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

Ran Zhang; Shengqiang Liu

doi:10.3934/dcdsb.2020159

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2021, Volume 26, Issue 2: 1197-1204. Doi: 10.3934/dcdsb.2020159

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

Ran Zhang^1, and
Shengqiang Liu^2, ,

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
^* Corresponding author: Shengqiang Liu

Received: August 31, 2019

Revised: January 31, 2020

Early access: May 2020

Published: February 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 37L60, 92D25; Secondary: 35C07.

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