A spatial SIS model in heterogeneous environments with vary advective rate

We study a spatial susceptible-infected-susceptible(SIS) model in heterogeneous environments with vary advective rate. We establish the asymptotic stability of the unique disease-free equilibrium(DFE) when $ \mathcal{R}_0 < 1 $ and the existence of the endemic equilibrium when $ \mathcal{R}_0 > 1 $. Here $ \mathcal{R}_0 $ is the basic reproduction number. We also discuss the effect of diffusion on the stability of the DFE.