# American Institute of Mathematical Sciences

• Previous Article
A new numerical method for level set motion in normal direction used in optical flow estimation
• DCDS-S Home
• This Issue
• Next Article
Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction
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  March 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
##### References:

show all references

##### References:
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) $d = 1$
 [1] Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057 [2] Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. doi: 10.3934/dcdsb.2019245 [3] Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039 [4] Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138 [5] Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536 [6] Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267 [7] Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597 [8] Safia Slimani, Paul Raynaud de Fitte, Islam Boussaada. Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5003-5039. doi: 10.3934/dcdsb.2019042 [9] Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2189-2219. doi: 10.3934/dcdsb.2021129 [10] Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737 [11] Shanbing Li, Jianhua Wu. Effect of cross-diffusion in the diffusion prey-predator model with a protection zone. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1539-1558. doi: 10.3934/dcds.2017063 [12] Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133 [13] Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129 [14] Shuping Li, Weinian Zhang. Bifurcations of a discrete prey-predator model with Holling type II functional response. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 159-176. doi: 10.3934/dcdsb.2010.14.159 [15] Shu Li, Zhenzhen Li, Binxiang Dai. Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022025 [16] R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423 [17] Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180 [18] Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244 [19] Meng Zhao, Wantong Li, Yihong Du. The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4599-4620. doi: 10.3934/cpaa.2020208 [20] Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161

2020 Impact Factor: 2.425