# American Institute of Mathematical Sciences

October  2021, 20(10): 3299-3318. doi: 10.3934/cpaa.2021106

## 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 Published  October 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 & Applied Analysis, 2021, 20 (10) : 3299-3318. doi: 10.3934/cpaa.2021106
##### References:

show all references

##### 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 $\mu_{j} = e^{-\tau}$
 [1] Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059 [2] Chengxia Lei, Fujun Li, Jiang Liu. Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4499-4517. doi: 10.3934/dcdsb.2018173 [3] C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837 [4] Bin-Guo Wang, Wan-Tong Li, Lizhong Qiang. An almost periodic epidemic model in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 271-289. doi: 10.3934/dcdsb.2016.21.271 [5] Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291 [6] Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291 [7] Yu Yang, Jinling Zhou, Cheng-Hsiung Hsu. Critical traveling wave solutions for a vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021087 [8] Wan-Tong Li, Guo Lin, Cong Ma, Fei-Ying Yang. Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 467-484. doi: 10.3934/dcdsb.2014.19.467 [9] Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1001-1022. doi: 10.3934/cpaa.2015.14.1001 [10] Yang Yang, Yun-Rui Yang, Xin-Jun Jiao. Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence. Electronic Research Archive, 2020, 28 (1) : 1-13. doi: 10.3934/era.2020001 [11] Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1797-1809. doi: 10.3934/dcdsb.2021028 [12] Mahin Salmani, P. van den Driessche. A model for disease transmission in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 185-202. doi: 10.3934/dcdsb.2006.6.185 [13] Yu Yang, Dongmei Xiao. Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 195-211. doi: 10.3934/dcdsb.2010.13.195 [14] Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867 [15] Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 61-74. doi: 10.3934/dcdsb.2011.15.61 [16] Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207 [17] Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082 [18] Daozhou Gao, Yijun Lou, Shigui Ruan. A periodic Ross-Macdonald model in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3133-3145. doi: 10.3934/dcdsb.2014.19.3133 [19] Xia Wang, Shengqiang Liu. Global properties of a delayed SIR epidemic model with multiple parallel infectious stages. Mathematical Biosciences & Engineering, 2012, 9 (3) : 685-695. doi: 10.3934/mbe.2012.9.685 [20] Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971

2020 Impact Factor: 1.916