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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

Xuefeng San and  Yuan He ,

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.

Keywords: SIR epidemic model, bilinear incidence, a patchy environment, traveling wave solutions, latent period, global interaction.
Mathematics Subject Classification: 34A33, 35C07, 35B40, 92D30.
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 and Applied Analysis, 2021, 20 (10) : 3299-3318. doi: 10.3934/cpaa.2021106
References:
[1]

X. Chen, S. C. Fu and J. S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824.

[2]

X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.

[3]

Y. Y. Chen, J. S. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359.  doi: 10.1088/1361-6544/aa6b0a.

[4]

Y. Y. Chen, J. S. Guo and C. H. Yao, Traveling wave solutions for a continuous and discrete diffusive predator-prey model, J. Math. Anal. Appl., 445 (2017), 212-239.  doi: 10.1016/j.jmaa.2016.07.071.

[5]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.

[6]

A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8.

[7]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.

[8]

S. C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069.

[9]

S. C. Fu, J. S. Guo and C. C. Wu, Traveling wave solutions for a discrete diffusive epidemic model, J. Nonlinear Convex Anal., 17 (2016), 1739-1751. 

[10]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.

[11]

X. F. San and Z. C. Wang, Traveling waves for a two-group epidemic model with latent period in a patchy environment, J. Math. Anal. Appl., 475 (2019), 1502-1531.  doi: 10.1016/j.jmaa.2019.03.029.

[12]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[13]

Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.

[14]

P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.  doi: 10.1093/imamat/68.4.409.

[15]

C. C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differ. Equ., 262 (2017), 272-282.  doi: 10.1016/j.jde.2016.09.022.

[16]

F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.

[17]

F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.

[18]

R. Zhang and S. Liu, On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1197-1204.  doi: 10.3934/dcdsb.2020159.

[19]

L. Zhao, Z. C. Wang and S. Ruan, Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325.  doi: 10.1088/1361-6544/aa59ae.

[20]

L. Zhao, Z. C. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915.  doi: 10.1007/s00285-018-1227-9.

[21]

J. Zhou, L. Song and J. Wei, Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay, J. Differ. Equ., 268 (2020), 4491-4524.  doi: 10.1016/j.jde.2019.10.034.

[22]

J. Zhou, L. Song, J. Wei and H. Xu, Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 476 (2019), 522-538.  doi: 10.1016/j.jmaa.2019.03.066.

show all references

References:
[1]

X. Chen, S. C. Fu and J. S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824.

[2]

X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.

[3]

Y. Y. Chen, J. S. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359.  doi: 10.1088/1361-6544/aa6b0a.

[4]

Y. Y. Chen, J. S. Guo and C. H. Yao, Traveling wave solutions for a continuous and discrete diffusive predator-prey model, J. Math. Anal. Appl., 445 (2017), 212-239.  doi: 10.1016/j.jmaa.2016.07.071.

[5]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.

[6]

A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8.

[7]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.

[8]

S. C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069.

[9]

S. C. Fu, J. S. Guo and C. C. Wu, Traveling wave solutions for a discrete diffusive epidemic model, J. Nonlinear Convex Anal., 17 (2016), 1739-1751. 

[10]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.

[11]

X. F. San and Z. C. Wang, Traveling waves for a two-group epidemic model with latent period in a patchy environment, J. Math. Anal. Appl., 475 (2019), 1502-1531.  doi: 10.1016/j.jmaa.2019.03.029.

[12]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[13]

Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.

[14]

P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.  doi: 10.1093/imamat/68.4.409.

[15]

C. C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differ. Equ., 262 (2017), 272-282.  doi: 10.1016/j.jde.2016.09.022.

[16]

F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.

[17]

F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.

[18]

R. Zhang and S. Liu, On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1197-1204.  doi: 10.3934/dcdsb.2020159.

[19]

L. Zhao, Z. C. Wang and S. Ruan, Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325.  doi: 10.1088/1361-6544/aa59ae.

[20]

L. Zhao, Z. C. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915.  doi: 10.1007/s00285-018-1227-9.

[21]

J. Zhou, L. Song and J. Wei, Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay, J. Differ. Equ., 268 (2020), 4491-4524.  doi: 10.1016/j.jde.2019.10.034.

[22]

J. Zhou, L. Song, J. Wei and H. Xu, Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 476 (2019), 522-538.  doi: 10.1016/j.jmaa.2019.03.066.

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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