Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

Abstract

In this paper, we consider a Kermack-McKendrick epidemic model withnonlocal dispersal. We find that the existence and nonexistence oftraveling wave solutions are determined by the reproduction number.To prove the existence of nontrivial traveling wave solutions, weconstruct an invariant cone in a bounded domain with initialfunctions being defined on, and apply Schauder's fixed point theoremas well as limitingargument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, thenonexistence of traveling wave solutions is obtained by Laplace transform if the speed isless than the critical velocity.