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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

Ran Zhang 1, and  Shengqiang Liu 2,,

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.

Keywords: Lyapunov functional, Traveling wave solution, Asymptotic behavior, Diffusive epidemic model, Lattice dynamical system.
Mathematics Subject Classification: Primary: 37L60, 92D25; Secondary: 35C07.
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. 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.  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. Fang, J. 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. Li, W.-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. Li, W.-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. 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.  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. Zhang, W. 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. 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.  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. Fang, J. 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. Li, W.-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. Li, W.-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. 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.  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. Zhang, W. 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) $
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