Asymptotic profiles of the steady states for an SIS epidemic reactiondiffusion model
Linda J. S. Allen  Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 794091042, United States (email) Abstract: To understand the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, a spatial SIS reactiondiffusion model is studied, with the focus on the existence, uniqueness and particularly the asymptotic profile of the steadystates. First, the basic reproduction number $\R_{0}$ is defined for this SIS PDE model. It is shown that if $\R_{0} < 1$, the unique diseasefree equilibrium is globally asymptotic stable and there is no endemic equilibrium. If $\R_{0} > 1$, the diseasefree 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 diseasefree equilibrium is always unstable $(\R_{0} > 1)$ for highrisk domains. For lowrisk domains, the diseasefree 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 diseasefree equilibrium as the mobility of susceptible individuals tends to zero. Surprisingly, the density of susceptibles for this limiting diseasefree 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.
Keywords: Spatial heterogeneity, dispersal, basic reproduction number, diseasefree equilibrium, endemic equilibrium.
Received: January 2007; Revised: October 2007; Available Online: February 2008. 
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