  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 the initial data stand for the spatial concentrations of
infectious agent and 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 the 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 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 solutions. Finally, some applications are presented to
illustrate the theoretical results.
