A Nonlocal Dispersal SIS Epidemic Model in Heterogeneous Environment
  This article is concerned with a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Neumann boundary condition, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous and the total population number is constant. We first introduce the basic reproduction number $R_0$ and then discuss the existence, uniqueness and stability of steady states of the nonlocal dispersal SIS epidemic model in terms of $R_0$. In particular, we also consider the impacts of the large diffusion rates of the susceptible and infected population on the persistence and extinction of the disease, and these results imply that the nonlocal movement of the susceptible or infected individuals will enhance the persistence of the disease. Additionally, our analytical results also suggest that the spatial heterogeneity tends to boost the spread of the disease. We have to emphasize that the main difficulty is that a lack of regularizing effect occurs. 
Fei-Ying Yang, Wan-Tong Li and Liang Zhang
math.AP