
<volume>407</volume>
<issue>6803</issue>
<fpage>487</fpage>
<lpage>490</lpage>
<pub-id pub-id-type="pmid">11028994</pub-id>
</element-citation>
</ref>
<ref id="CR79">
<label>79.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Helbing</surname>
<given-names>Dirk</given-names>
</name>
<name>
<surname>Keltsch</surname>
<given-names>Joachim</given-names>
</name>
<name>
<surname>Molnár</surname>
<given-names>Péter</given-names>
</name>
</person-group>
<article-title>Modelling the evolution of human trail systems</article-title>
<source>Nature</source>
<year>1997</year>
<volume>388</volume>
<issue>6637</issue>
<fpage>47</fpage>
<lpage>50</lpage>
<pub-id pub-id-type="pmid">9214501</pub-id>
</element-citation>
</ref>
<ref id="CR80">
<label>80.</label>
<mixed-citation publication-type="other">Herrera-Valdez, M.A., M. Cruz-Aponte, and C. Castillo-Chavez (2011) Multiple outbreaks for the same pandemic: Local transportation and social distancing explain the different “waves” of A-H1N1pdm cases observed in méxico during 2009, Math. Biosc. Eng.
<bold>8</bold>
: 21–48.</mixed-citation>
</ref>
<ref id="CR81">
<label>81.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hethcote</surname>
<given-names>Herbert W.</given-names>
</name>
</person-group>
<article-title>The Mathematics of Infectious Diseases</article-title>
<source>SIAM Review</source>
<year>2000</year>
<volume>42</volume>
<issue>4</issue>
<fpage>599</fpage>
<lpage>653</lpage>
</element-citation>
</ref>
<ref id="CR82">
<label>82.</label>
<mixed-citation publication-type="other">Hethcote, H.W. and J. W. Van Ark (2013) Modeling HIV Transmission and AIDS in the United States, volume 95. Springer Science & Business Media.</mixed-citation>
</ref>
<ref id="CR83">
<label>83.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hethcote</surname>
<given-names>H.W.</given-names>
</name>
<name>
<surname>Yi</surname>
<given-names>Li</given-names>
</name>
<name>
<surname>Zhujun</surname>
<given-names>Jing</given-names>
</name>
</person-group>
<article-title>Hopf bifurcation in models for pertussis epidemiology</article-title>
<source>Mathematical and Computer Modelling</source>
<year>1999</year>
<volume>30</volume>
<issue>11-12</issue>
<fpage>29</fpage>
<lpage>45</lpage>
</element-citation>
</ref>
<ref id="CR84">
<label>84.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Honório</surname>
<given-names>Nildimar Alves</given-names>
</name>
<name>
<surname>Silva</surname>
<given-names>Wellington da Costa</given-names>
</name>
<name>
<surname>Leite</surname>
<given-names>Paulo José</given-names>
</name>
<name>
<surname>Gonçalves</surname>
<given-names>Jaylei Monteiro</given-names>
</name>
<name>
<surname>Lounibos</surname>
<given-names>Leon Philip</given-names>
</name>
<name>
<surname>Lourenço-de-Oliveira</surname>
<given-names>Ricardo</given-names>
</name>
</person-group>
<article-title>Dispersal of Aedes aegypti and Aedes albopictus (Diptera: Culicidae) in an urban endemic dengue area in the State of Rio de Janeiro, Brazil</article-title>
<source>Memórias do Instituto Oswaldo Cruz</source>
<year>2003</year>
<volume>98</volume>
<issue>2</issue>
<fpage>191</fpage>
<lpage>198</lpage>
</element-citation>
</ref>
<ref id="CR85">
<label>85.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hsieh</surname>
<given-names>Ying-Hen</given-names>
</name>
<name>
<surname>Liu</surname>
<given-names>Junli</given-names>
</name>
<name>
<surname>Tzeng</surname>
<given-names>Yun-Huei</given-names>
</name>
<name>
<surname>Wu</surname>
<given-names>Jianhong</given-names>
</name>
</person-group>
<article-title>Impact of visitors and hospital staff on nosocomial transmission and spread to community</article-title>
<source>Journal of Theoretical Biology</source>
<year>2014</year>
<volume>356</volume>
<fpage>20</fpage>
<lpage>29</lpage>
<pub-id pub-id-type="pmid">24727185</pub-id>
</element-citation>
</ref>
<ref id="CR86">
<label>86.</label>
<mixed-citation publication-type="other">Hutson, V. (1984) A theorem on average Liapunov functions, Monatshefte für Mathematik,
<bold>98</bold>
: 267–275.</mixed-citation>
</ref>
<ref id="CR87">
<label>87.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Iggidr</surname>
<given-names>Aberrahman</given-names>
</name>
<name>
<surname>Sallet</surname>
<given-names>Gauthier</given-names>
</name>
<name>
<surname>Souza</surname>
<given-names>Max O.</given-names>
</name>
</person-group>
<article-title>On the dynamics of a class of multi-group models for vector-borne diseases</article-title>
<source>Journal of Mathematical Analysis and Applications</source>
<year>2016</year>
<volume>441</volume>
<issue>2</issue>
<fpage>723</fpage>
<lpage>743</lpage>
</element-citation>
</ref>
<ref id="CR88">
<label>88.</label>
<mixed-citation publication-type="other">Kareiva, P. (1990) Population dynamics in spatially complex environments: theory and data, Phil. Trans. Roy. Soc. Lond. B
<bold>330</bold>
: 175–190.</mixed-citation>
</ref>
<ref id="CR89">
<label>89.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Karesh</surname>
<given-names>William B.</given-names>
</name>
<name>
<surname>Cook</surname>
<given-names>Robert A.</given-names>
</name>
<name>
<surname>Bennett</surname>
<given-names>Elizabeth L.</given-names>
</name>
<name>
<surname>Newcomb</surname>
<given-names>James</given-names>
</name>
</person-group>
<article-title>Wildlife Trade and Global Disease Emergence</article-title>
<source>Emerging Infectious Diseases</source>
<year>2005</year>
<volume>11</volume>
<issue>7</issue>
<fpage>1000</fpage>
<lpage>1002</lpage>
<pub-id pub-id-type="pmid">16022772</pub-id>
</element-citation>
</ref>
<ref id="CR90">
<label>90.</label>
<mixed-citation publication-type="other">Kermack, W.O. & A.G. McKendrick (1927) A contribution to the mathematical theory of epidemics, Proc. Royal Soc. London,
<bold>1</bold>
15:700–721.</mixed-citation>
</ref>
<ref id="CR91">
<label>91.</label>
<mixed-citation publication-type="other">Kermack, W.O. & A.G. McKendrick (1932) Contributions to the mathematical theory of epidemics, part. II, Proc. Roy. Soc. London,
<bold>1</bold>
38:55–83.</mixed-citation>
</ref>
<ref id="CR92">
<label>92.</label>
<mixed-citation publication-type="other">Kermack, W.O. & A.G. McKendrick (1933) Contributions to the mathematical theory of epidemics, part. III, Proc. Roy. Soc. London,
<bold>1</bold>
41:94–112.</mixed-citation>
</ref>
<ref id="CR93">
<label>93.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Khan</surname>
<given-names>Kamran</given-names>
</name>
<name>
<surname>Arino</surname>
<given-names>Julien</given-names>
</name>
<name>
<surname>Hu</surname>
<given-names>Wei</given-names>
</name>
<name>
<surname>Raposo</surname>
<given-names>Paulo</given-names>
</name>
<name>
<surname>Sears</surname>
<given-names>Jennifer</given-names>
</name>
<name>
<surname>Calderon</surname>
<given-names>Felipe</given-names>
</name>
<name>
<surname>Heidebrecht</surname>
<given-names>Christine</given-names>
</name>
<name>
<surname>Macdonald</surname>
<given-names>Michael</given-names>
</name>
<name>
<surname>Liauw</surname>
<given-names>Jessica</given-names>
</name>
<name>
<surname>Chan</surname>
<given-names>Angie</given-names>
</name>
<name>
<surname>Gardam</surname>
<given-names>Michael</given-names>
</name>
</person-group>
<article-title>Spread of a Novel Influenza A (H1N1) Virus via Global Airline Transportation</article-title>
<source>New England Journal of Medicine</source>
<year>2009</year>
<volume>361</volume>
<issue>2</issue>
<fpage>212</fpage>
<lpage>214</lpage>
<pub-id pub-id-type="pmid">19564630</pub-id>
</element-citation>
</ref>
<ref id="CR94">
<label>94.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Kilpatrick</surname>
<given-names>A. M.</given-names>
</name>
<name>
<surname>Chmura</surname>
<given-names>A. A.</given-names>
</name>
<name>
<surname>Gibbons</surname>
<given-names>D. W.</given-names>
</name>
<name>
<surname>Fleischer</surname>
<given-names>R. C.</given-names>
</name>
<name>
<surname>Marra</surname>
<given-names>P. P.</given-names>
</name>
<name>
<surname>Daszak</surname>
<given-names>P.</given-names>
</name>
</person-group>
<article-title>Predicting the global spread of H5N1 avian influenza</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>2006</year>
<volume>103</volume>
<issue>51</issue>
<fpage>19368</fpage>
<lpage>19373</lpage>
</element-citation>
</ref>
<ref id="CR95">
<label>95.</label>
<mixed-citation publication-type="other">Kimball A.N. (2016) Risky Trade: Infectious Disease in the Era of Global Trade, Routledge.</mixed-citation>
</ref>
<ref id="CR96">
<label>96.</label>
<mixed-citation publication-type="other">Kirschner, D. (1999) Dynamics of co-infection with tuberculosis and HIV-1, Theor. Pop. Biol.
<bold>55</bold>
: 94–109.</mixed-citation>
</ref>
<ref id="CR97">
<label>97.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>KLEIN</surname>
<given-names>EILI</given-names>
</name>
<name>
<surname>LAXMINARAYAN</surname>
<given-names>RAMANAN</given-names>
</name>
<name>
<surname>SMITH</surname>
<given-names>DAVID L.</given-names>
</name>
<name>
<surname>GILLIGAN</surname>
<given-names>CHRISTOPHER A.</given-names>
</name>
</person-group>
<article-title>Economic incentives and mathematical models of disease</article-title>
<source>Environment and Development Economics</source>
<year>2007</year>
<volume>12</volume>
<issue>5</issue>
<fpage>707</fpage>
<lpage>732</lpage>
</element-citation>
</ref>
<ref id="CR98">
<label>98.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Koram</surname>
<given-names>K.A.</given-names>
</name>
<name>
<surname>Bennett</surname>
<given-names>S.</given-names>
</name>
<name>
<surname>Adiamah</surname>
<given-names>J.H.</given-names>
</name>
<name>
<surname>Greenwood</surname>
<given-names>B.M.</given-names>
</name>
</person-group>
<article-title>Socio-economic risk factors for malaria in a peri-urban area of The Gambia</article-title>
<source>Transactions of the Royal Society of Tropical Medicine and Hygiene</source>
<year>1995</year>
<volume>89</volume>
<issue>2</issue>
<fpage>146</fpage>
<lpage>150</lpage>
<pub-id pub-id-type="pmid">7778137</pub-id>
</element-citation>
</ref>
<ref id="CR99">
<label>99.</label>
<mixed-citation publication-type="other">Laxminarayan, R. (2001) Bacterial resistance and optimal use of antibiotics, Discussion Papers dp-01-23 Resources for the Future.</mixed-citation>
</ref>
<ref id="CR100">
<label>100.</label>
<mixed-citation publication-type="other">Legrand, J., R. F. Grais, P.-Y. Boelle, A.-J. Valleron, and A. Flahault (2007) Understanding the dynamics of Ebola epidemics, Epidemiology & Infection
<bold>135</bold>
: 610–621.</mixed-citation>
</ref>
<ref id="CR101">
<label>101.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Levin</surname>
<given-names>Simon A.</given-names>
</name>
</person-group>
<article-title>The Problem of Pattern and Scale in Ecology: The Robert H. MacArthur Award Lecture</article-title>
<source>Ecology</source>
<year>1992</year>
<volume>73</volume>
<issue>6</issue>
<fpage>1943</fpage>
<lpage>1967</lpage>
</element-citation>
</ref>
<ref id="CR102">
<label>102.</label>
<mixed-citation publication-type="other">Levin, S.A. (2001) Fragile Dominion: Complexity and the Commons, volume 18. Springer.</mixed-citation>
</ref>
<ref id="CR103">
<label>103.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Levin</surname>
<given-names>S. A.</given-names>
</name>
<name>
<surname>Paine</surname>
<given-names>R. T.</given-names>
</name>
</person-group>
<article-title>Disturbance, Patch Formation, and Community Structure</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>1974</year>
<volume>71</volume>
<issue>7</issue>
<fpage>2744</fpage>
<lpage>2747</lpage>
</element-citation>
</ref>
<ref id="CR104">
<label>104.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Levins</surname>
<given-names>R.</given-names>
</name>
</person-group>
<article-title>Some Demographic and Genetic Consequences of Environmental Heterogeneity for Biological Control</article-title>
<source>Bulletin of the Entomological Society of America</source>
<year>1969</year>
<volume>15</volume>
<issue>3</issue>
<fpage>237</fpage>
<lpage>240</lpage>
</element-citation>
</ref>
<ref id="CR105">
<label>105.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Levinthal</surname>
<given-names>Daniel A.</given-names>
</name>
<name>
<surname>March</surname>
<given-names>James G.</given-names>
</name>
</person-group>
<article-title>The myopia of learning</article-title>
<source>Strategic Management Journal</source>
<year>1993</year>
<volume>14</volume>
<issue>S2</issue>
<fpage>95</fpage>
<lpage>112</lpage>
</element-citation>
</ref>
<ref id="CR106">
<label>106.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Lipsitch</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Bergstrom</surname>
<given-names>C. T.</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>B. R.</given-names>
</name>
</person-group>
<article-title>The epidemiology of antibiotic resistance in hospitals: Paradoxes and prescriptions</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>2000</year>
<volume>97</volume>
<issue>4</issue>
<fpage>1938</fpage>
<lpage>1943</lpage>
</element-citation>
</ref>
<ref id="CR107">
<label>107.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Lipsitch</surname>
<given-names>Marc</given-names>
</name>
<name>
<surname>Cohen</surname>
<given-names>Ted</given-names>
</name>
<name>
<surname>Murray</surname>
<given-names>Megan</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>Bruce R</given-names>
</name>
</person-group>
<article-title>Antiviral Resistance and the Control of Pandemic Influenza</article-title>
<source>PLoS Medicine</source>
<year>2007</year>
<volume>4</volume>
<issue>1</issue>
<fpage>e15</fpage>
<pub-id pub-id-type="pmid">17253900</pub-id>
</element-citation>
</ref>
<ref id="CR108">
<label>108.</label>
<mixed-citation publication-type="other">Macdonald, G. (1956) Epidemiological basis of malaria control, Bull.
<bold>15</bold>
: 613.</mixed-citation>
</ref>
<ref id="CR109">
<label>109.</label>
<mixed-citation publication-type="other">Macdonald, G. (1956) Theory of the eradication of malaria, Bull. WHO
<bold>15</bold>
: 369.</mixed-citation>
</ref>
<ref id="CR110">
<label>110.</label>
<mixed-citation publication-type="other">Meloni, S., N. Perra, A. Arenas, S. Gómez, Y. Moreno, and A. Vespignani (2011) Modeling human mobility responses to the large-scale spreading of infectious diseases, Scientific reports
<bold>1</bold>
: 62.</mixed-citation>
</ref>
<ref id="CR111">
<label>111.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Moghadas</surname>
<given-names>Seyed M</given-names>
</name>
</person-group>
<article-title>Management of drug resistance in the population: influenza as a case study</article-title>
<source>Proceedings of the Royal Society B: Biological Sciences</source>
<year>2008</year>
<volume>275</volume>
<issue>1639</issue>
<fpage>1163</fpage>
<lpage>1169</lpage>
</element-citation>
</ref>
<ref id="CR112">
<label>112.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>MOGHADAS</surname>
<given-names>SEYED M.</given-names>
</name>
</person-group>
<article-title>Emergence of Resistance in Influenza With Compensatory Mutations</article-title>
<source>Mathematical Population Studies</source>
<year>2011</year>
<volume>18</volume>
<issue>2</issue>
<fpage>106</fpage>
<lpage>121</lpage>
</element-citation>
</ref>
<ref id="CR113">
<label>113.</label>
<mixed-citation publication-type="other">Moghadas, S.M., C. S. Bowman, G. Röst, D. N. Fisman, and J. Wu (2008) Post-exposure prophylaxis during pandemic outbreaks, BMC Medicine
<bold>7</bold>
: 73.</mixed-citation>
</ref>
<ref id="CR114">
<label>114.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Moghadas</surname>
<given-names>Seyed M.</given-names>
</name>
<name>
<surname>Bowman</surname>
<given-names>Christopher S.</given-names>
</name>
<name>
<surname>Röst</surname>
<given-names>Gergely</given-names>
</name>
<name>
<surname>Wu</surname>
<given-names>Jianhong</given-names>
</name>
</person-group>
<article-title>Population-Wide Emergence of Antiviral Resistance during Pandemic Influenza</article-title>
<source>PLoS ONE</source>
<year>2008</year>
<volume>3</volume>
<issue>3</issue>
<fpage>e1839</fpage>
<pub-id pub-id-type="pmid">18350174</pub-id>
</element-citation>
</ref>
<ref id="CR115">
<label>115.</label>
<mixed-citation publication-type="other">Morin, B.R., A. Kinzig, S. Levin, and C. Perrings (2017) Economic incentives in the socially optimal management of infectious disease: When r_{0} is not enough, EcoHealth, pages 1–16, 2017.</mixed-citation>
</ref>
<ref id="CR116">
<label>116.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>MORIN</surname>
<given-names>BENJAMIN R.</given-names>
</name>
<name>
<surname>FENICHEL</surname>
<given-names>ELI P.</given-names>
</name>
<name>
<surname>CASTILLO-CHAVEZ</surname>
<given-names>CARLOS</given-names>
</name>
</person-group>
<article-title>SIR DYNAMICS WITH ECONOMICALLY DRIVEN CONTACT RATES</article-title>
<source>Natural Resource Modeling</source>
<year>2013</year>
<volume>26</volume>
<issue>4</issue>
<fpage>505</fpage>
<lpage>525</lpage>
<pub-id pub-id-type="pmid">25152563</pub-id>
</element-citation>
</ref>
<ref id="CR117">
<label>117.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Morin</surname>
<given-names>Benjamin R.</given-names>
</name>
<name>
<surname>Perrings</surname>
<given-names>Charles</given-names>
</name>
<name>
<surname>Kinzig</surname>
<given-names>Ann</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>Simon</given-names>
</name>
</person-group>
<article-title>The social benefits of private infectious disease-risk mitigation</article-title>
<source>Theoretical Ecology</source>
<year>2015</year>
<volume>8</volume>
<issue>4</issue>
<fpage>467</fpage>
<lpage>479</lpage>
<pub-id pub-id-type="pmid">26858777</pub-id>
</element-citation>
</ref>
<ref id="CR118">
<label>118.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Morin</surname>
<given-names>Benjamin R.</given-names>
</name>
<name>
<surname>Perrings</surname>
<given-names>Charles</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>Simon</given-names>
</name>
<name>
<surname>Kinzig</surname>
<given-names>Ann</given-names>
</name>
</person-group>
<article-title>Disease risk mitigation: The equivalence of two selective mixing strategies on aggregate contact patterns and resulting epidemic spread</article-title>
<source>Journal of Theoretical Biology</source>
<year>2014</year>
<volume>363</volume>
<fpage>262</fpage>
<lpage>270</lpage>
<pub-id pub-id-type="pmid">25150459</pub-id>
</element-citation>
</ref>
<ref id="CR119">
<label>119.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Naresh</surname>
<given-names>R.</given-names>
</name>
<name>
<surname>Tripathi</surname>
<given-names>A.</given-names>
</name>
</person-group>
<article-title>MODELLING AND ANALYSIS OF HIV‐TB CO‐INFECTION IN A VARIABLE SIZE POPULATION</article-title>
<source>Mathematical Modelling and Analysis</source>
<year>2005</year>
<volume>10</volume>
<issue>3</issue>
<fpage>275</fpage>
<lpage>286</lpage>
</element-citation>
</ref>
<ref id="CR120">
<label>120.</label>
<mixed-citation publication-type="other">Newman, M.E. (2002) Spread of epidemic disease on networks, Phys. Rev.
<bold>66</bold>
: 016128.</mixed-citation>
</ref>
<ref id="CR121">
<label>121.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Newman</surname>
<given-names>M. E. J.</given-names>
</name>
</person-group>
<article-title>The Structure and Function of Complex Networks</article-title>
<source>SIAM Review</source>
<year>2003</year>
<volume>45</volume>
<issue>2</issue>
<fpage>167</fpage>
<lpage>256</lpage>
</element-citation>
</ref>
<ref id="CR122">
<label>122.</label>
<mixed-citation publication-type="other">Nowak, M. and R. M. May (2000) Virus Dynamics: Mathematical Principles of Immunology and Virology: Mathematical Principles of Immunology and Virology, Oxford University Press, UK.</mixed-citation>
</ref>
<ref id="CR123">
<label>123.</label>
<mixed-citation publication-type="other">Nuno, M., C. Castillo-Chavez, Z. Feng, and M. Martcheva (2008) Mathematical models of influenza: the role of cross-immunity, quarantine and age-structure, In Mathematical Epidemiology, pages 349–364. Springer, 2008.</mixed-citation>
</ref>
<ref id="CR124">
<label>124.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Nuno</surname>
<given-names>M.</given-names>
</name>
<name>
<surname>Reichert</surname>
<given-names>T. A.</given-names>
</name>
<name>
<surname>Chowell</surname>
<given-names>G.</given-names>
</name>
<name>
<surname>Gumel</surname>
<given-names>A. B.</given-names>
</name>
</person-group>
<article-title>Protecting residential care facilities from pandemic influenza</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>2008</year>
<volume>105</volume>
<issue>30</issue>
<fpage>10625</fpage>
<lpage>10630</lpage>
</element-citation>
</ref>
<ref id="CR125">
<label>125.</label>
<mixed-citation publication-type="other">Okubo, A. (1980) Diffusion and ecological problems:(mathematical models), Biomathematics.</mixed-citation>
</ref>
<ref id="CR126">
<label>126.</label>
<mixed-citation publication-type="other">Okubo, A. and S. A. Levin (2013) Diffusion and ecological problems: modern perspectives, volume 14. Springer Science & Business Media.</mixed-citation>
</ref>
<ref id="CR127">
<label>127.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Paine</surname>
<given-names>R. T.</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>Simon A.</given-names>
</name>
</person-group>
<article-title>Intertidal Landscapes: Disturbance and the Dynamics of Pattern</article-title>
<source>Ecological Monographs</source>
<year>1981</year>
<volume>51</volume>
<issue>2</issue>
<fpage>145</fpage>
<lpage>178</lpage>
</element-citation>
</ref>
<ref id="CR128">
<label>128.</label>
<mixed-citation publication-type="other">Pastor-Satorras, R. and A. Vespignani (2001) Epidemic dynamics and endemic states in complex networks, Phys. Rev. E
<bold>63</bold>
: 066117.</mixed-citation>
</ref>
<ref id="CR129">
<label>129.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Pavlin</surname>
<given-names>Boris I.</given-names>
</name>
<name>
<surname>Schloegel</surname>
<given-names>Lisa M.</given-names>
</name>
<name>
<surname>Daszak</surname>
<given-names>Peter</given-names>
</name>
</person-group>
<article-title>Risk of Importing Zoonotic Diseases through Wildlife Trade, United States</article-title>
<source>Emerging Infectious Diseases</source>
<year>2009</year>
<volume>15</volume>
<issue>11</issue>
<fpage>1721</fpage>
<lpage>1726</lpage>
<pub-id pub-id-type="pmid">19891857</pub-id>
</element-citation>
</ref>
<ref id="CR130">
<label>130.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Pennings</surname>
<given-names>Joost M.E.</given-names>
</name>
<name>
<surname>Wansink</surname>
<given-names>Brian</given-names>
</name>
<name>
<surname>Meulenberg</surname>
<given-names>Matthew T.G.</given-names>
</name>
</person-group>
<article-title>A note on modeling consumer reactions to a crisis: The case of the mad cow disease</article-title>
<source>International Journal of Research in Marketing</source>
<year>2002</year>
<volume>19</volume>
<issue>1</issue>
<fpage>91</fpage>
<lpage>100</lpage>
</element-citation>
</ref>
<ref id="CR131">
<label>131.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Perra</surname>
<given-names>Nicola</given-names>
</name>
<name>
<surname>Balcan</surname>
<given-names>Duygu</given-names>
</name>
<name>
<surname>Gonçalves</surname>
<given-names>Bruno</given-names>
</name>
<name>
<surname>Vespignani</surname>
<given-names>Alessandro</given-names>
</name>
</person-group>
<article-title>Towards a Characterization of Behavior-Disease Models</article-title>
<source>PLoS ONE</source>
<year>2011</year>
<volume>6</volume>
<issue>8</issue>
<fpage>e23084</fpage>
<pub-id pub-id-type="pmid">21826228</pub-id>
</element-citation>
</ref>
<ref id="CR132">
<label>132.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Perrings</surname>
<given-names>Charles</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
<name>
<surname>Chowell</surname>
<given-names>Gerardo</given-names>
</name>
<name>
<surname>Daszak</surname>
<given-names>Peter</given-names>
</name>
<name>
<surname>Fenichel</surname>
<given-names>Eli P.</given-names>
</name>
<name>
<surname>Finnoff</surname>
<given-names>David</given-names>
</name>
<name>
<surname>Horan</surname>
<given-names>Richard D.</given-names>
</name>
<name>
<surname>Kilpatrick</surname>
<given-names>A. Marm</given-names>
</name>
<name>
<surname>Kinzig</surname>
<given-names>Ann P.</given-names>
</name>
<name>
<surname>Kuminoff</surname>
<given-names>Nicolai V.</given-names>
</name>
<name>
<surname>Levin</surname>
<given-names>Simon</given-names>
</name>
<name>
<surname>Morin</surname>
<given-names>Benjamin</given-names>
</name>
<name>
<surname>Smith</surname>
<given-names>Katherine F.</given-names>
</name>
<name>
<surname>Springborn</surname>
<given-names>Michael</given-names>
</name>
</person-group>
<article-title>Merging Economics and Epidemiology to Improve the Prediction and Management of Infectious Disease</article-title>
<source>EcoHealth</source>
<year>2014</year>
<volume>11</volume>
<issue>4</issue>
<fpage>464</fpage>
<lpage>475</lpage>
<pub-id pub-id-type="pmid">25233829</pub-id>
</element-citation>
</ref>
<ref id="CR133">
<label>133.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Porco</surname>
<given-names>Travis C.</given-names>
</name>
<name>
<surname>Small</surname>
<given-names>Peter M.</given-names>
</name>
<name>
<surname>Blower</surname>
<given-names>Sally M.</given-names>
</name>
</person-group>
<article-title>Amplification Dynamics: Predicting the Effect of HIV on Tuberculosis Outbreaks</article-title>
<source>JAIDS Journal of Acquired Immune Deficiency Syndromes</source>
<year>2001</year>
<volume>28</volume>
<issue>5</issue>
<fpage>437</fpage>
<lpage>444</lpage>
<pub-id pub-id-type="pmid">11744831</pub-id>
</element-citation>
</ref>
<ref id="CR134">
<label>134.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Preisser</surname>
<given-names>Evan L.</given-names>
</name>
<name>
<surname>Bolnick</surname>
<given-names>Daniel I.</given-names>
</name>
</person-group>
<article-title>The Many Faces of Fear: Comparing the Pathways and Impacts of Nonconsumptive Predator Effects on Prey Populations</article-title>
<source>PLoS ONE</source>
<year>2008</year>
<volume>3</volume>
<issue>6</issue>
<fpage>e2465</fpage>
<pub-id pub-id-type="pmid">18560575</pub-id>
</element-citation>
</ref>
<ref id="CR135">
<label>135.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Raimundo</surname>
<given-names>Silvia Martorano</given-names>
</name>
<name>
<surname>Engel</surname>
<given-names>Alejandro B.</given-names>
</name>
<name>
<surname>Yang</surname>
<given-names>Hyun Mo</given-names>
</name>
<name>
<surname>Bassanezi</surname>
<given-names>Rodney Carlos</given-names>
</name>
</person-group>
<article-title>An Approach to Estimating the Transmission Coefficients for AIDS and for Tuberculosis Using Mathematical Models</article-title>
<source>Systems Analysis Modelling Simulation</source>
<year>2003</year>
<volume>43</volume>
<issue>4</issue>
<fpage>423</fpage>
<lpage>442</lpage>
</element-citation>
</ref>
<ref id="CR136">
<label>136.</label>
<mixed-citation publication-type="other">Rass, L. and J. Radcliffe (2003) Spatial Deterministic Epidemics, volume 102. Am. Math. Soc.</mixed-citation>
</ref>
<ref id="CR137">
<label>137.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Reluga</surname>
<given-names>Timothy C.</given-names>
</name>
</person-group>
<article-title>Game Theory of Social Distancing in Response to an Epidemic</article-title>
<source>PLoS Computational Biology</source>
<year>2010</year>
<volume>6</volume>
<issue>5</issue>
<fpage>e1000793</fpage>
<pub-id pub-id-type="pmid">20523740</pub-id>
</element-citation>
</ref>
<ref id="CR138">
<label>138.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Rimmelzwaan</surname>
<given-names>G. F.</given-names>
</name>
</person-group>
<article-title>Full restoration of viral fitness by multiple compensatory co-mutations in the nucleoprotein of influenza A virus cytotoxic T-lymphocyte escape mutants</article-title>
<source>Journal of General Virology</source>
<year>2005</year>
<volume>86</volume>
<issue>6</issue>
<fpage>1801</fpage>
<lpage>1805</lpage>
<pub-id pub-id-type="pmid">15914859</pub-id>
</element-citation>
</ref>
<ref id="CR139">
<label>139.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>RodrÍguez</surname>
<given-names>D</given-names>
</name>
</person-group>
<article-title>Models of Infectious Diseases in Spatially Heterogeneous Environments</article-title>
<source>Bulletin of Mathematical Biology</source>
<year>2001</year>
<volume>63</volume>
<issue>3</issue>
<fpage>547</fpage>
<lpage>571</lpage>
<pub-id pub-id-type="pmid">11374305</pub-id>
</element-citation>
</ref>
<ref id="CR140">
<label>140.</label>
<mixed-citation publication-type="other">Roeger, L.I.W., Z. Feng, C. Castillo-Chavez, et al. (2009) Modeling TB and HIV co-infections, Math. Biosc. Eng.
<bold>6</bold>
: 815–837.</mixed-citation>
</ref>
<ref id="CR141">
<label>141.</label>
<mixed-citation publication-type="other">Ross, R. (1911) The Prevention of Malaria, John Murray, London.</mixed-citation>
</ref>
<ref id="CR142">
<label>142.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Ruktanonchai</surname>
<given-names>Nick W.</given-names>
</name>
<name>
<surname>Smith</surname>
<given-names>David L.</given-names>
</name>
<name>
<surname>De Leenheer</surname>
<given-names>Patrick</given-names>
</name>
</person-group>
<article-title>Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control</article-title>
<source>Mathematical Biosciences</source>
<year>2016</year>
<volume>279</volume>
<fpage>90</fpage>
<lpage>101</lpage>
<pub-id pub-id-type="pmid">27436636</pub-id>
</element-citation>
</ref>
<ref id="CR143">
<label>143.</label>
<mixed-citation publication-type="other">Rvachev, L.A. and I. M. Longini Jr. (1985) A mathematical model for the global spread of influenza, Math. Biosc.
<bold>75</bold>
: 3–22.</mixed-citation>
</ref>
<ref id="CR144">
<label>144.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>SCHULZER</surname>
<given-names>MICHAEL</given-names>
</name>
<name>
<surname>RADHAMANI</surname>
<given-names>M P</given-names>
</name>
<name>
<surname>GRZYBOWSKI</surname>
<given-names>STEFAN</given-names>
</name>
<name>
<surname>MAK</surname>
<given-names>EDWIN</given-names>
</name>
<name>
<surname>FITZGERALD</surname>
<given-names>J MARK</given-names>
</name>
</person-group>
<article-title>A Mathematical Model for the Prediction of the Impact of HIV Infection on Tuberculosis</article-title>
<source>International Journal of Epidemiology</source>
<year>1994</year>
<volume>23</volume>
<issue>2</issue>
<fpage>400</fpage>
<lpage>407</lpage>
<pub-id pub-id-type="pmid">8082969</pub-id>
</element-citation>
</ref>
<ref id="CR145">
<label>145.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Smith</surname>
<given-names>David L</given-names>
</name>
<name>
<surname>Dushoff</surname>
<given-names>Jonathan</given-names>
</name>
<name>
<surname>McKenzie</surname>
<given-names>F. Ellis</given-names>
</name>
</person-group>
<article-title>The Risk of a Mosquito-Borne Infectionin a Heterogeneous Environment</article-title>
<source>PLoS Biology</source>
<year>2004</year>
<volume>2</volume>
<issue>11</issue>
<fpage>e368</fpage>
<pub-id pub-id-type="pmid">15510228</pub-id>
</element-citation>
</ref>
<ref id="CR146">
<label>146.</label>
<mixed-citation publication-type="other">Smith, H. (1986) Cooperative systems of differential equations with concave nonlinearities, Nonlinear Analysis: Theory, Methods & Applications
<bold>10</bold>
: 1037–1052.</mixed-citation>
</ref>
<ref id="CR147">
<label>147.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Tatem</surname>
<given-names>Andrew J.</given-names>
</name>
</person-group>
<article-title>The worldwide airline network and the dispersal of exotic species: 2007-2010</article-title>
<source>Ecography</source>
<year>2009</year>
<volume>32</volume>
<issue>1</issue>
<fpage>94</fpage>
<lpage>102</lpage>
<pub-id pub-id-type="pmid">20300170</pub-id>
</element-citation>
</ref>
<ref id="CR148">
<label>148.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Tatem</surname>
<given-names>A. J.</given-names>
</name>
<name>
<surname>Hay</surname>
<given-names>S. I.</given-names>
</name>
<name>
<surname>Rogers</surname>
<given-names>D. J.</given-names>
</name>
</person-group>
<article-title>Global traffic and disease vector dispersal</article-title>
<source>Proceedings of the National Academy of Sciences</source>
<year>2006</year>
<volume>103</volume>
<issue>16</issue>
<fpage>6242</fpage>
<lpage>6247</lpage>
</element-citation>
</ref>
<ref id="CR149">
<label>149.</label>
<mixed-citation publication-type="other">Tatem, A.J., D. J. Rogers, and S. Hay (2006) Global transport networks and infectious disease spread, Advances in Parasitology
<bold>62</bold>
: 293–343.</mixed-citation>
</ref>
<ref id="CR150">
<label>150.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Towers</surname>
<given-names>Sherry</given-names>
</name>
<name>
<surname>Afzal</surname>
<given-names>Shehzad</given-names>
</name>
<name>
<surname>Bernal</surname>
<given-names>Gilbert</given-names>
</name>
<name>
<surname>Bliss</surname>
<given-names>Nadya</given-names>
</name>
<name>
<surname>Brown</surname>
<given-names>Shala</given-names>
</name>
<name>
<surname>Espinoza</surname>
<given-names>Baltazar</given-names>
</name>
<name>
<surname>Jackson</surname>
<given-names>Jasmine</given-names>
</name>
<name>
<surname>Judson-Garcia</surname>
<given-names>Julia</given-names>
</name>
<name>
<surname>Khan</surname>
<given-names>Maryam</given-names>
</name>
<name>
<surname>Lin</surname>
<given-names>Michael</given-names>
</name>
<name>
<surname>Mamada</surname>
<given-names>Robert</given-names>
</name>
<name>
<surname>Moreno</surname>
<given-names>Victor M.</given-names>
</name>
<name>
<surname>Nazari</surname>
<given-names>Fereshteh</given-names>
</name>
<name>
<surname>Okuneye</surname>
<given-names>Kamaldeen</given-names>
</name>
<name>
<surname>Ross</surname>
<given-names>Mary L.</given-names>
</name>
<name>
<surname>Rodriguez</surname>
<given-names>Claudia</given-names>
</name>
<name>
<surname>Medlock</surname>
<given-names>Jan</given-names>
</name>
<name>
<surname>Ebert</surname>
<given-names>David</given-names>
</name>
<name>
<surname>Castillo-Chavez</surname>
<given-names>Carlos</given-names>
</name>
</person-group>
<article-title>Mass Media and the Contagion of Fear: The Case of Ebola in America</article-title>
<source>PLOS ONE</source>
<year>2015</year>
<volume>10</volume>
<issue>6</issue>
<fpage>e0129179</fpage>
<pub-id pub-id-type="pmid">26067433</pub-id>
</element-citation>
</ref>
<ref id="CR151">
<label>151.</label>
<mixed-citation publication-type="other">Vivas-Barber, A.L., C. Castillo-Chavez, and E. Barany (2014) Dynamics of an SAIQR influenza model. Biomath
<bold>3</bold>
: 1–13.</mixed-citation>
</ref>
<ref id="CR152">
<label>152.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Webb</surname>
<given-names>Glenn</given-names>
</name>
<name>
<surname>Blaser</surname>
<given-names>Martin</given-names>
</name>
<name>
<surname>Zhu</surname>
<given-names>Huaiping</given-names>
</name>
<name>
<surname>Ardal</surname>
<given-names>Sten</given-names>
</name>
<name>
<surname>Wu</surname>
<given-names>Jianhong</given-names>
</name>
</person-group>
<article-title>Critical Role of Nosocomial Transmission in the Toronto SARS Outbreak</article-title>
<source>Mathematical Biosciences and Engineering</source>
<year>2004</year>
<volume>1</volume>
<issue>1</issue>
<fpage>1</fpage>
<lpage>13</lpage>
<pub-id pub-id-type="pmid">20369956</pub-id>
</element-citation>
</ref>
<ref id="CR153">
<label>153.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wesolowski</surname>
<given-names>A.</given-names>
</name>
<name>
<surname>Eagle</surname>
<given-names>N.</given-names>
</name>
<name>
<surname>Tatem</surname>
<given-names>A. J.</given-names>
</name>
<name>
<surname>Smith</surname>
<given-names>D. L.</given-names>
</name>
<name>
<surname>Noor</surname>
<given-names>A. M.</given-names>
</name>
<name>
<surname>Snow</surname>
<given-names>R. W.</given-names>
</name>
<name>
<surname>Buckee</surname>
<given-names>C. O.</given-names>
</name>
</person-group>
<article-title>Quantifying the Impact of Human Mobility on Malaria</article-title>
<source>Science</source>
<year>2012</year>
<volume>338</volume>
<issue>6104</issue>
<fpage>267</fpage>
<lpage>270</lpage>
<pub-id pub-id-type="pmid">23066082</pub-id>
</element-citation>
</ref>
<ref id="CR154">
<label>154.</label>
<mixed-citation publication-type="other">West, R.W. and J. R. Thompson (1997) Modeling the impact of HIV on the spread of tuberculosis in the United States, Math. Biosc.
<bold>143</bold>
: 35–60.</mixed-citation>
</ref>
<ref id="CR155">
<label>155.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Wilson</surname>
<given-names>Edward O.</given-names>
</name>
</person-group>
<article-title>Group Selection and Its Significance for Ecology</article-title>
<source>BioScience</source>
<year>1973</year>
<volume>23</volume>
<issue>11</issue>
<fpage>631</fpage>
<lpage>638</lpage>
</element-citation>
</ref>
<ref id="CR156">
<label>156.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Xiao</surname>
<given-names>Yanyu</given-names>
</name>
<name>
<surname>Brauer</surname>
<given-names>Fred</given-names>
</name>
<name>
<surname>Moghadas</surname>
<given-names>Seyed M.</given-names>
</name>
</person-group>
<article-title>Can treatment increase the epidemic size?</article-title>
<source>Journal of Mathematical Biology</source>
<year>2015</year>
<volume>72</volume>
<issue>1-2</issue>
<fpage>343</fpage>
<lpage>361</lpage>
<pub-id pub-id-type="pmid">25925242</pub-id>
</element-citation>
</ref>
<ref id="CR157">
<label>157.</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>YORKE</surname>
<given-names>JAMES A.</given-names>
</name>
<name>
<surname>HETHCOTE</surname>
<given-names>HERBERT W.</given-names>
</name>
<name>
<surname>NOLD</surname>
<given-names>ANNETT</given-names>
</name>
</person-group>
<article-title>Dynamics and Control of the Transmission of Gonorrhea</article-title>
<source>Sexually Transmitted Diseases</source>
<year>1978</year>
<volume>5</volume>
<issue>2</issue>
<fpage>51</fpage>
<lpage>56</lpage>
<pub-id pub-id-type="pmid">10328031</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 000096 | SxmlIndent | more

Ou

HfdSelect -h $EXPLOR_AREA/Data/Pmc/Corpus/biblio.hfd -nk 000096 | 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:7123038
   |texte=   Challenges, Opportunities and Theoretical Epidemiology
}}

Pour générer des pages wiki

HfdIndexSelect -h $EXPLOR_AREA/Data/Pmc/Corpus/RBID.i   -Sk "pubmed:NONE" \
       | HfdSelect -Kh $EXPLOR_AREA/Data/Pmc/Corpus/biblio.hfd   \
       | NlmPubMed2Wicri -a SrasV1 

Wicri

This area was generated with Dilib version V0.6.33.
Data generation: Tue Apr 28 14:49:16 2020. Site generation: Sat Mar 27 22:06:49 2021