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June  2014, 19(4): 1155-1170. doi: 10.3934/dcdsb.2014.19.1155

Persistence in some periodic epidemic models with infection age or constant periods of infection

Carlota Rebelo 1, , Alessandro Margheri 1, and  Nicolas Bacaër 2,

1. 

Centro de Matemática e Aplicações Fundamentais, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal, Portugal
2. 

IRD and University Paris 6, Research group UMMISCO, Bondy, France

Received  June 2013 Revised  February 2014 Published  April 2014

Much recent work has focused on persistence for epidemic models with periodic coefficients. But the case where the infected compartments satisfy a delay differential equation or a partial differential equation does not seem to have been considered so far. The purpose of this paper is to provide a framework for proving persistence in such a case. Some examples are presented, such as a periodic SIR model structured by time since infection and a periodic SIS delay model.
Keywords: seasonality, epidemiological models, delay differential equation., Persistence.
Mathematics Subject Classification: Primary: 92D30, 45J05; Secondary: 54H2.
Citation: Carlota Rebelo, Alessandro Margheri, Nicolas Bacaër. Persistence in some periodic epidemic models with infection age or constant periods of infection. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1155-1170. doi: 10.3934/dcdsb.2014.19.1155
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show all references

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[1]

Kluwer, Dordrecht, 2000.  Google Scholar

[2]

Bull. Math. Biol., 69 (2007), 1067-1091. doi: 10.1007/s11538-006-9166-9.  Google Scholar

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SIAM J. Appl. Math., 53 (1993), 1447-1479. doi: 10.1137/0153068.  Google Scholar

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Math. Biosci., 31 (1976), 87-104. doi: 10.1016/0025-5564(76)90042-0.  Google Scholar

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Comment. Math. Univ. Carolinae, 41 (2000), 459-467.  Google Scholar

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J. Dynam. Differ. Equat., 18 (2006), 485-523. doi: 10.1007/s10884-006-9021-6.  Google Scholar

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[13]

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[14]

Proc. Amer. Math. Soc., 107 (1989), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.  Google Scholar

[15]

Springer, Berlin, 1995.  Google Scholar

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Electron. J. Differ. Equat., 65 (2001), 1-35.  Google Scholar

[17]

Applic. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.  Google Scholar

[18]

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[19]

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[20]

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[24]

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[25]

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[26]

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J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

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