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

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.
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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

R. Drnovšek, Bounds for the spectral radius of positive operators,, Comment. Math. Univ. Carolinae, 41 (2000), 459.

[10]

A. Fonda, Uniformly persistent semidynamical systems,, Proc. Amer. Math. Soc., 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2.

[11]

J. Hale, Dissipation and compact attractors,, J. Dynam. Differ. Equat., 18 (2006), 485. doi: 10.1007/s10884-006-9021-6.

[12]

J. Hale, Asymptotic behavior of dissipative systems,, American Mathematical Society, (1988).

[13]

J. Hofbauer, A unified approach to persistence,, Acta Applicandae Math., 14 (1989), 11. doi: 10.1007/BF00046670.

[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.

[15]

T. Kato, Perturbation theory for linear operators,, Springer, (1995).

[16]

P. Magal, Compact attractors for time-periodic age-structured population models,, Electron. J. Differ. Equat., 65 (2001), 1.

[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.

[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.

[19]

R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations,, SIAM J. Math. Anal., 9 (1978), 356. doi: 10.1137/0509024.

[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.

[21]

H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics,, J. Math. Biol., 4 (1977), 69. doi: 10.1007/BF00276353.

[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.

[23]

H. L. Smith and H. R. Thieme, Dynamical systems and population persistence,, American Mathematical Society, (2011).

[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.

[25]

H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Integral Equat., 7 (1984), 253.

[26]

G. F. Webb, Theory of nonlinear age-dependent population dynamics,, Marcel Dekker, (1985).

[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.

[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.

show all references

References:
[1]

S. Anita, Analysis and control of age-dependent population dynamics,, Kluwer, (2000).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

R. Drnovšek, Bounds for the spectral radius of positive operators,, Comment. Math. Univ. Carolinae, 41 (2000), 459.

[10]

A. Fonda, Uniformly persistent semidynamical systems,, Proc. Amer. Math. Soc., 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2.

[11]

J. Hale, Dissipation and compact attractors,, J. Dynam. Differ. Equat., 18 (2006), 485. doi: 10.1007/s10884-006-9021-6.

[12]

J. Hale, Asymptotic behavior of dissipative systems,, American Mathematical Society, (1988).

[13]

J. Hofbauer, A unified approach to persistence,, Acta Applicandae Math., 14 (1989), 11. doi: 10.1007/BF00046670.

[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.

[15]

T. Kato, Perturbation theory for linear operators,, Springer, (1995).

[16]

P. Magal, Compact attractors for time-periodic age-structured population models,, Electron. J. Differ. Equat., 65 (2001), 1.

[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.

[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.

[19]

R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations,, SIAM J. Math. Anal., 9 (1978), 356. doi: 10.1137/0509024.

[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.

[21]

H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics,, J. Math. Biol., 4 (1977), 69. doi: 10.1007/BF00276353.

[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.

[23]

H. L. Smith and H. R. Thieme, Dynamical systems and population persistence,, American Mathematical Society, (2011).

[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.

[25]

H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Integral Equat., 7 (1984), 253.

[26]

G. F. Webb, Theory of nonlinear age-dependent population dynamics,, Marcel Dekker, (1985).

[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.

[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.

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