Stationary distribution and extinction of a stochastic SEI epidemic model with logistic growth and nonlinear perturbation

Zeyu Xu; Liang Wang; Zeyu Xu; Liang Wang

doi:10.3934/math.20251254

AIMS Mathematics

2025, Volume 10, Issue 12: 28488-28513. doi: 10.3934/math.20251254

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Stationary distribution and extinction of a stochastic SEI epidemic model with logistic growth and nonlinear perturbation

Zeyu Xu ,
Liang Wang ^,

School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China

Received: 24 September 2025 Revised: 11 November 2025 Accepted: 24 November 2025 Published: 03 December 2025
MSC : 92D25, 92D30, 60H10

In this paper, we proposed and studied a stochastic SEI (Susceptible-Exposed-Infectious) epidemic model with logistic growth and nonlinear perturbation. By constructing a series of suitable stochastic Lyapunov functions, we derived sufficient conditions for the existence of a unique ergodic stationary distribution. Furthermore, we obtained criteria which ensured the disease approached extinction at an exponential rate. At the same time, under these criteria, the distribution of susceptible individuals converged weakly to a unique invariant probability measure. The numerical simulations supported the theoretical results.
- stochastic SEI epidemic model,
- logistic growth,
- nonlinear perturbation,
- ergodicity,
- extinction
Citation: Zeyu Xu, Liang Wang. Stationary distribution and extinction of a stochastic SEI epidemic model with logistic growth and nonlinear perturbation[J]. AIMS Mathematics, 2025, 10(12): 28488-28513. doi: 10.3934/math.20251254

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Abstract

In this paper, we proposed and studied a stochastic SEI (Susceptible-Exposed-Infectious) epidemic model with logistic growth and nonlinear perturbation. By constructing a series of suitable stochastic Lyapunov functions, we derived sufficient conditions for the existence of a unique ergodic stationary distribution. Furthermore, we obtained criteria which ensured the disease approached extinction at an exponential rate. At the same time, under these criteria, the distribution of susceptible individuals converged weakly to a unique invariant probability measure. The numerical simulations supported the theoretical results.

References

[1]	W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics–Ⅰ, B. Math. Biol., 53 (1991), 33–55. https://doi.org/10.1007/bf02464423 doi: 10.1007/bf02464423
[2]	M. De la Sen, S. A. Quesada, A. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 270 (2015), 953–976. https://doi.org/10.1016/j.amc.2015.08.099 doi: 10.1016/j.amc.2015.08.099
[3]	E. C. Gabrick, P. R. Protachevicz, A. M. Batista, K. C. Iarosz, S. L. T. de Souza, A. C. L. Almeida, et al., Effect of two vaccine doses in the SEIR epidemic model using a stochastic cellular automaton, Physica A, 597 (2022), 127258. https://doi.org/10.1016/j.physa.2022.127258 doi: 10.1016/j.physa.2022.127258
[4]	M. Mugnaine, E. C. Gabrick, P. R. Protachevicz, K. C. Iarosz, S. L. T. de Souza, A. C. L. Almeida, et al., Control attenuation and temporary immunity in a cellular automata SEIR epidemic model, Chaos Soliton. Fract., 155 (2022), 111784. https://doi.org/10.1016/j.chaos.2021.111784 doi: 10.1016/j.chaos.2021.111784
[5]	B. Han, B. Zhou, D. Jiang, T. Hayat, A. Alsaedi, Stationary solution, extinction and density function for a high-dimensional stochastic SEI epidemic model with general distributed delay, Appl. Math. Comput., 405 (2021), 126236. https://doi.org/10.1016/j.amc.2021.126236 doi: 10.1016/j.amc.2021.126236
[6]	K. I. Kim, Z. Lin, Asymptotic behavior of an SEI epidemic model with diffusion, Math. Comput. Model., 47 (2008), 1314–1322. https://doi.org/10.1016/j.mcm.2007.08.004 doi: 10.1016/j.mcm.2007.08.004
[7]	Z. Cao, W. Feng, X. Wen, L. Zu, Dynamical behavior of a stochastic SEI epidemic model with saturation incidence and logistic growth, Physica A, 523 (2019), 894–907. https://doi.org/10.1016/j.physa.2019.04.228 doi: 10.1016/j.physa.2019.04.228
[8]	V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43–61. https://doi.org/10.1016/0025-5564(78)90006-8 doi: 10.1016/0025-5564(78)90006-8
[9]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Stationary distribution of a stochastic delayed SVEIR epidemic model with vaccination and saturation incidence, Physica A, 512 (2018), 849–863. https://doi.org/10.1016/j.physa.2018.08.054 doi: 10.1016/j.physa.2018.08.054
[10]	B. Øksendal, Stochastic differential equations: An introduction with applications, Heidelberg, New York: Springer, 2000.
[11]	J. E. Truscott, C. A. Gilligan, Response of a deterministic epidemiological system to a stochastically varying environment, P. Natl. Acad. Sci., 100 (2003), 9067–9072. https://doi.org/10.1073/pnas.1436273100 doi: 10.1073/pnas.1436273100
[12]	S. Hussain, E. N. Madi, H. Khan, H. Gulzar, S. Etemad, S. Rezapour, et al., On the stochastic modeling of COVID-19 under the environmental white noise, J. Funct. Space., 2022 (2022), 4320865. https://doi.org/10.1155/2022/4320865 doi: 10.1155/2022/4320865
[13]	D. Jiang, J. Yu, C. Ji, N. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Model., 54 (2011), 221–232. https://doi.org/10.1016/j.mcm.2011.02.004 doi: 10.1016/j.mcm.2011.02.004
[14]	Y. Zhao, D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718–727. https://doi.org/10.1016/j.amc.2014.05.124 doi: 10.1016/j.amc.2014.05.124
[15]	B. Han, D. Jiang, B. Zhou, T. Hayat, A. Alsaedi, Stationary distribution and probability density function of a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth, Chaos Soliton. Fract., 142 (2021), 110519. https://doi.org/10.1016/j.chaos.2020.110519 doi: 10.1016/j.chaos.2020.110519
[16]	Q. Liu, D. Jiang, Threshold behavior in a stochastic SIR epidemic model with Logistic birth, Physica A, 540 (2020), 123488. https://doi.org/10.1016/j.physa.2019.123488 doi: 10.1016/j.physa.2019.123488
[17]	W. Wei, W. Xu, J. Liu, Y. Song, S. Zhang, Dynamical behavior of a stochastic regime-switching epidemic model with logistic growth and saturated incidence rate, Chaos Soliton. Fract., 173 (2023), 113663. https://doi.org/10.1016/j.chaos.2023.113663 doi: 10.1016/j.chaos.2023.113663
[18]	D. Li, J. Cui, M. Liu, S. Liu, The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate, B. Math. Biol., 77 (2015), 1705–1743. https://doi.org/10.1007/s11538-015-0101-9 doi: 10.1007/s11538-015-0101-9
[19]	Q. Liu, Q. Chen, Dynamics of a stochastic SIR epidemic model with saturated incidence, Appl. Math. Comput., 282 (2016), 155–166. https://doi.org/10.1016/j.amc.2016.02.022 doi: 10.1016/j.amc.2016.02.022
[20]	E. Beretta, V. Kolmanovskii, L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simulat., 45 (1998), 269–277. https://doi.org/10.1016/S0378-4754(97)00106-7 doi: 10.1016/S0378-4754(97)00106-7
[21]	M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Math. Biosci., 175 (2002), 117–131. https://doi.org/10.1016/S0025-5564(01)00089-X doi: 10.1016/S0025-5564(01)00089-X
[22]	L. Imhof, S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations, 217 (2005), 26–53. https://doi.org/10.1016/j.jde.2005.06.017 doi: 10.1016/j.jde.2005.06.017
[23]	B. Han, D. Jiang, T. Hayat, A. Alsaedi, B. Ahmad, Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation, Chaos Soliton. Fract., 140 (2020), 110238. https://doi.org/10.1016/j.chaos.2020.110238 doi: 10.1016/j.chaos.2020.110238
[24]	Q. Liu, D. Jiang, T. Hayat, B. Ahmad, Stationary distribution and extinction of a stochastic predator-prey model with additional food and nonlinear perturbation, Appl. Math. Comput., 320 (2018), 226–239. https://doi.org/10.1016/j.amc.2017.09.030 doi: 10.1016/j.amc.2017.09.030
[25]	C. Lu, H. Liu, D. Zhang, Dynamics and simulations of a second order stochastically perturbed SEIQV epidemic model with saturated incidence rate, Chaos Soliton. Fract., 152 (2021), 111312. https://doi.org/10.1016/j.chaos.2021.111312 doi: 10.1016/j.chaos.2021.111312
[26]	Q. Liu, D. Jiang, Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation, Appl. Math. Lett., 73 (2017), 8–15. https://doi.org/10.1016/j.aml.2017.04.021 doi: 10.1016/j.aml.2017.04.021
[27]	X. Mao, Stochastic differential equations and applications, 2 Eds., Chichester: Woodhead Publishing, 2008. https://doi.org/10.1533/9780857099402
[28]	H. Liu, X. Li, Q. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805–810. https://doi.org/10.1016/j.sysconle.2013.06.002 doi: 10.1016/j.sysconle.2013.06.002
[29]	R. Z. Has'minskii, Stochastic stability of differential equations, Alphen aan den Rijn: Sijthoff and Noordhoff, 1980.
[30]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type Ⅱ functional response, J. Nonlinear. Sci., 28 (2018), 1151–1187. https://doi.org/10.1007/s00332-018-9444-3 doi: 10.1007/s00332-018-9444-3
[31]	N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, Amsterdam: North-Holland, 1981.
[32]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. https://doi.org/10.1137/S0036144500378302 doi: 10.1137/S0036144500378302
[33]	W. Huang, M. Han, K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7 (2010), 51–66. https://doi.org/10.3934/mbe.2010.7.51 doi: 10.3934/mbe.2010.7.51
[34]	C. Liu, R. Cui, Qualitative analysis on an SIRS reaction-diffusion epidemic model with saturation infection mechanism, Nonlinear Anal.-Real, 62 (2021), 103364. https://doi.org/10.1016/j.nonrwa.2021.103364 doi: 10.1016/j.nonrwa.2021.103364
[35]	C. Yang, J. Wang, Basic reproduction numbers for a class of reaction-diffusion epidemic models, B. Math. Biol., 82 (2020), 1–25. https://doi.org/10.1007/s11538-020-00788-x doi: 10.1007/s11538-020-00788-x
[36]	A. Din, The stochastic bifurcation analysis and stochastic delayed optimal control for epidemic model with general incidence function, Chaos, 31 (2021), 123101. https://doi.org/10.1063/5.0063050 doi: 10.1063/5.0063050
[37]	B. Berrhazi, M. E. Fatini, T. Caraballo, R. Pettersson, A stochastic SIRI epidemic model with Lévy noise, Discret. Cont. Dyn.-B, 23 (2018), 2415–2431. https://doi.org/10.3934/dcdsb.2018057 doi: 10.3934/dcdsb.2018057
[38]	N. Privault, L. Wang, Stochastic SIR Lévy jump model with heavy-tailed increments, J. Nonlinear Sci., 31 (2021), 15. https://doi.org/10.1007/s00332-020-09670-5 doi: 10.1007/s00332-020-09670-5
[39]	J. Yang, S. Ma, D. Wei, Dynamical analysis of SIR model with Gamma distribution delay driven by Lévy noise and switching, Electron. Res. Arch., 33 (2025), 3158–3176. https://doi.org/10.3934/era.2025138 doi: 10.3934/era.2025138
[40]	X. Yuan, Y. Yao, H. Wu, M. Feng, Impacts of physical-layer information on epidemic spreading in cyber-physical networked systems, IEEE T. Circuits-I, 72 (2025), 5957–5969. https://doi.org/10.1109/TCSI.2025.3550386 doi: 10.1109/TCSI.2025.3550386
[41]	L. Zhang, C. Guo, M. Feng, Effect of local and global information on the dynamical interplay between awareness and epidemic transmission in multiplex networks, Chaos, 32 (2022), 083138. https://doi.org/10.1063/5.0092464 doi: 10.1063/5.0092464
[42]	S. Pandey, D. Das, U. Ghosh, S. Chakraborty, Bifurcation and onset of chaos in an eco-epidemiological system with the influence of time delay, Chaos, 34 (2024), 063122. https://doi.org/10.1063/5.0177410 doi: 10.1063/5.0177410
[43]	N. Tuncer, M. Martcheva, Modeling seasonality in avian influenza H5N1, J. Biol. Syst., 21 (2013), 1340004. https://doi.org/10.1142/S0218339013400044 doi: 10.1142/S0218339013400044
[44]	E. C. Gabrick, E. Sayari, P. R. Protachevicz, J. D. Szezech Jr., K. C. Iarosz, S. L. T. de Souza, et al., Unpredictability in seasonal infectious diseases spread, Chaos Soliton. Fract., 166 (2023), 113001. https://doi.org/10.1016/j.chaos.2022.113001 doi: 10.1016/j.chaos.2022.113001

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