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Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure
doi: 10.3934/cpaa.2021106
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## Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

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

* Corresponding author

Received  December 2020 Revised  April 2021 Early access June 2021

Fund Project: 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)

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.

Citation: Xuefeng San, Yuan He. Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021106
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##### References:
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 $\mu_{j} = e^{-\tau}$
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 $\mu_{j} = e^{-\tau}$
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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