DYNAMIC BEHAVIOR AND PARAMETER ESTIMATION OF A STOCHASTIC SIS DISEASE TRANSMISSION MODEL WITH LOGISTIC POPULATION INPUT AND DEGENERATE DIFFUSION TERM

Xuexin Li; Yanan Zhao; Tong Wu

doi:10.11948/20240033

2026 Volume 16 Issue 6

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Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(6): 3071-3097

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Xuexin Li, Yanan Zhao, Tong Wu. DYNAMIC BEHAVIOR AND PARAMETER ESTIMATION OF A STOCHASTIC SIS DISEASE TRANSMISSION MODEL WITH LOGISTIC POPULATION INPUT AND DEGENERATE DIFFUSION TERM[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3071-3097. doi: 10.11948/20240033

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Xuexin Li, Yanan Zhao, Tong Wu. DYNAMIC BEHAVIOR AND PARAMETER ESTIMATION OF A STOCHASTIC SIS DISEASE TRANSMISSION MODEL WITH LOGISTIC POPULATION INPUT AND DEGENERATE DIFFUSION TERM[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3071-3097. doi: 10.11948/20240033

DYNAMIC BEHAVIOR AND PARAMETER ESTIMATION OF A STOCHASTIC SIS DISEASE TRANSMISSION MODEL WITH LOGISTIC POPULATION INPUT AND DEGENERATE DIFFUSION TERM

1.
School of Science, Changchun University, Changchun 130021, China

Author Bio: Email: 18834855595@163.com(X. Li); Email: tutu123aaa@163.com(T. Wu)
Corresponding author: Email: zhaoyn111@163.com(Y. Zhao)
Fund Project: The work was supported by Science and Technology Development Project of Jilin Province (YDZJ202401359ZYTS)

Abstract

Abstract

To account for population growth constrained by environmental capacity, we formulate a stochastic SIS model with logistic population input and standard incidence to describe disease transmission.Due to the degenerate structure of the stochastic differential equation, we employ the Lyapunov function method and Markov semigroup theory to establish a threshold theorem, including the extinction and ergodicity of the stochastic system.In addition, since the parameters in stochastic models are usually unknown in practice, we study parameter estimation for the proposed model under discrete observations by comparing least squares estimation, pseudo-maximum likelihood estimation (pseudo-MLE), and Bayesian posterior mean estimation. We further investigate how the observation step size, time horizon, and noise intensity affect the widths of confidence intervals and the accuracy of point estimates. Numerical simulations are provided to illustrate the theoretical results and to compare the performance of the three estimation methods.The results complement the dynamical analysis by providing a comparative study of parameter identification for this stochastic SIS model.
- Stochastic SIS model /
- logistic population input /
- degenerate diffusion /
- threshold theorem /
- parameter estimation
MSC: 37A30, 47D07, 60H10, 92D30

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References

[1]	M. A. Achterberg, B. Prasse and P. Van Mieghem, Analysis of continuous-time Markovian $\varepsilon$-SIS epidemics on networks, Phys. Rev. E, 2022, 105(5), 054305. $\varepsilon$-SIS epidemics on networks" target="_blank">Google Scholar
[2]	R. Anderson and R. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. Google Scholar
[3]	D. R. Bell, The Malliavin Calculus, Courier Corporation, New York, 2012. Google Scholar
[4]	J. P. N. Bishwal, Parameter Estimation in Stochastic Differential Equations, Springer, Berlin, 2007. Google Scholar
[5]	G. E. P. Box and G. C. Tiao, Bayesian Inference in Statistical Analysis, John Wiley & Sons, New York, 2011. Google Scholar
[6]	F. Brauer, Mathematical approaches to problems in resource management and epidemiology, Proceedings of a Conference Held at Ithaca, 1987, 109–123. Google Scholar
[7]	J. Cao and L. Hu, Asymptotic properties of a pseudo-MLE for CIR model, in: Proc. 2010 Int. Conf. ⅡGSS-CPS, Nanjing, China, 2010, 28–31. Google Scholar
[8]	L. Q. Gao and H. W. Hethcote, Disease transmission models with density-dependent demographics, J. Math. Biol., 1992, 30(7), 717–731. Google Scholar
[9]	A. Gray, D. Greenhalgh, L. Hu, et al., A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 2011, 71(3), 876–902. doi: 10.1137/10081856X CrossRef Google Scholar
[10]	S. He, S. Tang, Y. Cai, et al., A stochastic epidemic model coupled with seasonal air pollution: Analysis and data fitting, Stoch. Environ. Res. Risk Assess., 2020, 34(12), 2245–2257. doi: 10.1007/s00477-020-01856-3 CrossRef Google Scholar
[11]	S. He and Z. Xiang, Dynamics of stochastic SIQS model based on nonlinear incidence: Disease extinction and stationary distribution under degenerate diffusion, Electron. Res. Arch., 2025, 33(7), 4259–4283. doi: 10.3934/era.2025193 CrossRef Google Scholar
[12]	S. M. Iacus, Simulation and Inference for Stochastic Differential Equations: With R Examples, Springer, New York, 2008. Google Scholar
[13]	M. C. M. de Jong, O. Diekmann, et al., How does transmission of infection depend on population size?, Epidemic Models: Their Structure and Relation to Data, 1995, 84–94. Google Scholar
[14]	R. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. Google Scholar
[15]	N. R. Kristensen, H. Madsen and S. B. Jørgensen, Parameter estimation in stochastic grey-box models, Automatica, 2004, 40(2), 225–237. doi: 10.1016/j.automatica.2003.10.001 CrossRef Google Scholar
[16]	S. Li and S. Guo, Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions, Discrete Contin. Dyn. Syst. B, 2021, 26(5), 2693–2719. doi: 10.3934/dcdsb.2020201 CrossRef Google Scholar
[17]	Q. Liu, D. Jiang, T. Hayat, et al., Threshold dynamics of a stochastic SIS epidemic model with nonlinear incidence rate, Phys. A, 2019, 526, 120946. doi: 10.1016/j.physa.2019.04.182 CrossRef Google Scholar
[18]	Q. Liu, D. Jiang, T. Hayat, et al., A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate, Phys. A, 2020, 551, 124152. doi: 10.1016/j.physa.2020.124152 CrossRef Google Scholar
[19]	X. Mao, Stochastic Differential Equations and Applications, Elsevier, Horwood Publishing, 2007. Google Scholar
[20]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. Google Scholar
[21]	H. Nemoto and Y. Shimizu, Statistical inference for discretely sampled stochastic functional differential equations with small noise, Stat. Inference Stoch. Process., 2024, 27(2), 427–456. doi: 10.1007/s11203-023-09299-7 CrossRef Google Scholar
[22]	J. Pan, A. Gray, D. Greenhalgh, et al., Parameter estimation for the stochastic SIS epidemic model, Stat. Inference Stoch. Process., 2014, 17(1), 75–98. doi: 10.1007/s11203-014-9091-8 CrossRef Google Scholar
[23]	K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl., 1997, 215(1), 56–74. doi: 10.1006/jmaa.1997.5609 CrossRef Google Scholar
[24]	J. O. Rawlings, S. G. Pantula and D. A. Dickey, Applied Regression Analysis: A Research Tool, Springer, New York, 1998. Google Scholar
[25]	R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stochastic Process. Appl., 2003, 108(1), 93–107. doi: 10.1016/S0304-4149(03)00090-5 CrossRef Google Scholar
[26]	R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems, Math. Biosci., 2007, 206(1), 108–119. doi: 10.1016/j.mbs.2006.03.006 CrossRef Google Scholar
[27]	R. Rudnicki, K. Pichór and M. Tyran-Kamińska, Markov semigroups and their applications, Springer, Berlin, 2002, 215–238. Google Scholar
[28]	J. Timmer, Parameter estimation in nonlinear stochastic differential equations, Chaos Solitons Fractals, 2000, 11(15), 2571–2578. doi: 10.1016/S0960-0779(00)00015-1 CrossRef Google Scholar
[29]	H. Wang, G. Zhang, T. Chen, et al., Threshold analysis of a stochastic SIRS epidemic model with logistic birth and nonlinear incidence, Mathematics, 2023, 11(7), 1737. doi: 10.3390/math11071737 CrossRef Google Scholar
[30]	L. Wang, C. Gao, R. Rifhat, et al., Stationary distribution and bifurcation analysis for a stochastic SIS model with nonlinear incidence and degenerate diffusion, Chaos Solitons Fractals, 2024, 182, 114872. Google Scholar
[31]	Z. Xu and L. Wang, Stationary distribution and extinction of a stochastic SEI epidemic model with logistic growth and nonlinear perturbation, AIMS Math., 2025, 10(12), 28488–28513. doi: 10.3934/math.20251254 CrossRef Google Scholar
[32]	Z. Xu, L. Wang and D. Jiang, The threshold of a stochastic SIR epidemic model with logistic growth, J. Appl. Anal. Comput., 2026, 16(3), 1187–1206. Google Scholar
[33]	T. Xue, X. Fan and Z. Chang, Dynamics of a stochastic SIRS epidemic model with standard incidence and vaccination, Math. Biosci. Eng., 2022, 19(10), 10618–10636. Google Scholar
[34]	A. Zellner, Applications of Bayesian analysis in econometrics, J. R. Stat. Soc. Ser. D Stat., 1983, 32(1), 23–34. Google Scholar
[35]	X. Zhang, Z. Zhang, J. Tong, et al., Ergodicity of stochastic smoking model and parameter estimation, Adv. Differ. Equ., 2016, 2016(1), 1–20. Google Scholar
[36]	Y. Zhao, D. Jiang, X. Mao, et al., The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete Contin. Dyn. Syst. B, 2015, 20(4), 1277–1295. Google Scholar
[37]	C. Zhou and Z. Li, Parameter estimation of stochastic SEIR epidemic model using particle MCMC, Chaos, 2025, 35(5), 053139. Google Scholar