Spreading speed for an epidemic model with time delay in a patchy environment
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Abstract:
In this paper, we study the spreading speed for an epidemic model with time delay in a patchy environment. By employing the auxiliary equation and analyzing the upper-lower bounds of $s$-component, the spreading speed is obtained.References
- D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 5–49. MR 0427837
- Feng Cao and Wenxian Shen, Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media, Discrete Contin. Dyn. Syst. 37 (2017), no. 9, 4697–4727. MR 3661816, DOI 10.3934/dcds.2017202
- Arnaud Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations 260 (2016), no. 12, 8316–8357. MR 3482685, DOI 10.1016/j.jde.2016.02.023
- Arnaud Ducrot, Thomas Giletti, and Hiroshi Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations 58 (2019), no. 4, Paper No. 137, 34. MR 3984097, DOI 10.1007/s00526-019-1576-2
- Stephen P. Ellner, Akira Sasaki, Yoshihiro Haraguchi, and Hirotsugu Matsuda, Speed of invasion in lattice population models: pair-edge approximation, J. Math. Biol. 36 (1998), no. 5, 469–484. MR 1625099, DOI 10.1007/s002850050109
- Jian Fang, Junjie Wei, and Xiao-Qiang Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), no. 2119, 1919–1934. MR 2652726, DOI 10.1098/rspa.2009.0577
- Bingtuan Li, Hans F. Weinberger, and Mark A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci. 196 (2005), no. 1, 82–98. MR 2156610, DOI 10.1016/j.mbs.2005.03.008
- Xing Liang and Xiao-Qiang Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math. 60 (2007), no. 1, 1–40. MR 2270161, DOI 10.1002/cpa.20154
- Xing Liang and Xiao-Qiang Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal. 259 (2010), no. 4, 857–903. MR 2652175, DOI 10.1016/j.jfa.2010.04.018
- Xing Liang and Tao Zhou, Spreading speeds of KPP-type lattice systems in heterogeneous media, Commun. Contemp. Math. 22 (2020), no. 1, 1850083, 42. MR 4064907, DOI 10.1142/S0219199718500839
- Shiwang Ma, Peixuan Weng, and Xingfu Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal. 65 (2006), no. 10, 1858–1890. MR 2258473, DOI 10.1016/j.na.2005.10.042
- D. Smets and M. Willem, Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal. 149 (1997), no. 1, 266–275. MR 1471108, DOI 10.1006/jfan.1996.3121
- Xue-Feng San and Zhi-Cheng Wang, Traveling waves for a two-group epidemic model with latent period in a patchy environment, J. Math. Anal. Appl. 475 (2019), no. 2, 1502–1531. MR 3944385, DOI 10.1016/j.jmaa.2019.03.029
- Horst R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol. 8 (1979), no. 2, 173–187. MR 648984, DOI 10.1007/BF00279720
- P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48. John A. Jacquez memorial volume. MR 1950747, DOI 10.1016/S0025-5564(02)00108-6
- Zongyi Wang, Traveling waves and spreading speed on a lattice model with age structure, Electron. J. Differential Equations (2012), No. 161, 15. MR 2991395
- H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982), no. 3, 353–396. MR 653463, DOI 10.1137/0513028
- Hans F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), no. 6, 511–548. MR 1943224, DOI 10.1007/s00285-002-0169-3
- Peixuan Weng, Huaxiong Huang, and Jianhong Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math. 68 (2003), no. 4, 409–439. MR 1988153, DOI 10.1093/imamat/68.4.409
- Peixuan Weng and Yanling Tian, Asymptotic speed of propagation and traveling wave solutions for a lattice integral equation, Nonlinear Anal. 70 (2009), no. 1, 159–175. MR 2468226, DOI 10.1016/j.na.2007.11.043
- Fuzhen Wu, Asymptotic speed of spreading in a delay lattice differential equation without quasimonotonicity, Electron. J. Differential Equations (2014), No. 213, 10. MR 3273096
- Zhiting Xu and Peixuan Weng, Asymptotic speed of propagation and traveling wavefronts for a lattice vector disease model, Nonlinear Anal. Real World Appl. 12 (2011), no. 6, 3621–3641. MR 2832997, DOI 10.1016/j.nonrwa.2011.06.020
- Taishan Yi, Yuming Chen, and Jianhong Wu, Unimodal dynamical systems: comparison principles, spreading speeds and travelling waves, J. Differential Equations 254 (2013), no. 8, 3538–3572. MR 3020887, DOI 10.1016/j.jde.2013.01.031
- Zhi-Xian Yu, Weiguo Zhang, and Xiaoming Wang, Spreading speeds and travelling waves for non-monotone time-delayed 2D lattice systems, Math. Comput. Modelling 58 (2013), no. 7-8, 1510–1521. MR 3143380, DOI 10.1016/j.mcm.2013.06.009
- Kate Fang Zhang and Xiao-Qiang Zhao, Spreading speed and travelling waves for a spatially discrete SIS epidemic model, Nonlinearity 21 (2008), no. 1, 97–112. MR 2376319, DOI 10.1088/0951-7715/21/1/005
Additional Information
- Xue-Feng San
- Affiliation: School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China
- Jian-Wen Sun
- Affiliation: School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China
- MR Author ID: 939090
- ORCID: 0000-0002-8384-553X
- Email: jianwensun@lzu.edu.cn
- Received by editor(s): June 12, 2020
- Published electronically: November 30, 2020
- Additional Notes: This work was supported by NSF of China (11401277,11731005).
The second author is the corresponding author. - Communicated by: Wenxian Shen
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 739-746
- MSC (2010): Primary 34A33, 35C07, 35B40
- DOI: https://doi.org/10.1090/proc/15303
- MathSciNet review: 4198079