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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 |
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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Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.
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Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.
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S. C. Fu, J. S. Guo and C. C. Wu,
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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.
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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. |



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