February  2021, 26(2): 1197-1204. doi: 10.3934/dcdsb.2020159

On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model

1. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

School of Mathematical Sciences, Tiangong University, Tianjin 300387, China

* Corresponding author: Shengqiang Liu

Received  September 2019 Revised  February 2020 Published  May 2020

A recent paper [Y.-Y. Chen, J.-S. Guo, F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359] presented a discrete diffusive Kermack-McKendrick epidemic model, and the authors proved the existence of traveling wave solutions connecting the disease-free equilibrium to the endemic equilibrium. However, the boundary asymptotic behavior of the traveling waves converge to the endemic equilibrium at $ +\infty $ is still an open problem. In this paper, we investigate the above open problem and completely solve it by constructing suitable Lyapunov functional and employing Lebesgue dominated convergence theorem.

Citation: Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159
References:
[1]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  Google Scholar

[2]

M. Brucal-Hallare and E. V. Vleck, Traveling wavefronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Syst., 10 (2011), 921-959.  doi: 10.1137/100819461.  Google Scholar

[3]

Y.-Y. ChenJ.-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.  Google Scholar

[4]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012.  Google Scholar

[5]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.  Google Scholar

[6]

J. FangJ. Wei and X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. A-Math. Phys. Eng. Sci., 466 (2010), 1919-1934.  doi: 10.1098/rspa.2009.0577.  Google Scholar

[7]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.  Google Scholar

[8]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1.  Google Scholar

[9]

Y. LiW.-T. Li and G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pure Appl. Anal., 14 (2015), 1001-1022.  doi: 10.3934/cpaa.2015.14.1001.  Google Scholar

[10]

Y. LiW.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.  Google Scholar

[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.  Google Scholar

[12]

Z. Yang and G. Zhang, Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity, Sci. China Math., 61 (2018), 1789-1806.  doi: 10.1007/s11425-017-9175-2.  Google Scholar

[13]

L. ZhaoZ.-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.  Google Scholar

[14]

R. Zhang and S. Liu, Traveling waves for SVIR epidemic model with nonlocal dispersal, Math. Biosci. Eng., 16 (2019), 1654-1682.  doi: 10.3934/mbe.2019079.  Google Scholar

[15]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differential Equations, 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.  Google Scholar

show all references

References:
[1]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  Google Scholar

[2]

M. Brucal-Hallare and E. V. Vleck, Traveling wavefronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Syst., 10 (2011), 921-959.  doi: 10.1137/100819461.  Google Scholar

[3]

Y.-Y. ChenJ.-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.  Google Scholar

[4]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012.  Google Scholar

[5]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.  Google Scholar

[6]

J. FangJ. Wei and X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. A-Math. Phys. Eng. Sci., 466 (2010), 1919-1934.  doi: 10.1098/rspa.2009.0577.  Google Scholar

[7]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.  Google Scholar

[8]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1.  Google Scholar

[9]

Y. LiW.-T. Li and G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pure Appl. Anal., 14 (2015), 1001-1022.  doi: 10.3934/cpaa.2015.14.1001.  Google Scholar

[10]

Y. LiW.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.  Google Scholar

[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.  Google Scholar

[12]

Z. Yang and G. Zhang, Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity, Sci. China Math., 61 (2018), 1789-1806.  doi: 10.1007/s11425-017-9175-2.  Google Scholar

[13]

L. ZhaoZ.-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.  Google Scholar

[14]

R. Zhang and S. Liu, Traveling waves for SVIR epidemic model with nonlocal dispersal, Math. Biosci. Eng., 16 (2019), 1654-1682.  doi: 10.3934/mbe.2019079.  Google Scholar

[15]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differential Equations, 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.  Google Scholar

Figure 1.  The monotonicity of function $ g(x) $
[1]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[2]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[3]

Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020360

[4]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[5]

Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021007

[6]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017

[7]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357

[8]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[9]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[10]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[11]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378

[12]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047

[13]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

[14]

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  doi: 10.3934/dcdsb.2021028

[15]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[16]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[17]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[18]

Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021016

[19]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[20]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]