Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 92D30, 45J05; Secondary: 54H20.


    \begin{equation} \\ \end{equation}
  • [1]

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


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    R. Drnovšek, Bounds for the spectral radius of positive operators, Comment. Math. Univ. Carolinae, 41 (2000), 459-467.


    A. Fonda, Uniformly persistent semidynamical systems, Proc. Amer. Math. Soc., 104 (1988), 111-116.doi: 10.1090/S0002-9939-1988-0958053-2.


    J. Hale, Dissipation and compact attractors, J. Dynam. Differ. Equat., 18 (2006), 485-523.doi: 10.1007/s10884-006-9021-6.


    J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society, Providence RI, 1988.


    J. Hofbauer, A unified approach to persistence, Acta Applicandae Math., 14 (1989), 11-22.doi: 10.1007/BF00046670.


    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.


    T. Kato, Perturbation theory for linear operators, Springer, Berlin, 1995.


    P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differ. Equat., 65 (2001), 1-35.


    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.


    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.


    R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations, SIAM J. Math. Anal., 9 (1978), 356-376.doi: 10.1137/0509024.


    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.


    H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics, J. Math. Biol., 4 (1977), 69-80.doi: 10.1007/BF00276353.


    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.


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


    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.


    H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equat., 7 (1984), 253-277.


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


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(168) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint