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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
| 1. | School of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
| 2. | School of Mathematical Sciences, Tiangong University, Tianjin 300387, China |
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 and Continuous Dynamical Systems - B,
2021, 26
(2)
: 1197-1204.
doi: 10.3934/dcdsb.2020159
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Traveling wavefronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Syst., 10 (2011), 921-959.
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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. |
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A. Ducrot and P. Magal,
Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911.
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Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. A-Math. Phys. Eng. Sci., 466 (2010), 1919-1934.
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Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.
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Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.
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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. |
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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. |
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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.
doi: 10.1016/j.jmaa.2019.03.029. |
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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. |
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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. |
| [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. |
| [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. |
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. |
| [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. |
| [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] |
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. |
| [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. |
| [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. |
| [7] |
R. Kapral,
Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.
doi: 10.1007/BF01192578. |
| [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. |
| [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. |
| [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. |
| [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] |
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. |
| [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. |
| [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. |
| [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. |
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