A diffusive SIS epidemic model with saturated incidence function in a heterogeneous environment

In this paper, we propose a diffusive susceptible-infected-susceptible epidemic model in a spatiotemporally heterogeneous environment. We consider a saturated incidence function of the form SIm+S+I , where m is a nonnegative function. When there is a positive disease-induced mortality rate everywhere, we demonstrate that the disease will always become extinct and the susceptible population will stabilize at zero or a positive constant. In the case where the disease-induced mortality rate is negligible, we examine the model in a time-periodic and spatially heterogeneous environment and establish the threshold dynamics between disease extinction and persistence in terms of the basic reproduction number. Under certain conditions, we show the global attractivity of both the disease-free equilibrium and endemic equilibrium by constructing appropriate Lyapunov functions. Moreover, we determine the spatial distribution of the disease when the diffusion rate of the susceptible or infected population (or both) is sufficiently small. Our findings suggest that the presence of a saturation effect reduces the transmission risk, allows for the total population size to have a substantial impact on disease dynamics, and may significantly alter the spatial distribution of the disease under certain circumstances.