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
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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