Dynamics of the discrete-time man-environment-man epidemic model with a free boundary

Jian Feng; Meng Zhao; Jian Feng; Meng Zhao

doi:10.3934/nhm.2025049

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1145-1174. doi: 10.3934/nhm.2025049

Previous Article Next Article

Research article Special Issues

Dynamics of the discrete-time man-environment-man epidemic model with a free boundary

Jian Feng ,
Meng Zhao ^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 09 April 2025 Revised: 06 October 2025 Accepted: 22 October 2025 Published: 27 October 2025

In this paper, we investigate the discrete-time man-environment-man epidemic model with a free boundary, which can be viewed as a time-discrete version of the free boundary model studied by Ahn, Beak, and Lin. First, applying the properties of the principal eigenvalue of the corresponding eigenvalue problem, we obtain the global dynamics of the corresponding fixed boundary problem. Then, we solve the problem step by step and establish the well-posedness of the solution. Moreover, we provide some sufficient conditions for the diseases spreading and vanishing by using the modified comparison principle. Finally, we give the long-time behavior of the solution by making use of the above results about the corresponding fixed boundary problem.
- discrete-time,
- epidemic model,
- free boundary,
- spreading-vanishing dichotomy
Citation: Jian Feng, Meng Zhao. Dynamics of the discrete-time man-environment-man epidemic model with a free boundary[J]. Networks and Heterogeneous Media, 2025, 20(4): 1145-1174. doi: 10.3934/nhm.2025049

Related Papers:

Abstract

In this paper, we investigate the discrete-time man-environment-man epidemic model with a free boundary, which can be viewed as a time-discrete version of the free boundary model studied by Ahn, Beak, and Lin. First, applying the properties of the principal eigenvalue of the corresponding eigenvalue problem, we obtain the global dynamics of the corresponding fixed boundary problem. Then, we solve the problem step by step and establish the well-posedness of the solution. Moreover, we provide some sufficient conditions for the diseases spreading and vanishing by using the modified comparison principle. Finally, we give the long-time behavior of the solution by making use of the above results about the corresponding fixed boundary problem.

References

[1]	V. Capasso, S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. d'pidemiol. Sante Publique, 27 (1979), 121–132.
[2]	V. Capasso, L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981), 173–184. https://doi.org/10.1007/BF00275212 doi: 10.1007/BF00275212
[3]	X. Q. Zhao, W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117–1128. https://doi.org/10.3934/dcdsb.2004.4.1117 doi: 10.3934/dcdsb.2004.4.1117
[4]	I. Ahn, S. Baek, Z. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Modell., 40 (2016), 7082–7101. https://doi.org/10.1016/j.apm.2016.02.038 doi: 10.1016/j.apm.2016.02.038
[5]	Q. Chen, F. Li, Z. Teng, F. Wang, Global dynamics and asymptotic spreading speeds for a partially degenerate epidemic model with time delay and free boundaries, J. Dyn. Differ. Equ., 34 (2022), 1209–1236. https://doi.org/10.1007/s10884-020-09934-4 doi: 10.1007/s10884-020-09934-4
[6]	R. Wang, Y. Du, Long-time dynamics of a diffusive epidemic model with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2201–2238. https://doi.org/10.3934/dcdsb.2020360 doi: 10.3934/dcdsb.2020360
[7]	T. Y. Chang, Y. Du, Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries, Electron. Res. Arch., 30 (2022), 289–313. https://doi.org/10.3934/era.2022016 doi: 10.3934/era.2022016
[8]	X. Li, L. Li, M. Wang, A free boundary problem of an epidemic model with nonlocal diffusion and nonlocal infective rate, Commun. Pure Appl. Anal., 24 (2025), 36–59. https://doi.org/10.3934/cpaa.2024076 doi: 10.3934/cpaa.2024076
[9]	G. Bunting, Y. Du, K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks Heterog. Media, 7 (2012), 583–603. https://doi.org/10.3934/nhm.2012.7.583 doi: 10.3934/nhm.2012.7.583
[10]	Y. Du, Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405. https://doi.org/10.1137/090771089 doi: 10.1137/090771089
[11]	J. F. Cao, W. T. Li, F. Y. Yang, Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 247–266. https://doi.org/10.3934/dcdsb.2017013 doi: 10.3934/dcdsb.2017013
[12]	K. I. Kim, Z. Lin, Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992–2001. https://doi.org/10.1016/j.nonrwa.2013.02.003 doi: 10.1016/j.nonrwa.2013.02.003
[13]	L. Li, S. Liu, M. Wang, A viral propagation model with a nonlinear infection rate and free boundaries, Sci. China Math., 64 (2021), 1971–1992. https://doi.org/10.1007/s11425-020-1680-0 doi: 10.1007/s11425-020-1680-0
[14]	Z. Lin, H. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381–1409. https://doi.org/10.1007/s00285-017-1124-7 doi: 10.1007/s00285-017-1124-7
[15]	M. Zhao, Dynamics of a reaction-diffusion waterborne pathogen model with free boundaries, Nonlinear Anal. Real World Appl., 77 (2024), 104043. https://doi.org/10.1016/j.nonrwa.2023.104043 doi: 10.1016/j.nonrwa.2023.104043
[16]	Y. Du, Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105–3132. https://doi.org/10.3934/dcdsb.2014.19.3105 doi: 10.3934/dcdsb.2014.19.3105
[17]	Y. Du, M. Wang, M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253–287. https://doi.org/10.1016/j.matpur.2016.06.005 doi: 10.1016/j.matpur.2016.06.005
[18]	Y. Tang, B. Dai, Z. Li, Dynamics of a Lotka-Volterra weak competition model with time delays and free boundaries, Z. Angew. Math. Phys., 73 (2022), 143. https://doi.org/10.1007/s00033-022-01788-8 doi: 10.1007/s00033-022-01788-8
[19]	M. Wang, On some free boundary problems of the prey-predator model, J. Differ. Equ., 256 (2014), 3365–3394. https://doi.org/10.1016/j.jde.2014.02.013 doi: 10.1016/j.jde.2014.02.013
[20]	H. Zhang, L. Li, M. Wang, Free boundary problems for the local-nonlocal diffusive model with different moving parameters, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 474–498. https://doi.org/10.3934/dcdsb.2022085 doi: 10.3934/dcdsb.2022085
[21]	F. Lutscher, Integrodifference Equations in Spatial Ecology, Switzerland: Springer International Publishing, 2019. https://doi.org/10.1007/978-3-030-29294-2
[22]	L. J. S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83–105. https://doi.org/10.1016/0025-5564(94)90025-6
[23]	L. J. S. Allen, M. A. Jones, C. F. Martin, A discrete-time model with vaccination for a measles epidemic, Math. Biosci., 105 (1991), 111–131. https://doi.org/10.1016/0025-5564(91)90051-J doi: 10.1016/0025-5564(91)90051-J
[24]	M. A. Lewis, J. Rencławowicz, P. van den Driessche, M. Wonham, A comparison of continuous and discrete-time West Nile virus models, Bull. Math. Biol., 68 (2006), 491–509. https://doi.org/10.1007/s11538-005-9039-7 doi: 10.1007/s11538-005-9039-7
[25]	H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353–396. https://doi.org/10.1137/0513028 doi: 10.1137/0513028
[26]	P. A. Naik, Z. Eskandari, A. Madzvamuse, Z. Avazzadeh, J. Zu, Complex dynamics of a discrete-time seasonally forced SIR epidemic model, Math. Methods Appl. Sci., 46 (2023), 7045–7059. https://doi.org/10.1002/mma.8955 doi: 10.1002/mma.8955
[27]	Y. Li, Z. Guo, J. Liu, Longtime behavior for solutions to a temporally discrete diffusion equation with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 29 (2024), 4361–4379. https://doi.org/10.3934/dcdsb.2024046 doi: 10.3934/dcdsb.2024046
[28]	Q. Bian, W. Zhang, Z. Yu, Temporally discrete three-species Lotka-Volterra competitive systems with time delays, Taiwan. J. Math., 20 (2016), 49–75. https://doi.org/10.11650/tjm.20.2016.5597 doi: 10.11650/tjm.20.2016.5597
[29]	H. Guo, Z. Guo, Y. Li, Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain, Discrete Contin. Dyn. Syst. Ser. B, 29 (2024), 198–213. https://doi.org/10.3934/dcdsb.2023093 doi: 10.3934/dcdsb.2023093
[30]	G. Lin, W. T. Li, Traveling wavefronts in temporally discrete reaction-diffusion equations with delay, Nonlinear Anal. Real World Appl., 9 (2008), 197–205. https://doi.org/10.1016/j.nonrwa.2006.11.003 doi: 10.1016/j.nonrwa.2006.11.003
[31]	J. Wang, J. F. Cao, The spreading frontiers in partially degenerate reaction–diffusion systems, Nonlinear Anal. Theory Methods Appl., 122 (2015), 215–238. https://doi.org/10.1016/j.na.2015.04.003 doi: 10.1016/j.na.2015.04.003
[32]	W. Wang, X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652–1673. https://doi.org/10.1137/120872942 doi: 10.1137/120872942
[33]	Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Vol. 1: Maximum Principles and Applications, World Scientific, Hackensack, NJ, 2006. https://doi.org/10.1142/5999
[34]	X. Q. Zhao, Dynamical Systems in Population Biology, New York: Springer, 2003.
[35]	Y. Yuan, Z. Guo, Monotone methods and stability results for nonlocal reaction-diffusion equations with time delay, J. Appl. Anal. Comput., 8 (2018), 1342–1368. https://doi.org/10.11948/2018.1342 doi: 10.11948/2018.1342

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)