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

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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

Abstract
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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

References:

[1]	S. Anita, Analysis and control of age-dependent population dynamics,, Kluwer, (2000). Google Scholar
[2]	N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067. doi: 10.1007/s11538-006-9166-9. Google Scholar
[3]	N. Bacaër and E. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,, J. Math. Biol., 62 (2011), 741. doi: 10.1007/s00285-010-0354-8. Google Scholar
[4]	N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0. Google Scholar
[5]	N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Math. Biosci., 210 (2007), 647. doi: 10.1016/j.mbs.2007.07.005. Google Scholar
[6]	C. Castillo-Chavez and H.R. Thieme, How may infection-age dependent infectivity affect the dynamics of HIV/AIDS?,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068. Google Scholar
[7]	K. Cooke and J. L. Kaplan, A periodicity threshold theorem for epidemics and population growth,, Math. Biosci., 31 (1976), 87. doi: 10.1016/0025-5564(76)90042-0. Google Scholar
[8]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity,, J. Math. Anal. Appl., 338 (2008), 101. doi: 10.1016/j.jmaa.2007.05.011. Google Scholar
[9]	R. Drnovšek, Bounds for the spectral radius of positive operators,, Comment. Math. Univ. Carolinae, 41 (2000), 459. Google Scholar
[10]	A. Fonda, Uniformly persistent semidynamical systems,, Proc. Amer. Math. Soc., 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2. Google Scholar
[11]	J. Hale, Dissipation and compact attractors,, J. Dynam. Differ. Equat., 18 (2006), 485. doi: 10.1007/s10884-006-9021-6. Google Scholar
[12]	J. Hale, Asymptotic behavior of dissipative systems,, American Mathematical Society, (1988). Google Scholar
[13]	J. Hofbauer, A unified approach to persistence,, Acta Applicandae Math., 14 (1989), 11. doi: 10.1007/BF00046670. Google Scholar
[14]	J. Hofbauer and J. W. H. So, Uniform persistence and repellors for maps,, Proc. Amer. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4. Google Scholar
[15]	T. Kato, Perturbation theory for linear operators,, Springer, (1995). Google Scholar
[16]	P. Magal, Compact attractors for time-periodic age-structured population models,, Electron. J. Differ. Equat., 65 (2001), 1. Google Scholar
[17]	P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infectious-age model,, Applic. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. Google Scholar
[18]	R. D. Nussbaum, Periodic solutions of some integral equations from the theory of epidemics, in Nonlinear systems and applications* (ed. V. Lakshmikantham)*, (1977), 235. Google Scholar
[19]	R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations,, SIAM J. Math. Anal., 9 (1978), 356. doi: 10.1137/0509024. Google Scholar
[20]	C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models,, J. Math. Biol., 64 (2012), 933. doi: 10.1007/s00285-011-0440-6. Google Scholar
[21]	H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics,, J. Math. Biol., 4 (1977), 69. doi: 10.1007/BF00276353. Google Scholar
[22]	H. L. Smith, An introduction to delay differential equations with applications to the life sciences,, Springer, (2011). doi: 10.1007/978-1-4419-7646-8. Google Scholar
[23]	H. L. Smith and H. R. Thieme, Dynamical systems and population persistence,, American Mathematical Society, (2011). Google Scholar
[24]	H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. doi: 10.1137/080732870. Google Scholar
[25]	H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Integral Equat., 7 (1984), 253. Google Scholar
[26]	G. F. Webb, Theory of nonlinear age-dependent population dynamics,, Marcel Dekker, (1985). Google Scholar
[27]	G. F. Webb, E. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals,, Proc. Nat. Acad. Sci., 102 (2005), 13343. doi: 10.1073/pnas.0504053102. Google Scholar
[28]	F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496. doi: 10.1016/j.jmaa.2006.01.085. Google Scholar

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References:

[1]	S. Anita, Analysis and control of age-dependent population dynamics,, Kluwer, (2000). Google Scholar
[2]	N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067. doi: 10.1007/s11538-006-9166-9. Google Scholar
[3]	N. Bacaër and E. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,, J. Math. Biol., 62 (2011), 741. doi: 10.1007/s00285-010-0354-8. Google Scholar
[4]	N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0. Google Scholar
[5]	N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Math. Biosci., 210 (2007), 647. doi: 10.1016/j.mbs.2007.07.005. Google Scholar
[6]	C. Castillo-Chavez and H.R. Thieme, How may infection-age dependent infectivity affect the dynamics of HIV/AIDS?,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068. Google Scholar
[7]	K. Cooke and J. L. Kaplan, A periodicity threshold theorem for epidemics and population growth,, Math. Biosci., 31 (1976), 87. doi: 10.1016/0025-5564(76)90042-0. Google Scholar
[8]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity,, J. Math. Anal. Appl., 338 (2008), 101. doi: 10.1016/j.jmaa.2007.05.011. Google Scholar
[9]	R. Drnovšek, Bounds for the spectral radius of positive operators,, Comment. Math. Univ. Carolinae, 41 (2000), 459. Google Scholar
[10]	A. Fonda, Uniformly persistent semidynamical systems,, Proc. Amer. Math. Soc., 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2. Google Scholar
[11]	J. Hale, Dissipation and compact attractors,, J. Dynam. Differ. Equat., 18 (2006), 485. doi: 10.1007/s10884-006-9021-6. Google Scholar
[12]	J. Hale, Asymptotic behavior of dissipative systems,, American Mathematical Society, (1988). Google Scholar
[13]	J. Hofbauer, A unified approach to persistence,, Acta Applicandae Math., 14 (1989), 11. doi: 10.1007/BF00046670. Google Scholar
[14]	J. Hofbauer and J. W. H. So, Uniform persistence and repellors for maps,, Proc. Amer. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4. Google Scholar
[15]	T. Kato, Perturbation theory for linear operators,, Springer, (1995). Google Scholar
[16]	P. Magal, Compact attractors for time-periodic age-structured population models,, Electron. J. Differ. Equat., 65 (2001), 1. Google Scholar
[17]	P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infectious-age model,, Applic. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. Google Scholar
[18]	R. D. Nussbaum, Periodic solutions of some integral equations from the theory of epidemics, in Nonlinear systems and applications* (ed. V. Lakshmikantham)*, (1977), 235. Google Scholar
[19]	R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations,, SIAM J. Math. Anal., 9 (1978), 356. doi: 10.1137/0509024. Google Scholar
[20]	C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models,, J. Math. Biol., 64 (2012), 933. doi: 10.1007/s00285-011-0440-6. Google Scholar
[21]	H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics,, J. Math. Biol., 4 (1977), 69. doi: 10.1007/BF00276353. Google Scholar
[22]	H. L. Smith, An introduction to delay differential equations with applications to the life sciences,, Springer, (2011). doi: 10.1007/978-1-4419-7646-8. Google Scholar
[23]	H. L. Smith and H. R. Thieme, Dynamical systems and population persistence,, American Mathematical Society, (2011). Google Scholar
[24]	H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. doi: 10.1137/080732870. Google Scholar
[25]	H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Integral Equat., 7 (1984), 253. Google Scholar
[26]	G. F. Webb, Theory of nonlinear age-dependent population dynamics,, Marcel Dekker, (1985). Google Scholar
[27]	G. F. Webb, E. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals,, Proc. Nat. Acad. Sci., 102 (2005), 13343. doi: 10.1073/pnas.0504053102. Google Scholar
[28]	F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496. doi: 10.1016/j.jmaa.2006.01.085. Google Scholar

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