Traveling waves and entire solutions for an epidemic model with asymmetric dispersal

Wan-Tong Li; Wen-Bing Xu; Li Zhang

doi:10.3934/dcds.2017107

Article Contents

2017, Volume 37, Issue 5: 2483-2512. Doi: 10.3934/dcds.2017107

This issue Previous Article Existence of solutions for a class of abstract neutral differential equations Next Article Population dynamical behavior of a two-predator one-prey stochastic model with time delay

Traveling waves and entire solutions for an epidemic model with asymmetric dispersal

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Author Bio: xuwb08@lzu.edu.cn(W.-B. Xu); zhangl13@lzu.edu.cn(L. Zhang)
Corresponding author
Corresponding author

Received: May 31, 2016

Revised: November 30, 2016

Published: May 2017

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Abstract

This paper is concerned with traveling waves and entire solutions of one epidemic model with asymmetric dispersal kernel function arising from the spread of an epidemic by oral-faecal transmission. The asymmetry of the kernel function will have an influence on two aspects: (ⅰ) the minimal wave speed of traveling wave fronts may be nonpositive, but we give a new restrictive condition on the kernel function to guarantee it is positive; (ⅱ) the two traveling wave solutions with the same speed spreading from right and left of $x$-axis may be different in shape, which further makes that the entire solutions with five or four parameters may be asymmetric and the entire solutions with three parameters increasing in $x$ may be different from those decreasing in $x$ in shape. As for traveling wave solutions, we get the existence, asymptotic behavior and uniqueness of the two traveling wave solutions spreading from right and left of $x$-axis, respectively. We further construct three new entire solutions with five, four or three parameters. Two comparison principles also be established.

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Mathematics Subject Classification: 35K57, 35R20, 92D25.

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