Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 2

Stability and uniqueness of traveling waves of a non-local dispersal SIR epidemic model

Pages: 87 – 123

DOI: http://dx.doi.org/10.4310/DPDE.2017.v14.n2.a1

Authors

Yan Li (School of Mathematics and Statistics, Xidian University, Xi’an, Shanxi, China)

Wan-Tong Li (School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, China)

Guo-Bao Zhang (College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, China)

Abstract

This paper is mainly concerned with the exponential stability and uniqueness of traveling waves of a delayed nonlocal dispersal SIR epidemic model. We first prove the stability of traveling waves by using the weighted energy method, where the traveling waves are allowed to be non-monotone. Next we establish the exact asymptotic behavior of traveling waves at $- \infty$ by using Ikehara’s theorem. Then the uniqueness of traveling waves is obtained by the stability result. Finally, we discuss how the non-local dispersal affects the stability of traveling waves. The conclusion shows that the non-local dispersal slows down the convergence rate of the solution to the traveling waves.

Keywords

weighted energy method, traveling waves, exponential stability and uniqueness, nonlocal dispersal, the convergence rate

2010 Mathematics Subject Classification

Primary 37-xx, 70-xx, 76-xx, 92-xx. Secondary 34-xx, 35-xx, 80-xx, 82-xx.

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