A free boundary problem of an epidemic model with nonlocal diffusion and nonlocal infective rate

This paper is concerned with an epidemic model with free boundaries, where the infective rate of humans is a nonlocal term due to the nonlocal diffusion of the infectious agents. This is different from the models in [19] where the infective rate depends only on the agents at location $ x $. We show that the longtime behaviors of our model is the same with those in [19], but the methods used here are different. Particularly, we get the exponential stability for the vanishing case that is a new result for such kind of problem. Moreover, by analyzing an eigenvalue problem, we discuss in detail the impact of initial habitat $ [-h_0, h_0] $ and diffusion coefficient $ d $ on spreading.