{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T03:09:08Z","timestamp":1790910548269,"version":"4.1.0"},"reference-count":42,"publisher":"Elsevier BV","issue":"2","license":[{"start":{"date-parts":[[2011,12,1]],"date-time":"2011-12-01T00:00:00Z","timestamp":1322697600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2011,12,1]],"date-time":"2011-12-01T00:00:00Z","timestamp":1322697600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2011,12,1]],"date-time":"2011-12-01T00:00:00Z","timestamp":1322697600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2011,12,1]],"date-time":"2011-12-01T00:00:00Z","timestamp":1322697600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2011,12,1]],"date-time":"2011-12-01T00:00:00Z","timestamp":1322697600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2011,12,1]],"date-time":"2011-12-01T00:00:00Z","timestamp":1322697600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2011,12,1]],"date-time":"2011-12-01T00:00:00Z","timestamp":1322697600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Biosciences"],"published-print":{"date-parts":[[2011,12]]},"DOI":"10.1016\/j.mbs.2011.09.003","type":"journal-article","created":{"date-parts":[[2011,10,3]],"date-time":"2011-10-03T08:20:10Z","timestamp":1317630010000},"page":"118-126","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":114,"title":["Global dynamics of cholera models with differential infectivity"],"prefix":"10.1016","volume":"234","author":[{"given":"Zhisheng","family":"Shuai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P.","family":"van den Driessche","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.mbs.2011.09.003_b0005","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1016\/S0140-6736(11)60273-0","article-title":"Transmission dynamics and control of cholera in Haiti: an epidemic model","volume":"377","author":"Andrews","year":"2011","journal-title":"Lancet"},{"key":"10.1016\/j.mbs.2011.09.003_b0010","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/S0304-3800(98)00009-X","article-title":"Simple models for the impact of rabbit calicivirus disease (RCD) on Australasian rabbits","volume":"109","author":"Barlow","year":"1998","journal-title":"Ecol. Model."},{"key":"10.1016\/j.mbs.2011.09.003_b0015","series-title":"Nonnegative Matrices in the Mathematical Sciences","author":"Berman","year":"1979"},{"key":"10.1016\/j.mbs.2011.09.003_b0020","article-title":"Dynamical Systems: Stability Theory and Applications","volume":"vol. 35","author":"Bhatia","year":"1967"},{"key":"10.1016\/j.mbs.2011.09.003_b0025","first-page":"121","article-title":"A mathematical model for the 1973 cholera epidemic in the European Mediterranean region","volume":"27","author":"Capasso","year":"1979","journal-title":"Rev. Epid. San. Publ."},{"key":"10.1016\/j.mbs.2011.09.003_b0030","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1006\/aima.1997.1673","article-title":"A polynomial counterexample to the Markus-Yamabe conjecture","volume":"131","author":"Cima","year":"1997","journal-title":"Adv. Math."},{"issue":"1","key":"10.1016\/j.mbs.2011.09.003_b0035","doi-asserted-by":"crossref","DOI":"10.1186\/1471-2334-1-1","article-title":"Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir","volume":"1","author":"Code\u00e7o","year":"2001","journal-title":"BMC Infect. Dis."},{"key":"10.1016\/j.mbs.2011.09.003_b0040","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/BF00178324","article-title":"On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations","volume":"28","author":"Diekmann","year":"1990","journal-title":"J. Math. Biol."},{"key":"10.1016\/j.mbs.2011.09.003_b0045","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1289\/ehp.02110783","article-title":"Disease transmission models for public health decision making: analysis of epidemic and endemic conditions caused by waterborne pathogens","volume":"110","author":"Eisenberg","year":"2002","journal-title":"Environ. Health Perspect."},{"key":"10.1016\/j.mbs.2011.09.003_b0050","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1007\/BF02218848","article-title":"Uniform persistence and flows near a closed positively invariant set","volume":"6","author":"Freedman","year":"1994","journal-title":"J. Dynam. Differ. Eqn."},{"key":"10.1016\/j.mbs.2011.09.003_b0055","doi-asserted-by":"crossref","first-page":"513","DOI":"10.3934\/mbe.2006.3.513","article-title":"Global dynamics of a staged progression model for infectious diseases","volume":"3","author":"Guo","year":"2006","journal-title":"Math. Biosci. Eng."},{"key":"10.1016\/j.mbs.2011.09.003_b0060","first-page":"259","article-title":"Global stability of the endemic equilibrium of multigroup SIR epidemic models","volume":"14","author":"Guo","year":"2006","journal-title":"Can. Appl. Math. Q."},{"key":"10.1016\/j.mbs.2011.09.003_b0065","doi-asserted-by":"crossref","first-page":"2793","DOI":"10.1090\/S0002-9939-08-09341-6","article-title":"A graph-theoretic approach to the method of global Lyapunov functions","volume":"136","author":"Guo","year":"2008","journal-title":"Proc. Amer. Math. Soc."},{"key":"10.1016\/j.mbs.2011.09.003_b0070","first-page":"63","article-title":"Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?","volume":"3","author":"Hartley","year":"2006","journal-title":"PLOS Med."},{"key":"10.1016\/j.mbs.2011.09.003_b0075","series-title":"Control of Communicable Diseases Manual","year":"2008"},{"key":"10.1016\/j.mbs.2011.09.003_b0080","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/S0025-5564(98)10057-3","article-title":"The differential infectivity and staged progression models for the transmission of HIV","volume":"155","author":"Hyman","year":"1999","journal-title":"Math. Biosci."},{"key":"10.1016\/j.mbs.2011.09.003_b0085","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1111\/j.1524-4733.2009.00562.x","article-title":"Cost-effectiveness of new-generation oral cholera vaccines: a multisite analysis","volume":"12","author":"Jeuland","year":"2009","journal-title":"Val. Health"},{"key":"10.1016\/j.mbs.2011.09.003_b0090","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1093\/wber\/lhp006","article-title":"A cost-benefit analysis of cholera vaccination programs in Beira, Mozambique","volume":"23","author":"Jeuland","year":"2009","journal-title":"World Bank Econ. Rev."},{"key":"10.1016\/j.mbs.2011.09.003_b0095","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1007\/s11538-008-9384-4","article-title":"Dynamics of indirectly transmitted infectious diseases with immunological threshold","volume":"71","author":"Joh","year":"2009","journal-title":"Bull. Math. Biol."},{"key":"10.1016\/j.mbs.2011.09.003_b0100","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1038\/nature07084","article-title":"Inapparent infections and cholera dynamics","volume":"454","author":"King","year":"2008","journal-title":"Nature"},{"key":"10.1016\/j.mbs.2011.09.003_b0105","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1016\/j.bulm.2004.02.001","article-title":"Global properties of basic virus dynamics models","volume":"66","author":"Korobeinikov","year":"2004","journal-title":"Bull. Math. Biol."},{"key":"10.1016\/j.mbs.2011.09.003_b0110","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1007\/s11538-005-9037-9","article-title":"Lyapunov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission","volume":"68","author":"Korobeinikov","year":"2006","journal-title":"Bull. Math. Biol."},{"key":"10.1016\/j.mbs.2011.09.003_b0115","doi-asserted-by":"crossref","first-page":"1871","DOI":"10.1007\/s11538-007-9196-y","article-title":"Global properties of infectious disease models with nonlinear incidence","volume":"69","author":"Korobeinikov","year":"2007","journal-title":"Bull. Math. Biol."},{"key":"10.1016\/j.mbs.2011.09.003_b0120","doi-asserted-by":"crossref","first-page":"57","DOI":"10.3934\/mbe.2004.1.57","article-title":"A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence","volume":"1","author":"Korobeinikov","year":"2004","journal-title":"Math. Biosci. Eng."},{"key":"10.1016\/j.mbs.2011.09.003_b0125","series-title":"The Stability of Dynamical Systems","author":"LaSalle","year":"1976"},{"key":"10.1016\/j.mbs.2011.09.003_b0130","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S0025-5564(99)00030-9","article-title":"Global dynamics of a SEIR model with varying total population size","volume":"160","author":"Li","year":"1999","journal-title":"Math. Biosci."},{"key":"10.1016\/j.mbs.2011.09.003_b0135","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jde.2009.09.003","article-title":"Global-stability problems for coupled systems of differential equations on networks","volume":"248","author":"Li","year":"2010","journal-title":"J. Differ. Eqn."},{"key":"10.1016\/j.mbs.2011.09.003_b0140","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1111\/j.1708-8305.2008.00270.x","article-title":"Cost-benefit of WC\/rBS oral cholera vaccine for vaccination against ETEC-caused travelers\u2019 diarrhea","volume":"16","author":"Lundkvist","year":"2009","journal-title":"J. Travel Med."},{"key":"10.1016\/j.mbs.2011.09.003_b0145","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1007\/s11538-005-9047-7","article-title":"Generality of the final size formula for an epidemic of a newly invading infectious disease","volume":"68","author":"Ma","year":"2006","journal-title":"Bull. Math. Biol."},{"key":"10.1016\/j.mbs.2011.09.003_b0150","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.mbs.2003.08.003","article-title":"Modeling the long-term control of viremia in HIV-1 infected patients treated with antiretroviral therapy","volume":"188","author":"Michele","year":"2004","journal-title":"Math. Biosci."},{"key":"10.1016\/j.mbs.2011.09.003_b0155","doi-asserted-by":"crossref","first-page":"8767","DOI":"10.1073\/pnas.1019712108","article-title":"Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe","volume":"108","author":"Mukandavire","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"10.1016\/j.mbs.2011.09.003_b0160","doi-asserted-by":"crossref","first-page":"4398","DOI":"10.1073\/pnas.93.9.4398","article-title":"Viral dynamics in hepatitis B virus infection","volume":"93","author":"Nowak","year":"1996","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"10.1016\/j.mbs.2011.09.003_b0165","series-title":"Virus Dynamics: Mathematical Principles of Immunology and Virology","author":"Nowak","year":"2000"},{"key":"10.1016\/j.mbs.2011.09.003_b0170","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1111\/j.1574-6968.1998.tb12909.x","article-title":"Characterization of a 20-k Da pilus protein expressed by a diarrheogenic strain of non-O1\/non-O139 Vibrio cholerae","volume":"160","author":"Sengupta","year":"1998","journal-title":"FEMS Microbiol. Lett."},{"key":"10.1016\/j.mbs.2011.09.003_b0175","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1137\/0152030","article-title":"Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations","volume":"52","author":"Simon","year":"1992","journal-title":"SIAM J. Appl. Math."},{"key":"10.1016\/j.mbs.2011.09.003_b0180","doi-asserted-by":"crossref","first-page":"1313","DOI":"10.1137\/S0036139902406905","article-title":"Virus dynamics: a global analysis","volume":"63","author":"Smith","year":"2003","journal-title":"SIAM J. Appl. Math."},{"key":"10.1016\/j.mbs.2011.09.003_b0185","series-title":"The Theory of the Chemostat: Dynamics of Microbial Competition","author":"Smith","year":"1995"},{"key":"10.1016\/j.mbs.2011.09.003_b0190","unstructured":"J.P. Tian, S. Liao, J. Wang, Dynamical analysis and control strategies in modeling cholera, preprint. <www.math.ttu.edu\/past\/redraider2010\/Tian2.pdf>, 2010 (accessed 14.12.10)."},{"key":"10.1016\/j.mbs.2011.09.003_b0195","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.mbs.2011.04.001","article-title":"Global stability for cholera epidemic models","volume":"232","author":"Tian","year":"2011","journal-title":"Math. Biosci."},{"key":"10.1016\/j.mbs.2011.09.003_b0200","doi-asserted-by":"crossref","first-page":"1506","DOI":"10.1007\/s11538-010-9507-6","article-title":"Multiple transmission pathways and disease dynamics in a waterborne pathogen model","volume":"72","author":"Tien","year":"2010","journal-title":"Bull. Math. Biol."},{"key":"10.1016\/j.mbs.2011.09.003_b0205","doi-asserted-by":"crossref","first-page":"593","DOI":"10.7326\/0003-4819-154-9-201105030-00334","article-title":"2010: using a transmission model to explain spatial spread of disease and identify optimal control interventions","volume":"154","author":"Tuite","year":"2011","journal-title":"Ann. Internal Med."},{"key":"10.1016\/j.mbs.2011.09.003_b0210","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"van den Driessche","year":"2002","journal-title":"Math. Biosci."}],"container-title":["Mathematical Biosciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0025556411001441?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0025556411001441?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,9,21]],"date-time":"2025-09-21T01:18:59Z","timestamp":1758417539000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0025556411001441"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,12]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2011,12]]}},"alternative-id":["S0025556411001441"],"URL":"https:\/\/doi.org\/10.1016\/j.mbs.2011.09.003","relation":{},"ISSN":["0025-5564"],"issn-type":[{"value":"0025-5564","type":"print"}],"subject":[],"published":{"date-parts":[[2011,12]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Global dynamics of cholera models with differential infectivity","name":"articletitle","label":"Article Title"},{"value":"Mathematical Biosciences","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.mbs.2011.09.003","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"Copyright \u00a9 2011 Elsevier Inc. All rights reserved.","name":"copyright","label":"Copyright"}]}}