Approximation of the invariant probability density function for a stochastic SEIR epidemic model with infectivity during the latent, infectious, and immune periods

Qiumei Zhang; Qiumei Zhang

doi:10.3934/era.2026233

Electronic Research Archive

2026, Volume 34, Issue 8: 5236-5266. doi: 10.3934/era.2026233

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Approximation of the invariant probability density function for a stochastic SEIR epidemic model with infectivity during the latent, infectious, and immune periods

Qiumei Zhang ^,

School of Mathematics and Statistics, Changchun University, Changchun 130022, China

Received: 03 April 2026 Revised: 24 May 2026 Accepted: 01 June 2026 Published: 18 June 2026

In this paper, we proposed and studied a stochastic SEIR epidemic model with infectivity during the latent, infectious, and immune periods. First, the existence of a stationary distribution for the model was verified using the Lyapunov method. Then, by solving the corresponding Fokker-Planck equation, the explicit expression of the probability density function around the quasi-stable equilibrium point was obtained. Furthermore, the global approximation of the invariant probability density function for the model was explored. In addition, sufficient conditions for disease extinction were derived by constructing a suitable Lyapunov function. Finally, numerical simulations are presented to illustrate the analytical results and to reveal the impact of stochastic perturbations on disease transmission.
Citation: Qiumei Zhang. Approximation of the invariant probability density function for a stochastic SEIR epidemic model with infectivity during the latent, infectious, and immune periods[J]. Electronic Research Archive, 2026, 34(8): 5236-5266. doi: 10.3934/era.2026233

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Abstract

In this paper, we proposed and studied a stochastic SEIR epidemic model with infectivity during the latent, infectious, and immune periods. First, the existence of a stationary distribution for the model was verified using the Lyapunov method. Then, by solving the corresponding Fokker-Planck equation, the explicit expression of the probability density function around the quasi-stable equilibrium point was obtained. Furthermore, the global approximation of the invariant probability density function for the model was explored. In addition, sufficient conditions for disease extinction were derived by constructing a suitable Lyapunov function. Finally, numerical simulations are presented to illustrate the analytical results and to reveal the impact of stochastic perturbations on disease transmission.

References

[1]	Y. Cai, Y. Kang, W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221–240. https://doi.org/10.1016/j.amc.2017.02.003 doi: 10.1016/j.amc.2017.02.003
[2]	D. H. Nguyen, G. Yin, C. Zhu, Long-term analysis of a stochastic SIRS model with general incidence rates, SIAM J. Appl. Math., 80 (2020), 814–838. https://doi.org/10.1137/19M1246973 doi: 10.1137/19M1246973
[3]	G. Lan, S. Yuan, B. Song, The impact of hospital resources and environmental perturbations to the dynamics of SIRS model, J. Franklin Inst., 358 (2021), 2405–2433. https://doi.org/10.1016/j.jfranklin.2021.01.015 doi: 10.1016/j.jfranklin.2021.01.015
[4]	K. Qi, D. Jiang, H. Tasawar, A. Alsaedi, Virus dynamic behavior of a stochastic HIV/AIDS infection model including two kinds of target cell infections and CTL immune responses, Math. Comput. Simul., 188 (2021), 548–570. https://doi.org/10.1016/j.matcom.2021.05.009 doi: 10.1016/j.matcom.2021.05.009
[5]	X. Li, X. Mao, G. Yin, Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment and stability, IMA J. Numer. Anal., 39 (2019), 847–892. https://doi.org/10.1093/imanum/dry015 doi: 10.1093/imanum/dry015
[6]	G. Li, Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos, Solitons Fractals, 25 (2005), 1177–1184. https://doi.org/10.1016/j.chaos.2004.11.062 doi: 10.1016/j.chaos.2004.11.062
[7]	C. Sun, Y. H. Hsieh, Global analysis of an SEIR model with varying population size and vaccination, Appl. Math. Modell., 34 (2010), 2685–2697. https://doi.org/10.1016/j.apm.2009.12.005 doi: 10.1016/j.apm.2009.12.005
[8]	P. J. Witbooi, Stability of an SEIR epidemic model with independent stochastic perturbations, Physica A, 392 (2013), 4928–4936. https://doi.org/10.1016/j.physa.2013.06.025 doi: 10.1016/j.physa.2013.06.025
[9]	C. M. Cluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603–610. https://doi.org/10.3934/mbe.2009.6.603 doi: 10.3934/mbe.2009.6.603
[10]	M. De la Sen, A. Ibeas, S. Alonso-Quesada, R. Nistal, On a new epidemic model with asymptomatic and dead-infective subpopulations with feedback controls useful for Ebola disease, Discrete Dyn. Nat. Soc., 2017 (2017), 1–22. https://doi.org/10.1155/2017/4232971 doi: 10.1155/2017/4232971
[11]	B. Zhou, X. Zhang, D. Jiang, Dynamics and density function analysis of a stochastic SVI epidemic model with half saturated incidence rate, Chaos, Solitons Fractals, 137 (2020), 109865. https://doi.org/10.1016/j.chaos.2020.109865 doi: 10.1016/j.chaos.2020.109865
[12]	S. Rajasekar, M. Pitchaimani, Ergodic stationary distribution and extinction of a stochastic SIRS epidemic model with logistic growth and nonlinear incidence, Appl. Math. Comput., 377 (2020), 125143. https://doi.org/10.1016/j.amc.2020.125143 doi: 10.1016/j.amc.2020.125143
[13]	N. T. Dieu, V. H. Sam, N. H. Du, Threshold of a stochastic SIQS epidemic model with isolation, Discrete Contin. Dyn. Syst. - Ser. B, 27 (2022), 5009–5028. https://doi.org/10.3934/dcdsb.2021262 doi: 10.3934/dcdsb.2021262
[14]	M. De la Sena, S. Alonso-Quesadaa, 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
[15]	Q. Liu, D. Jiang, Stationary distribution of a stochastic cholera model with imperfect vaccination, Physica A, 550 (2020), 124031. https://doi.org/10.1016/j.physa.2019.124031 doi: 10.1016/j.physa.2019.124031
[16]	T. Caraballo, M. Fatini, M. Khalifi, R. Gerlach, R. Pettersson, Analysis of a stochastic distributed delay epidemic model with relapse and Gamma distribution kernel, Chaos, Solitons Fractals, 133 (2020), 109643. https://doi.org/10.1016/j.chaos.2020.109643 doi: 10.1016/j.chaos.2020.109643
[17]	D. H. Nguyen, N. N. Nguyen, G. Yin, Stochastic functional Kolmogorov equations Ⅱ: extinction, J. Differ. Equations, 294 (2021), 1–39. https://doi.org/10.1016/j.jde.2021.05.043 doi: 10.1016/j.jde.2021.05.043
[18]	B. Zhou, H. Wang, T. Wang, D. Jiang, Stochastic generalized Kolmogorov systems with small diffusion: Ⅰ. Explicit approximations for invariant probability density function, J. Differ. Equations, 382 (2024), 141–210. https://doi.org/10.1016/j.jde.2023.10.057 doi: 10.1016/j.jde.2023.10.057
[19]	S. Tian, Y. Zhou, D. Jiang, Dynamical behaviors of a stochastic SIRB model for cholera with general infection dynamics and Ornstein-Uhlenbeck process, Nonlinear Dyn., 113 (2025), 22043–22069. https://doi.org/10.1007/s11071-025-11227-8 doi: 10.1007/s11071-025-11227-8
[20]	R. Khasminskii, Stochastic Stability of Differential Equations, 2$^{nd}$ edition, Springer, Berlin, 2012. https://doi.org/10.1007/978-3-642-23280-0
[21]	T. C. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 1988.
[22]	G. Strang, Linear Algebra and Its Applications, 4$^{nd}$ edition, Cengage Learning Inc, 2005.
[23]	C. Zhu, G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155–1179. https://doi.org/10.1137/060649343 doi: 10.1137/060649343
[24]	Z. Ma, Y. Zhou, C. Li, Qualitative and stability methods for ordinary differential equations, Science Press, Beijing, 2015.
[25]	B. Han, B. Zhou, D. Jiang, H. Tasawar, 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
[26]	H. Roozen, An asymptotic solution to a two-dimensional exit problem arising in population dynamics, SIAM J. Appl. Math., 49 (1989), 1793. https://doi.org/10.1137/0149110 doi: 10.1137/0149110
[27]	C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1985.
[28]	H. Liu, Q. Yang, D. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820–825. https://doi.org/10.1016/j.automatica.2012.02.010 doi: 10.1016/j.automatica.2012.02.010
[29]	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

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