A reaction–diffusion SIS epidemic model in a time-periodic environment

In this paper, we consider a susceptible–infected–susceptible (SIS) reaction–diffusion model, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous and temporally periodic and the total population number is constant. We introduce a basic reproduction number and establish threshold-type results on the global dynamics in terms of . In particular, we obtain the asymptotic properties of with respect to the diffusion rate dI of the infected individuals, which exhibit the delicate influence of the time-periodic heterogeneous environment on the extinction and persistence of the infectious disease. Our analytical results suggest that the combination of spatial heterogeneity and temporal periodicity tends to enhance the persistence of the disease.

[1]  Xiao-Qiang Zhao,et al.  Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments , 2008 .

[2]  Nicholas D. Alikakos,et al.  LP Bounds of solutions of reaction-diffusion equations , 1979 .

[3]  L. Dung.  Dissipativity and global attractors for a class of quasilinear parabolic systems , 1997 .

[4]  Chris Cosner,et al.  Diffusive logistic equations with indefinite weights: population models in disrupted environments , 1991, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  Yuan Lou,et al.  Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model , 2008 .

[6]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[7]  Xiao-Qiang Zhao,et al.  Global Attractors and Steady States for Uniformly Persistent Dynamical Systems , 2005, SIAM J. Math. Anal..

[8]  Lionel Roques,et al.  Mathematical analysis of the optimal habitat configurations for species persistence. , 2007, Mathematical biosciences.

[9]  Herbert Amann,et al.  Linear and Quasilinear Parabolic Problems , 2019, Monographs in Mathematics.

[10]  Yuan Lou,et al.  Asymptotic Profiles of the Steady States for an SIS Epidemic Patch Model , 2007, SIAM J. Appl. Math..

[11]  P. C. Dunne,et al.  A semilinear parabolic system arising in the theory of superconductivity , 1981 .

[12]  Grégoire Nadin.  The principal eigenvalue of a space–time periodic parabolic operator , 2009 .

[13]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[14]  Rui Peng,et al.  Global stability of the steady states of an SIS epidemic reaction–diffusion model☆ , 2009 .

[15]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[16]  Nicolas Bacaër,et al.  The epidemic threshold of vector-borne diseases with seasonality , 2006, Journal of mathematical biology.

[17]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[18]  Rui Peng,et al.  Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I , 2009 .

[19]  Mingxin Wang,et al.  Strategy and stationary pattern in a three-species predator–prey model , 2004 .

[20]  Horst R. Thieme,et al.  Spectral Bound and Reproduction Number for Infinite-Dimensional Population Structure and Time Heterogeneity , 2009, SIAM J. Appl. Math..

[21]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[22]  Xiao-Qiang Zhao,et al.  A Nonlocal and Time-Delayed Reaction-Diffusion Model of Dengue Transmission , 2011, SIAM J. Appl. Math..

[23]  G. M. Lieberman.  SECOND ORDER PARABOLIC DIFFERENTIAL EQUATIONS , 1996 .

[24]  Rui Peng,et al.  Asymptotic profile of the positive steady state for an SIS epidemic reaction–diffusion model: Effects of epidemic risk and population movement , 2013 .

[25]  V. Hutson,et al.  The evolution of dispersal rates in a heterogeneous time-periodic environment , 2001, Journal of mathematical biology.

[26]  Herbert Amann,et al.  Linear and Quasilinear Parabolic Problems: Volume I: Abstract Linear Theory , 1995 .

[27]  Abstract evolution equations , 1969 .

[28]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[29]  Junping Shi,et al.  Optimal Spatial Harvesting Strategy and Symmetry-Breaking , 2008 .

[30]  Maoan Han,et al.  Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. , 2010, Mathematical biosciences and engineering : MBE.

[31]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .