# American Institute of Mathematical Sciences

January  2008, 21(1): 1-20. doi: 10.3934/dcds.2008.21.1

## Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model

 1 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States 2 Department of Zoology, University of Florida, Gainesville, FL 32611-8525, United States 3 Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210 4 Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, United States

Received  January 2007 Revised  October 2007 Published  February 2008

To understand the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, a spatial SIS reaction-diffusion model is studied, with the focus on the existence, uniqueness and particularly the asymptotic profile of the steady-states. First, the basic reproduction number $\R_{0}$ is defined for this SIS PDE model. It is shown that if $\R_{0} < 1$, the unique disease-free equilibrium is globally asymptotic stable and there is no endemic equilibrium. If $\R_{0} > 1$, the disease-free equilibrium is unstable and there is a unique endemic equilibrium. A domain is called high (low) risk if the average of the transmission rates is greater (less) than the average of the recovery rates. It is shown that the disease-free equilibrium is always unstable $(\R_{0} > 1)$ for high-risk domains. For low-risk domains, the disease-free equilibrium is stable $(\R_{0} < 1)$ if and only if infected individuals have mobility above a threshold value. The endemic equilibrium tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. Surprisingly, the density of susceptibles for this limiting disease-free equilibrium, which is always positive on the subdomain where the transmission rate is less than the recovery rate, must also be positive at some (but not all) places where the transmission rates exceed the recovery rates.
Citation: Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
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