Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model☆

Abstract This paper performs qualitative analysis on an SIS epidemic reaction–diffusion system with a linear source in spatially heterogeneous environment. The main feature of our model lies in that its total population number varies, compared to its counterpart proposed by Allen et al. [2] . The uniform bounds of solutions are derived, based on which, the threshold dynamics in terms of the basic reproduction number is established and the global stability of the unique endemic equilibrium is discussed when spatial environment is homogeneous. In particular, the asymptotic profile of endemic equilibria is determined if the diffusion rate of the susceptible or infected population is small or large. The theoretical results show that a varying total population can enhance persistence of infectious disease, and therefore the disease becomes more threatening and harder to control.

[1]  Z. Du,et al.  A priori L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} estimates for solutions of a clas , 2015, Journal of Mathematical Biology.

[2]  Herbert W. Hethcote,et al.  EPIDEMIOLOGY MODELS WITH VARIABLE POPULATION SIZE , 2008 .

[3]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

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

[5]  Gary M. Lieberman,et al.  Bounds for the Steady-State Sel'kov Model for Arbitrary p in Any Number of Dimensions , 2005, SIAM J. Math. Anal..

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

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

[8]  Shigui Ruan,et al.  An SIS patch model with variable transmission coefficients. , 2011, Mathematical biosciences.

[9]  W. Ni,et al.  On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type , 1986 .

[10]  Yuan Lou,et al.  Diffusion, Self-Diffusion and Cross-Diffusion , 1996 .

[11]  Huaiping Zhu,et al.  An SIS Infection Model Incorporating Media Coverage , 2008 .

[12]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

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

[14]  Yuan Lou,et al.  A spatial SIS model in advective heterogeneous environments , 2016 .

[15]  R. May,et al.  Population Biology of Infectious Diseases , 1982, Dahlem Workshop Reports.

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

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

[18]  Wei Ding,et al.  Traveling wave solutions for a diffusive sis epidemic model , 2013 .

[19]  Rui Peng,et al.  Effect of a protection zone in the diffusive Leslie predator-prey model , 2009 .

[20]  Yuan Lou,et al.  On the effects of migration and spatial heterogeneity on single and multiple species , 2006 .

[21]  Rui Peng,et al.  On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law , 2008 .

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

[23]  C. Vargas‐De‐León,et al.  On the global stability of SIS, SIR and SIRS epidemic models with standard incidence , 2011 .

[24]  Haim Brezis,et al.  Semi-linear second-order elliptic equations in L 1 , 1973 .

[25]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

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

[27]  Xingfu Zou,et al.  Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism , 2016 .

[28]  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 .

[29]  Keng Deng,et al.  Dynamics of a susceptible–infected–susceptible epidemic reaction–diffusion model , 2016 .