Electron. J. Differential Equations, Vol. 2017 (2017), No. 160, pp. 1-19.

Asymptotic behavior of traveling waves for a nonlocal epidemic model with delay

Haiqin Zhao

In this article we study the traveling wave solutions of a monostable nonlocal reaction-diffusion system with delay arising from the spread of an epidemic by oral-faecal transmission. From [23], there exists a minimal wave speed $c_*>0$ such that a traveling wave solution exists if and only if the wave speed is above $c_*>0$. In this article, we first establish the exact asymptotic behavior of the traveling waves at $\pm\infty$. Then, we construct some annihilating-front entire solutions which behave like a traveling wave front propagating from the left side (or the right side) on the x-axis or two traveling wave fronts propagating from both sides on the x-axis as $t\to-\infty$ and converge to the unique positive equilibrium as $t\to+\infty$. From the viewpoint of epidemiology, these results provide some new spread ways of the epidemic.

Submitted December 20, 2016. Published June 30, 2017.
Math Subject Classifications: 35K57, 35B05, 35B40, 92D30.
Key Words: Traveling wave front; epidemic model; reaction-diffusion system; monostable nonlinearity.

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Haiqin Zhao
School of Mathematics and Statistics
Xidian University
Xi'an, Shaanxi 710071, China
email: zhaohaiqin@xidian.edu.cn
Phone: 0086-02981891379

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