American Institute of Mathematical Sciences

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

 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, Dordrecht, 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), 1067-1091. 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), 741-762. 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), 421-436. 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), 647-658. 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), 1447-1479. 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), 87-104. 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), 101-110. 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), 459-467.  Google Scholar [10] A. Fonda, Uniformly persistent semidynamical systems, Proc. Amer. Math. Soc., 104 (1988), 111-116. doi: 10.1090/S0002-9939-1988-0958053-2.  Google Scholar [11] J. Hale, Dissipation and compact attractors, J. Dynam. Differ. Equat., 18 (2006), 485-523. doi: 10.1007/s10884-006-9021-6.  Google Scholar [12] J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society, Providence RI, 1988.  Google Scholar [13] J. Hofbauer, A unified approach to persistence, Acta Applicandae Math., 14 (1989), 11-22. 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), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.  Google Scholar [15] T. Kato, Perturbation theory for linear operators, Springer, Berlin, 1995.  Google Scholar [16] P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differ. Equat., 65 (2001), 1-35.  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), 1109-1140. 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), Academic Press, New York, (1977), 235-257.  Google Scholar [19] R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations, SIAM J. Math. Anal., 9 (1978), 356-376. 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), 933-949. 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), 69-80. doi: 10.1007/BF00276353.  Google Scholar [22] H. L. Smith, An introduction to delay differential equations with applications to the life sciences, Springer, Berlin, 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, Providence RI, 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), 188-211. doi: 10.1137/080732870.  Google Scholar [25] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equat., 7 (1984), 253-277.  Google Scholar [26] G. F. Webb, Theory of nonlinear age-dependent population dynamics, Marcel Dekker, New York, 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), 13343-13348. 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), 496-516. 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, Dordrecht, 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), 1067-1091. 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), 741-762. 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), 421-436. 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), 647-658. 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), 1447-1479. 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), 87-104. 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), 101-110. 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), 459-467.  Google Scholar [10] A. Fonda, Uniformly persistent semidynamical systems, Proc. Amer. Math. Soc., 104 (1988), 111-116. doi: 10.1090/S0002-9939-1988-0958053-2.  Google Scholar [11] J. Hale, Dissipation and compact attractors, J. Dynam. Differ. Equat., 18 (2006), 485-523. doi: 10.1007/s10884-006-9021-6.  Google Scholar [12] J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society, Providence RI, 1988.  Google Scholar [13] J. Hofbauer, A unified approach to persistence, Acta Applicandae Math., 14 (1989), 11-22. 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), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.  Google Scholar [15] T. Kato, Perturbation theory for linear operators, Springer, Berlin, 1995.  Google Scholar [16] P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differ. Equat., 65 (2001), 1-35.  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), 1109-1140. 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), Academic Press, New York, (1977), 235-257.  Google Scholar [19] R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations, SIAM J. Math. Anal., 9 (1978), 356-376. 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), 933-949. 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), 69-80. doi: 10.1007/BF00276353.  Google Scholar [22] H. L. Smith, An introduction to delay differential equations with applications to the life sciences, Springer, Berlin, 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, Providence RI, 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), 188-211. doi: 10.1137/080732870.  Google Scholar [25] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equat., 7 (1984), 253-277.  Google Scholar [26] G. F. Webb, Theory of nonlinear age-dependent population dynamics, Marcel Dekker, New York, 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), 13343-13348. 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), 496-516. doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar
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