THE THRESHOLD OF A STOCHASTIC SIR EPIDEMIC MODEL WITH LOGISTIC GROWTH

Zeyu Xu; Liang Wang; Daqing Jiang

doi:10.11948/20250208

2026 Volume 16 Issue 3

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Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(3): 1187-1206

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Zeyu Xu, Liang Wang, Daqing Jiang. THE THRESHOLD OF A STOCHASTIC SIR EPIDEMIC MODEL WITH LOGISTIC GROWTH[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1187-1206. doi: 10.11948/20250208

Citation:

Zeyu Xu, Liang Wang, Daqing Jiang. THE THRESHOLD OF A STOCHASTIC SIR EPIDEMIC MODEL WITH LOGISTIC GROWTH[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1187-1206. doi: 10.11948/20250208

THE THRESHOLD OF A STOCHASTIC SIR EPIDEMIC MODEL WITH LOGISTIC GROWTH

1.
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China
2.
College of Science, China University of Petroleum (East China), Qingdao 266580, China

Author Bio: Email: xuzeyu@njust.edu.cn(Z. Xu); Email: daqingjiang2010@hotmail.com(D. Jiang)
Corresponding author: Email: ntumacwangl@hotmail.com(L. Wang)
Fund Project: The authors were supported by National Natural Science Foundation of China (No. 12001271) and Fundamental Research Funds for the Central Universities (No. 30922010813)

Abstract

Abstract

In this paper, we propose and study a stochastic SIR epidemic model with logistic growth. The threshold dynamics of the stochastic model are governed by a parameter $ R_0^S $. Precisely, the disease goes to extinction at an exponential rate and the distribution of susceptible individuals converges weakly to a unique invariant probability measure provided that $ R_0^S<1 $; whereas if $ R_0^S>1 $, we show that there is a unique ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. Numerical simulations are introduced to illustrate our theoretical results. The aim of the work is to investigate the impact of environmental noise on the transmission of infectious diseases.
- SIR epidemic model /
- logistic growth /
- stationary distribution /
- ergodicity /
- extinction /
- threshold
MSC: 92D25, 92D30, 60H10

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References

[1]	M. Ahmad, A. Zada, M. Ghaderi, R. George and S. Rezapour, On the existence and stability of a neutral stochastic fractional differential system, Fractal Fract., 2022, 6(4), 203. doi: 10.3390/fractalfract6040203 CrossRef Google Scholar
[2]	I. Ahmed, G. U. Modu, A. Yusuf, P. Kumam and I. Yusuf, A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes, Results Phys., 2021, 21, 103776. doi: 10.1016/j.rinp.2020.103776 CrossRef Google Scholar
[3]	M. H. Akrami and A. Atabaigi, Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals, J. Appl. Math. Comput., 2020, 64(1), 615–633. Google Scholar
[4]	M. Barczy and G. Pap, Portmanteau theorem for unbounded measures, Stat. Probab. Lett., 2006, 76(17), 1831–1835. doi: 10.1016/j.spl.2006.04.025 CrossRef Google Scholar
[5]	J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Sci., 1977, 197(4302), 463–465. doi: 10.1126/science.197.4302.463 CrossRef Google Scholar
[6]	E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simul., 1998, 45(3–4), 269–277. doi: 10.1016/S0378-4754(97)00106-7 CrossRef Google Scholar
[7]	Z. Cao, W. Feng, X. Wen and L. Zu, Dynamical behavior of a stochastic SEI epidemic model with saturation incidence and logistic growth, Phys. A, 2019, 523, 894–907. doi: 10.1016/j.physa.2019.04.228 CrossRef Google Scholar
[8]	M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Math. Biosci., 2002, 175(2), 117–131. doi: 10.1016/S0025-5564(01)00089-X CrossRef Google Scholar
[9]	S. Chávez-Vázquez, J. E. Lavín-Delgado, J. F. Gómez-Aguilar, J. R. Razo-Hernández, S. Etemad and S. Rezapour, Trajectory tracking of Stanford robot manipulator by fractional-order sliding mode control, Appl. Math. Model., 2023, 120, 436–462. doi: 10.1016/j.apm.2023.04.001 CrossRef Google Scholar
[10]	N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 2007, 325(1), 36–53. doi: 10.1016/j.jmaa.2006.01.055 CrossRef Google Scholar
[11]	J. E. Lavín-Delgado, J. E. Solís-Pérez, J. F. Gómez-Aguilar, J. R. Razo-Hernández, S. Etemad and S. Rezapour, An improved object detection algorithm based on the Hessian matrix and conformable derivative, Circuits Syst. Signal Process., 2024, 43(8), 4991–5047. doi: 10.1007/s00034-024-02669-3 CrossRef Google Scholar
[12]	A. Din, The stochastic bifurcation analysis and stochastic delayed optimal control for epidemic model with general incidence function, Chaos An Interdiscip. J. Nonlinear Sci., 2021, 31, 123101. doi: 10.1063/5.0063050 CrossRef Google Scholar
[13]	N. H. Du, D. H. Nguyen and G. G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 2016, 53(1), 187–202. doi: 10.1017/jpr.2015.18 CrossRef Google Scholar
[14]	R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. Google Scholar
[15]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 2001, 43(3), 525–546. doi: 10.1137/S0036144500378302 CrossRef Google Scholar
[16]	S. Hussain, E. N. Madi, H. Khan, H. Gulzar, S. Etemad, S. Rezapour and M. K. A. Kaabar, On the stochastic modeling of COVID-19 under the environmental white noise, J. Funct. Spaces, 2022, 2022(1), 4320865. Google Scholar
[17]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981. Google Scholar
[18]	L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equ., 2005, 217(1), 26–53. doi: 10.1016/j.jde.2005.06.017 CrossRef Google Scholar
[19]	K. A. Kabir, K. Kuga and J. Tanimoto, Analysis of SIR epidemic model with information spreading of awareness, Chaos Solitons Fractals, 2019, 119, 118–125. doi: 10.1016/j.chaos.2018.12.017 CrossRef Google Scholar
[20]	T. Kanwal, A. Hussain, İ. Avcı, S. Etemad, S. Rezapour and D. F. M. Torres, Dynamics of a model of polluted lakes via fractal-fractional operators with two different numerical algorithms, Chaos Solitons Fractals, 2024, 181, 114653. doi: 10.1016/j.chaos.2024.114653 CrossRef Google Scholar
[21]	W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics–I, Bull. Math. Biol., 1991, 53(1–2), 33–55. Google Scholar
[22]	H. Khan, K. Alam, H. Gulzar, S. Etemad and S. Rezapour, A case study of fractal-fractional tuberculosis model in China: Existence and stability theories along with numerical simulations, Math. Comput. Simul., 2022, 198, 455–473. doi: 10.1016/j.matcom.2022.03.009 CrossRef Google Scholar
[23]	H. Khan, M. Aslam, A. H. Rajpar, Y. M. Chu, S. Etemad, S. Rezapour and H. Ahmad, A new fractal-fractional hybrid model for studying climate change on coastal ecosystems from the mathematical point of view, Fractals, 2024, 32(02), 2440015. doi: 10.1142/S0218348X24400152 CrossRef Google Scholar
[24]	H. Khan, A. H. Rajpar, J. Alzabut, M. Aslam, S. Etemad and S. Rezapour. On a fractal-fractional-based modeling for influenza and its analytical results, Qual. Theor. Dyn. Syst., 2024, 23(2), 70. Google Scholar
[25]	M. N. Khan, Z. Wang, N. A. Ahammad, S. Rezapour, M. Shutaywi, N. B. Ali and M. A. Elkotb, Mixed convective flow analysis of a Maxwell fluid with double diffusion theory on a vertically exponentially stretching surface, Appl. Water Sci., 2024, 14(8), 172. doi: 10.1007/s13201-024-02235-x CrossRef Google Scholar
[26]	T. Khan, A. Khan and G. Zaman, The extinction and persistence of the stochastic hepatitis B epidemic model, Chaos Solitons Fractals, 2018, 108, 123–128. doi: 10.1016/j.chaos.2018.01.036 CrossRef Google Scholar
[27]	D. Li, F. Wei and X. Mao, Stationary distribution and density function of a stochastic SVIR epidemic model, J. Frankl. Inst., 2022, 359(16), 9422–9449. doi: 10.1016/j.jfranklin.2022.09.026 CrossRef Google Scholar
[28]	Q. Liu and D. Jiang, Threshold behavior in a stochastic SIR epidemic model with logistic birth, Phys. A, 2020, 540, 123488. doi: 10.1016/j.physa.2019.123488 CrossRef Google Scholar
[29]	Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi, Periodic solution for a stochastic nonautonomous SIR epidemic model with logistic growth, Phys. A, 2016, 462, 816–826. doi: 10.1016/j.physa.2016.06.052 CrossRef Google Scholar
[30]	X. Liu, Y. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 2008, 253(1), 1–11. doi: 10.1016/j.jtbi.2007.10.014 CrossRef Google Scholar
[31]	Y. Liu, J. Li and D. Kuang, Dynamics of a stochastic SIR epidemic model with logistic growth, J. Nonlinear Model. Anal., 2023, 5(1), 73–94. Google Scholar
[32]	C. Lu, H. Liu and D. Zhang, Dynamics and simulations of a second order stochastically perturbed SEIQV epidemic model with saturated incidence rate, Chaos Solitons Fractals, 2021, 152, 111312. doi: 10.1016/j.chaos.2021.111312 CrossRef Google Scholar
[33]	G. Mani, B. Ramalingam, S. Etemad, İ. Avcı and S. Rezapour, On the menger probabilistic bipolar metric spaces: Fixed point theorems and applications, Qual. Theory Dyn. Syst., 2024, 23(3), 99. Google Scholar
[34]	X. Mao, Stochastic Differential Equations and Applications, second ed., Horwood, Chichester, United Kingdom, 2008. Google Scholar
[35]	H. Mohammadi, S. Kumar, S. Rezapour and S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Solitons Fractals, 2021, 144, 110668. doi: 10.1016/j.chaos.2021.110668 CrossRef Google Scholar
[36]	A. Mwasa and J. M. Tchuenche, Mathematical analysis of a cholera model with public health interventions, Biosyst., 2011, 105(3), 190–200. doi: 10.1016/j.biosystems.2011.04.001 CrossRef Google Scholar
[37]	B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Heidelberg, New York, 2000. Google Scholar
[38]	N. Privault and L. Wang, Stochastic SIR Lévy jump model with heavy-tailed increments, J. Nonlinear Sci., 2021, 31, 1–28. doi: 10.1007/s00332-020-09667-0 CrossRef Google Scholar
[39]	A. Qazza and R. Saadeh, On the analytical solution of fractional SIR epidemic model, Appl. Comput. Intell. Soft Comput., 2023, 2023(1), 6973734. Google Scholar
[40]	S. Rezapour, M. I. Abbas, S. Etemad and N. M. Dien, On a multi-point $p$-Laplacian fractional differential equation with generalized fractional derivatives, Math. Methods Appl. Sci., 2023, 46(7), 8390–8407. doi: 10.1002/mma.8301 CrossRef $p$-Laplacian fractional differential equation with generalized fractional derivatives" target="_blank">Google Scholar
[41]	S. Rezapour, S. T. M. Thabet, I. Kedim, M. Vivas-Cortez and M. Ghaderi, A computational method for investigating a quantum integrodifferential inclusion with simulations and heatmaps, AIMS Math., 2023, 8(11), 27241–27267. Google Scholar
[42]	S. Rezapour, S. T. M. Thabet, A. S. Rafeeq, I. Kedim, M. Vivas-Cortez and N. Aghazadeh, Topology degree results on a G-ABC implicit fractional differential equation under three-point boundary conditions, PLOS One, 2024, 19(7), e0300590. Google Scholar
[43]	H. Serrai, B. Tellab, S. Etemad, İ. Avcı and S. Rezapour, $\Psi$-Bielecki-type norm inequalities for a generalized Sturm-Liouville-Langevin differential equation involving $\Psi$-Caputo fractional derivative, Bound. Value Probl., 2024, 2024(1), 81. Google Scholar
[44]	S. Spencer, Stochastic Epidemic Models for Emerging Diseases, PhD thesis, University of Nottingham, Nottingham, United Kingdom, 2008. Google Scholar
[45]	S. T. M. Thabet, S. Al-Sa'di, I. Kedim, A. S. Rafeeq and S. Rezapour, Analysis study on multi-order $\rho$-Hilfer fractional pantograph implicit differential equation on unbounded domains, AIMS Math., 2023, 8(8), 18455–18473. doi: 10.3934/math.2023938 CrossRef $\rho$-Hilfer fractional pantograph implicit differential equation on unbounded domains" target="_blank">Google Scholar
[46]	E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A, 2005, 354, 111–126. doi: 10.1016/j.physa.2005.02.057 CrossRef Google Scholar
[47]	J. E. Truscott and C. A. Gilligan, Response of a deterministic epidemiological system to a stochastically varying environment, Proc. Natl. Acad. Sci., 2003, 100(15), 9067–9072. doi: 10.1073/pnas.1436273100 CrossRef Google Scholar
[48]	H. Waheed, A. Zada, I. L. Popa, S. Etemad and S. Rezapour, On a system of sequential Caputo-type $p$-Laplacian fractional BVPs with stability analysis, Qual. Theory Dyn. Syst., 2024, 23(3), 128. doi: 10.1007/s12346-024-00988-z CrossRef $p$-Laplacian fractional BVPs with stability analysis" target="_blank">Google Scholar
[49]	W. Wei, W. Xu, J. Liu, Y. Song and S. Zhang, Dynamical behavior of a stochastic regime-switching epidemic model with logistic growth and saturated incidence rate, Chaos Solitons Fractals, 2023, 173, 113663. doi: 10.1016/j.chaos.2023.113663 CrossRef Google Scholar
[50]	G. Zaman, Y. H. Kang and I. H. Jung, Stability analysis and optimal vaccination of an SIR epidemic model, Biosyst., 2008, 93(3), 240–249. doi: 10.1016/j.biosystems.2008.05.004 CrossRef Google Scholar
[51]	Z. Zhang, A. Zeb, S. Hussain and E. Alzahrani, Dynamics of COVID-19 mathematical model with stochastic perturbation, Adv. Differ. Equ., 2020, 2020, 1–12. doi: 10.1186/s13662-019-2438-0 CrossRef Google Scholar
[52]	Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 2014, 243, 718–727. Google Scholar