Asymptotic behavior of endemic equilibria for a SIS epidemic model in convective environments

Abstract

In this paper, we study a reaction-convection-diffusion SIS epidemic model with standard incidence function in a heterogeneous environment. The convection term is allowed to vary from positive to negative and a sign-changing function is used to specify convective direction. In particular, such a sign-changing function is allowed to be high-order degenerate at its critical points. We first establish the existence of endemic equilibria through the basic reproduction number $R_0$ and investigate the asymptotic profile of $R_0$ for large convection rate and small diffusion rate of the infectives, respectively. We further study the asymptotic behavior of endemic equilibria as convection approaches to infinity and the diffusion rate of infectives tends to zero, respectively. Our findings show that for large convection rate, both susceptible and infectious populations concentrate only at the critical points of the convection function, behaving exactly like a delta function; and for small diffusion rate of infectives, the density of susceptible population is positive while the total biomass of infectious population vanishes.