### Electronic Research Archive

2021, Issue 3: 2325-2358. doi: 10.3934/era.2020118

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

• Received: 01 August 2020 Revised: 01 October 2020 Published: 24 November 2020
• Primary: 35C07, 35B40, 35K57; Secondary: 92D30

• 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[J]. Electronic Research Archive, 2021, 29(3): 2325-2358. doi: 10.3934/era.2020118

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• 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.

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