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

To understand the impact of spatial heterogeneity of environment and movement of individuals \non the persistence and extinction of a disease, a spatial SIS reaction-diffusion model is studied, \nwith the focus on the existence, uniqueness and particularly the asymptotic profile of the steady- \nstates. First, the basic reproduction number R0 is defined for this SIS PDE model. It is shown \nthat if R0 < 1, the unique disease-free equilibrium is globally asymptotic stable and there is no \nendemic equilibrium. If R0 > 1, the disease-free equilibrium is unstable and there is a unique \nendemic equilibrium. \nA domain is called high (low) risk if the average of the transmission rates is greater (less) \nthan the average of the recovery rates. It is shown that the disease-free equilibrium is always \nunstable (R0 > 1) for high-risk domains. For low-risk domains, the disease-free equilibrium is \nstable (R0 < 1) if and only if infected individuals have mobility above a threshold value. The \nendemic equilibrium tends to a spatially inhomogeneous disease-free equilibrium as the mobility \nof susceptible individuals tends to zero. Surprisingly, the density of susceptible for this limiting \ndisease-free equilibrium, which is always positive on the subdomain where the transmission rate is \nless than the recovery rate, must also be positive at some, but not all, places where the transmission \nrates are greater than the recovery rates.