# American Institute of Mathematical Sciences

September  2013, 18(7): 1969-1993. doi: 10.3934/dcdsb.2013.18.1969

## Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China 2 School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received  March 2012 Revised  March 2013 Published  May 2013

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.
Citation: Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969
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