Spatial propagation in an epidemic model with nonlocal diffusion: The influences of initial data and dispersals

This paper studies an epidemic model with nonlocal dispersals. We focus on the influences of initial data and nonlocal dispersals on its spatial propagation. Here, initial data stand for the spatial concentrations of the infectious agent and the infectious human population when the epidemic breaks out and the nonlocal dispersals mean their diffusion strategies. Two types of initial data decaying to zero exponentially or faster are considered. For the first type, we show that spreading speeds are two constants whose signs change with the number of elements in some set. Moreover, we find an interesting phenomenon: the asymmetry of nonlocal dispersals can influence the propagating directions of the solutions and the stability of steady states. For the second type, we show that the spreading speed is decreasing with respect to the exponentially decaying rate of initial data, and further, its minimum value coincides with the spreading speed for the first type. In addition, we give some results about the nonexistence of traveling wave solutions and the monotone property of the solutions. Finally, some applications are presented to illustrate the theoretical results.