Threshold dynamics and asymptotic profiles of a time-periodic nonlocal dispersal SIS epidemic model with Dirichlet boundary conditions

In this paper, we consider a spatial heterogeneous and temporal periodic nonlocal susceptible-infected-susceptible epidemic model with Dirichlet boundary conditions, where the total population is varying. We first study a principal eigenvalue problem for a time-periodic nonlocal dispersal operator and obtain the limiting profile of principal eigenvalue. Next, we establish the existence, uniqueness and stability of steady states of the model in terms of the basic reproduction ratio. Finally, we discuss the impacts of small diffusion rates on the persistence and extinction of the disease. We show the asymptotic profiles of the disease-free equilibrium, endemic equilibrium and basic reproduction ratio as the diffusion rate tends to zero.