异质网络上随机 SIS 传染病模型的拟平稳分布与灭绝时间分析

Acta mathematica scientia,Series A ›› 2026, Vol. 46 ›› Issue (5): 2003-2016.

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A Stochastic SIS Epidemic Model in Heterogeneous Networks: Quasi-Stationarity and Extinction

Xiaojie Jing¹(), Guirong Liu²^,^*(), Zhen Jin³()

¹ School of Mathematics, North University of China, Taiyuan 030051
² School of Mathematics and Statistics, Shanxi University, Taiyuan 030006
³ Complex Systems Research Center, Shanxi University, Taiyuan 030006

Received:2025-02-21 Revised:2026-04-27 Online:2026-10-26 Published:2026-09-07
Contact: Guirong Liu E-mail:jingxiaojie@nuc.edu.cn;lgr5791@sxu.edu.cn;jinzhen@sxu.edu.cn
Supported by:
NSFC(12501698);NSFC(12371494);NSFC(12231012);Fundamental Research Program of Shanxi Province of China(202403021222164);Shanxi Provincial Key Research and Development Project(202202020101010)

Abstract

Abstract:

In this paper, the moment closure method is applied to the SIS epidemic model on heterogeneous networks. Assumptions of normal and log-normal distributions are made to capture the quasi-stationary behavior and the expected time to extinction of the disease starting from the quasi-stationary state. The approximate performance of the moment closure method is then tested by stochastic simulations. The results show that the approximation is in good agreement with the simulations for the quasi-stationary distribution, particularly for the log-normal distribution.

Key words: Epidemic spreading, Heterogeneous network, Quasi-stationary distribution, Time to extinction, Moment closure

CLC Number:

O175

Xiaojie Jing, Guirong Liu, Zhen Jin. A Stochastic SIS Epidemic Model in Heterogeneous Networks: Quasi-Stationarity and Extinction[J].Acta mathematica scientia,Series A, 2026, 46(5): 2003-2016.

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Figures/Tables 10

References 19

[1]	Dangerfield C E, Ross J V, Keeling M J. Integrating stochasticity and network structure into an epidemic model. J R Soc Interface, 2009, 6(38): 761-774 doi: 10.1098/rsif.2008.0410
[2]	Graham M, House T. Dynamics of stochastic epidemics on heterogeneous networks. J Math Biol, 2014, 68(7): 1583-1605 doi: 10.1007/s00285-013-0679-1 pmid: 23633042
[3]	Ball F, Britton T, Leung K Y, et al. A stochastic SIR network epidemic model with preventive dropping of edges. J Math Biol, 2019, 78(6): 1875-1951 doi: 10.1007/s00285-019-01329-4 pmid: 30868213
[4]	Achterberg M A, Van Mieghem P. Moment closure approximations of susceptible-infected-susceptible epidemics on adaptive networks. Phys Rev E, 2022, 106(1): Art 014308 doi: 10.1103/PhysRevE.106.014308
[5]	Nåsell I. An extension of the moment closure method. Theor Popul Biol, 2003, 64(2): 233-239 pmid: 12948684
[6]	Nåsell I. Moment closure and the stochastic logistic model. Theor Popul Biol, 2003, 63(2): 159-168 doi: 10.1016/s0040-5809(02)00060-6 pmid: 12615498
[7]	Lloyd A L. Estimating variability in models for recurrent epidemics: assessing the use of moment closure techniques. Theor Popul Biol, 2004, 65(1): 49-65 pmid: 14642344
[8]	Clancy D, Mendy S T. The effect of waning immunity on long-term behaviour of stochastic models for the spread of infection. J Math Biol, 2010, 61(4): 527-544 doi: 10.1007/s00285-009-0313-4 pmid: 19941137
[9]	Clancy D, Mendy S T. Approximating the quasi-stationary distribution of the SIS model for endemic infection. Methodol Comput Appl, 2011, 13(3): 603-618
[10]	Isham V. Assessing the variability of stochastic epidemics. Math Biosci, 1991, 107(2): 207-224
[11]	Whittle P. On the use of the normal approximation in the treatment of stochastic processes. J R Stat Soc B, 1957, 19(2): 268-281 doi: 10.1111/j.2517-6161.1957.tb00263.x
[12]	Allen, L J S. An Introduction to Stochastic Processes with Applications to Biology. Florida: CRC Press, 2010
[13]	Pastor-Satorras R, Vespignani A. Epidemic spreading in scale-free networks. Phys Rev Lett, 2001, 86(14): 3200-3203 pmid: 11290142
[14]	Pastor-Satorras R, Vespignani A. Epidemic dynamics in finite size scale-free networks. Phys Rev E, 2002, 65(3): Art 035108 doi: 10.1103/PhysRevE.65.035108
[15]	Nåsell I. An alternative to moment closure. Bull Math Biol, 2017, 79(9): 2088-2108 doi: 10.1007/s11538-017-0321-2
[16]	Keeling M J. Metapopulation moments: coupling, stochasticity and persistence. J Anim Ecol, 2000, 69(5): 725-736 doi: 10.1046/j.1365-2656.2000.00430.x pmid: 29314000
[17]	Keeling M J. Multiplicative moments and measures of persistence in ecology. J Theor Biol, 2000, 205(2): 269-281 pmid: 10873438
[18]	Britton T, Traoré A. A stochastic vector-borne epidemic model: Quasi-stationarity and extinction. Math Biosci, 2017, 289: 89-95 doi: S0025-5564(17)30268-7 pmid: 28511957
[19]	Darroch J N, Seneta E. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J Appl Probab, 1967, 4(1): 192-196 doi: 10.2307/3212311

A Stochastic SIS Epidemic Model in Heterogeneous Networks: Quasi-Stationarity and Extinction

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