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

Yoichi Enatsu; Emiko Ishiwata; Takeo Ushijima

doi:10.3934/dcdss.2020387

Article Contents

2021, Volume 14, Issue 3: 835-850. Doi: 10.3934/dcdss.2020387

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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
^* Corresponding author: Takeo Ushijima

Received: January 2019

Revised: March 2020

Early access: June 6, 2020

Published online: June 6, 2020

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Abstract

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.

Keywords:

Traveling wave,
free boundary,
epidemic model,
spreading phenomena,
reaction diffusion system of prey-predator type.

Mathematics Subject Classification: Primary: 35C07, 35R35; Secondary: 92D30.

Citation:

Full Text(HTML)

Figure 1.1. Parameter regions for existence and non existence of traveling wave ($ d = 0 $): (left) $ \gamma \lambda $ plane, (right) $ \gamma a $ plane.

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Figure 2.1. $ uv $ phase plane for (2.5), $ a>a^* $ (left), $ a < a^* $ (right).

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Figure 3.1. 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) $.

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Figure 3.2. 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.

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Figure 3.3. 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.

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Figure 3.4. 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 $.

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Figure 3.5. 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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