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doi: 10.3934/era.2020118

## Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment

 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

* Corresponding author: Hai-Feng Huo

Received  August 2020 Revised  October 2020 Published  November 2020

Fund Project: This work is supported by the National Natural Science Foundation of China (11861044 and 11661050), and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology

A reaction-diffusion SEIR model, including the self-protection for susceptible individuals, treatments for infectious individuals and constant recruitment, is introduced. The existence of traveling wave solution, which is determined by the basic reproduction number $R_0$ and wave speed $c,$ is firstly proved as $R_0>1$ and $c\geq c^*$ via the Schauder fixed point theorem, where $c^*$ is minimal wave speed. Asymptotic behavior of traveling wave solution at infinity is also proved by applying the Lyapunov functional. Furthermore, when $R_0\leq1$ or $R_0>1$ with $c\in(0,\ c^*),$ we derive the non-existence of traveling wave solution with utilizing two-sides Laplace transform. We take advantage of numerical simulations to indicate the existence of traveling wave, and show that self-protection and treatment can reduce the spread speed at last.

Citation: Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, doi: 10.3934/era.2020118
##### References:

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##### References:
The numerical simulations of existence for traveling wave solution of system (2)
Cross section curve of traveling wave solution for system (2) as $t = 200.$
Show the effects of self-protection $\sigma$ and treatment $\theta$ on minimal spread speed $c^*,$ where $\sigma$ and $\theta$ are taken from $0.05$ to $1.$
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