# American Institute of Mathematical Sciences

## On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model

 1 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 School of Mathematical Sciences, Tiangong University, Tianjin 300387, China

* Corresponding author: Shengqiang Liu

Received  September 2019 Revised  February 2020 Published  May 2020

A recent paper [Y.-Y. Chen, J.-S. Guo, F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359] presented a discrete diffusive Kermack-McKendrick epidemic model, and the authors proved the existence of traveling wave solutions connecting the disease-free equilibrium to the endemic equilibrium. However, the boundary asymptotic behavior of the traveling waves converge to the endemic equilibrium at $+\infty$ is still an open problem. In this paper, we investigate the above open problem and completely solve it by constructing suitable Lyapunov functional and employing Lebesgue dominated convergence theorem.

Citation: Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020159
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The monotonicity of function $g(x)$