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March  2021, 14(3): 835-850. doi: 10.3934/dcdss.2020387

Traveling wave solution for a diffusive simple epidemic model with a free boundary

 1 Department of Applied Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan 2 Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba 278-8510, Japan

* Corresponding author: Takeo Ushijima

Received  January 2019 Revised  March 2020 Published  June 2020

In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient $d$ of susceptible population is zero and the basic reproduction number is greater than 1. We obtained a curve in the parameter plane which is the boundary between the regions of existence and nonexistence of traveling wave. We numerically observed that in the region where the traveling wave exists the disease successfully propagate like traveling wave but in the region of no traveling wave disease stops to invade. We also numerically observed that as $d$ increases the speed of propagation slows down and the parameter region of propagation narrows down.

Citation: Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
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Parameter regions for existence and non existence of traveling wave ($d = 0$): (left) $\gamma \lambda$ plane, (right) $\gamma a$ plane
$uv$ phase plane for (2.5), $a>a^*$ (left), $a < a^*$ (right)
Numerical shooting in $uv$ phase plane (left), Profile of the traveling wave (right) $d = 0, \gamma = 0.3, a = 0.05 (\lambda \sim 0.05128)$
Evolution of the solutions in spreading case ($\gamma = 0.3$): Left column; $\lambda = 0.054159$, $T = 100$, ① $t = T$, ② $t = 2T$, ③ $t = 3T$, ④ $t = 4T$, ⑤ $t = 5T$, (top left) $d = 0$, (middle left) $d = 0.1$, (bottom left) $d = 1$. Right column; (top right) Profile of traveling wave $\lambda = 0.054159$, $d = 0$, (middle right) $\lambda = 0.254297$, $d = 0$, support of $I_0$ is the origin
Evolution of the solutions in vanishing case ($\gamma = 0.3$, $d = 0$): (left) $\lambda = 0.027$, (right) $\lambda = 0.281066$, support of $I_0$ is fairly large
Estimated speed of propagation: $\Box\ d = 0, \bigcirc\ 0.1, \ast\ 1$, (left) speed of propagation versus $\lambda$, $\gamma = 0.5$, (right) speed of propagation versus $\gamma$, $\lambda = 0.4$
Vanishing versus spreading diagram: $\times$ vanishing, $+$ spreading, horizontal axis $\gamma = 1/R_0$, vertical axis $\lambda = 1/\mu$, (top left) $d = 0$, (top right) $d = 0.1$, (bottom) $d = 1$
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