A time-periodic reaction–diffusion epidemic model with infection period

In this paper, we propose a time-periodic and diffusive SIR epidemic model with constant infection period. By introducing the basic reproduction number $${\mathcal{R}_0}$$R0 via a next generation operator for this model, we show that the disease goes extinction if $${\mathcal{R}_0 < 1}$$R0<1 ; while the disease is uniformly persistent if $${\mathcal{R}_0 > 1}$$R0>1.

[1]  Xiao-Qiang Zhao,et al.  Chain Transitivity, Attractivity, and Strong Repellors for Semidynamical Systems , 2001 .

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

[3]  F. M. Arscott,et al.  PERIODIC‐PARABOLIC BOUNDARY VALUE PROBLEMS AND POSITIVITY , 1992 .

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

[5]  Abstract evolution equations , 1969 .

[6]  Xiao-Qiang Zhao,et al.  An Epidemic Model with Population Dispersal and Infection Period , 2006, SIAM J. Appl. Math..

[7]  Hal L. Smith,et al.  Abstract functional-differential equations and reaction-diffusion systems , 1990 .

[8]  Alessandro Margheri,et al.  Persistence in some periodic epidemic models with infection age or constant periods of infection , 2014 .

[9]  G. Serio,et al.  A generalization of the Kermack-McKendrick deterministic epidemic model☆ , 1978 .

[10]  Xiao-Qiang Zhao,et al.  A Reaction-Diffusion Lyme Disease Model with Seasonality , 2013, SIAM J. Appl. Math..

[11]  Xiao-Qiang Zhao,et al.  Basic Reproduction Ratios for Periodic Compartmental Models with Time Delay , 2017 .

[12]  Nicolas Bacaër,et al.  On the biological interpretation of a definition for the parameter R0 in periodic population models , 2012, Journal of mathematical biology.

[13]  Rui Peng,et al.  A reaction–diffusion SIS epidemic model in a time-periodic environment , 2012 .

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

[15]  O. Diekmann,et al.  The Dynamics of Physiologically Structured Populations , 1986 .

[16]  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.

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

[18]  Hisashi Inaba,et al.  On a new perspective of the basic reproduction number in heterogeneous environments , 2011, Journal of Mathematical Biology.

[19]  Xiao-Qiang Zhao,et al.  Threshold dynamics in a time-delayed periodic sis epidemic model , 2009 .

[20]  Xiao-Qiang Zhao,et al.  Dynamical systems in population biology , 2003 .

[21]  Xiao-Qiang Zhao,et al.  Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period , 2015 .

[22]  R. Drnovšek.  Bounds for the spectral radius of positive operators , 2000 .

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

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

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

[26]  A L Lloyd,et al.  Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics. , 2001, Theoretical population biology.

[27]  Nicolas Bacaër,et al.  Genealogy with seasonality, the basic reproduction number, and the influenza pandemic , 2011, Journal of mathematical biology.