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

Hai-Feng Huo; Shi-Ke Hu; Hong Xiang; Hai-Feng Huo; Shi-Ke Hu; Hong Xiang

doi:10.3934/era.2020118

Electronic Research Archive

2021, Volume 29, Issue 3: 2325-2358. doi: 10.3934/era.2020118

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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

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.
- Self-protection,
- treatment,
- Schauder fixed point theorem,
- Lyapunov functional,
- two-sides Laplace transform
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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Abstract

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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