Travelling Waves Of Two Kinds Of Epidemic Systems With Nonlocal Dispersal

In this paper, the existence of travelling wave solutions for two kinds of epidemic systems with nonlocal dispersal are considered.First, the Capasso-Maddalena epidemic system with nonlocal dispersal is stud-ied. The existence of travelling wavefronts are obtained by using the super-lower solution technique combined with monotone iteration. Furthermore, a complicated system including the nonlocal effect from infected individual to infectious agent is discussed by means of the method mentioned above.Second, the SIR epidemic system with nonlocal dispersal and spatio-temporal delay is considered. Whenc> c*(c*denotes the critical speed of travelling wave), we construct a completely continuous operator on a bounded, closed and convex invariant cone, so that the existence of travelling wave solutions can be converted into the existence of fixed point of the operator on the invariant cone. That is, we use Schauder’s fixed point theorem to obtain the existence of travelling wave so-lutions. When0<c<c*. the non-existence of travelling wave solution is proved by using two-sided Laplace transform. Furthermore, threshold value determining whether epidemic disease can spread or not is obtained, the effect of spatial hetero-geneity (geographical movement), nonlocal interaction and time delay on c*are also considered.