A spatial SIS model in advective heterogeneous environments

We study the effects of diffusion and advection for a susceptible-infected-susceptible epidemic reaction–diffusion model in heterogeneous environments. The definition of the basic reproduction number R0R0 is given. If R0<1R0<1, the unique disease-free equilibrium (DFE) is globally asymptotically stable. Asymptotic behaviors of R0R0 for advection rate and mobility of the infected individuals (denoted by dIdI) are established, and the existence of the endemic equilibrium when R0>1R0>1 is studied. The effects of diffusion and advection rates on the stability of the DFE are further investigated. Among other things, we find that if the habitat is a low-risk domain, there may exist one critical value for the advection rate, under which the DFE changes its stability at least twice as dIdI varies from zero to infinity, while the DFE is unstable for any dIdI when the advection rate is larger than the critical value. These results are in strong contrast with the case of no advection, where the DFE changes its stability at most once as dIdI varies from zero to infinity.