Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

Xuefeng San; Yuan He

doi:10.3934/cpaa.2021106

Article Contents

2021, Volume 20, Issue 10: 3299-3318. Doi: 10.3934/cpaa.2021106 open access

This issue Previous Article Next Article Orbitally symmetric systems with applications to planar centers

Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

Xuefeng San and
Yuan He^,

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

^* Corresponding author
^* Corresponding author

Received: November 30, 2020

Revised: March 31, 2021

Early access: June 2021

Published: October 2021

The authors are grateful to the anonymous referees for their valuable comments and suggestions. The authors are also grateful to Professor Zhi-Cheng Wang for his insightful comments. This work is supported by NSF of China (11801241), the Fundamental Research Funds for the Central Universities (lzujbky-2017-165) and NSF of Gansu Province, China (1606RJZA069)

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Abstract

In this paper, we consider a two-group SIR epidemic model with bilinear incidence in a patchy environment. It is assumed that the infectious disease has a fixed latent period and spreads between two groups. Firstly, when the basic reproduction number $ \mathcal{R}_{0}>1 $ and speed $ c>c^{\ast} $, we prove that the system admits a nontrivial traveling wave solution, where $ c^{\ast} $ is the minimal wave speed. Next, when $ \mathcal{R}_{0}\leq1 $ and $ c>0 $, or $ \mathcal{R}_{0}>1 $ and $ c\in(0,c^{*}) $, we also show that there is no positive traveling wave solution, where $ k = 1,2 $. Finally, we discuss and simulate the dependence of the minimum speed $ c^{\ast} $ on the parameters.

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Mathematics Subject Classification: 34A33, 35C07, 35B40, 92D30.

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Figure 1. Relationship between $ c^{\ast} $ and $ D_{i} $ for $ i = 1,2 $. (a): $ D_{1} = x,\ D_{2} = y $, $ \beta_{11} = \beta_{22} = 0.08 $, $ \beta_{12} = \beta_{21} = 0.24 $, $ r_{1} = r_{2} = 1.1 $, $ \tau = 1 $ and $ \mu_{j} = e^{-\tau} $. (b): $ D_{i} = x,\ D_{j} = 1.1(i,j = 1,2) $ and $ i\neq j $, $ \beta_{11} = \beta_{22} = 0.08 $, $ \beta_{12} = \beta_{21} = 0.24 $, $ r_{1} = r_{2} = 1.1 $, $ \tau = 1 $ and $ \mu_{j} = e^{-\tau} $

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Figure 2. Relationship between $ c^{\ast} $ and $ \beta_{ij} $ for $ i = 1,2 $. (a): $ D_{1} = D_{2} = 1.2 $, $ \beta_{11} = x,\ \beta_{12} = y $, $ \beta_{21} = 0.24,\ \beta_{22} = 0.08 $, $ r_{1} = r_{2} = 1.1 $, $ \tau = 1 $ and $ \mu_{j} = e^{-\tau} $. (b): $ D_{1} = D_{2} = 1.2 $, $ \beta_{11} = x,\ \beta_{12} = \beta_{21} = 0.24,\ \beta_{22} = 0.08 $, $ r_{1} = r_{2} = 1.1 $, $ \tau = 1 $ and $ \mu_{j} = e^{-\tau} $. (c): $ D_{1} = D_{2} = 1.2 $, $ \beta_{11} = 0.08,\ \beta_{12} = 0.24 $, $ \beta_{21} = x,\ \beta_{22} = 0.08 $, $ r_{1} = r_{2} = 1.1 $, $ \tau = 1 $ and $ \mu_{j} = e^{-\tau} $

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Figure 3. Simulations of the dependence of $ c^{\ast} $ on $ \tau $ and $ r_{i} $ for $ i = 1,2 $. (a): $ D_{1} = D_{2} = 1.2 $, $ \beta_{11} = \beta_{22} = 0.08,\ \beta_{12} = \beta_{21} = 0.24 $, $ r_{1} = x,\ r_{2} = y $, $ \tau = 1 $ and $ \mu_{j} = e^{-\tau} $. (b): $ D_{1} = D_{2} = 1.2 $, $ \beta_{11} = \beta_{22} = 0.08,\ \beta_{12} = \beta_{21} = 0.24 $, $ r_{1} = x,\ r_{2} = 1.1 $, $ \tau = 1 $ and $ \mu_{j} = e^{-\tau} $.(c): $ D_{1} = D_{2} = 1.2 $, $ \beta_{11} = \beta_{22} = 0.08,\ \beta_{12} = \beta_{21} = 0.24 $, $ r_{1} = r_{2} = 1.1 $, $ \tau = x $ and $ \mu_{j} = e^{-\tau} $

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