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In this paper, we consider a Kermack-McKendrick epidemic model with nonlocal dispersal. We find that the existence and nonexistence of traveling wave solutions are determined by the reproduction number. To prove the existence of nontrivial traveling wave solutions, we construct an invariant cone in a bounded domain with initial functions being defined on, and apply Schauder's fixed point theorem as well as limiting argument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, the nonexistence of traveling wave solutions is obtained by Laplace transform if the speed is less than the critical velocity.2010 Mathematics Subject Classification. 35K57, 37C65, 92D30.

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This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model with standard incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding ordinary differential model and the minimal wave speed. These threshold dynamics are proved by constructing an invariant cone and applying Schauder's fixed point theorem on this cone and the Laplace transform. The main difficulties are the lack of an occurrence of a regularizing effect and the loss of the order-preserving property of this model. Introduction.The current paper is concerned with the existence and nonexistence of traveling wave solutions of the following nonlocal dispersal SIR model with standard incidence:where S, I and R denote the sizes of the susceptible, infected and removal individuals, respectively. The infection rate β and the removal rate γ are positive numbers. d i > 0 (i = 1, 2, 3) are dispersal rates for the susceptible, infected and removal individuals, respectively.