JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (6): 84-93.doi: 10.6040/j.issn.1671-9352.0.2021.793

Previous Articles    

Asymptotic behavior of a stochastic SIQS model with Markov switching

WANG Yan-mei1, LIU Gui-rong2*   

  1. 1. School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, Shanxi, China;
    2. School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China
  • Published:2022-06-10

PDF (PC)

253

Abstract: A stochastic SIQS epidemic model with Markovian switching and saturated incidence is investigated. First, the existence and uniqueness of the global positive solution of the model are proved by constructing the suitable Lyapunov functions. Then by using the ergodic property of Markov chains, the sufficient conditions of extinction and persistence in the mean of the disease are obtained. At last, the theoretical results are verified by numerical simulations. The results show that if one of the subsystems is stochastically persistent, and another is stochastically extinct, then the hybrid system may be either stochastically extinct or persistent, and the result depends on the probability that the Markov chain remains in each status. Telegraph noise has a major impact on disease transmission. It can be seen that isolation has an inhibitory effect on disease transmission, so the isolation of infected individuals is more helpful to control the spread of the disease.

Key words: stochastic epidemic model, Markov switching, stochastic extinction, stochastic persistence

CLC Number: 

  • O175.13
[1] HERBERT H, MA Zhien, LIAO Shengbing. Effects of quarantine in six endemic models for infectious diseases[J]. Mathematical Biosciences, 2002, 180(1/2):141-160.
[2] 马知恩, 周义仓, 王稳地, 等. 传染病动力学的数学建模与研究[M]. 北京: 科学出版社, 2004. MA Zhien, ZHOU Yicang, WANG Wendi, et al. Mathematical modeling and research of infectious disease dynamics[M]. Beijing: Science Press, 2004.
[3] CHEN Junjie. Local stability and global stability of SIQS models for disease[J]. International Journal of Biomathematics, 2004, 19(1):57-64.
[4] WEI Fengying, CHEN Fangxiang. Stochastic permanence of an SIQS epidemic model with saturated incidence and independent random perturbations[J]. Physica A: Statistical Mechanics and its Applications, 2016, 453:99-107.
[5] YANG Xiuxiang, LI Feng, CHENG Yuanji. Global stability analysis on the dynamics of an SIQ model with nonlinear incidence rate[J]. Advances in Intelligent and Soft Computing, 2012, 160:561-565.
[6] GRAY A, GREENHALGH D, HU Linfeng, et al. A stochastic differential equation SIS epidemic model[J]. SIAM Journal on Applied Mathematics, 2011, 71:876-902.
[7] LIN Yuguo, JIANG Daqing. Threshold behavior in a stochastic SIS epidemic model with standard incidence[J]. Journal of Dynamics and Differential Equations, 2014, 26:1079-1094.
[8] CAI Yongli, KANG Yun, WANG Weiming. A stochastic SIRS epidemic model with nonlinear incidence rate[J]. Applied Mathematics and Computation, 2017, 305:221-240.
[9] LIU Qun, JIANG Daqing, HAYAT T, et al. Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates[J]. Journal of the Franklin Institute, 2019, 356:2960-2993.
[10] ZHANG Xinhong, PENG Hao. Stationary distribution of a stochastic cholera epidemic model with vaccination under regime switching[J]. Applied Mathematics Letters, 2020, 102:106095.
[11] ZHANG Xinhong, JIANG Daqing, ALSAEDI A, et al. Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching[J]. Applied Mathematics Letters, 2016, 59:87-93.
[12] PHU N D, OREGAN D, TUONG T D. Longtime characterization for the general stochastic epidemic SIS model under regime-switching[J]. Nonlinear Analysis: Hybrid Systems, 2020, 38(3):100951.
[13] MAO Xuerong, YUAN Chenggui. Stochastic differential equations with Markovian Switching[M]. London: Imperial College Press, 2006.
[14] MAO Xuerong, MARION G, RENSHAW E. Environmental brownian noise suppresses explosions in population dynamics[J]. Stochastic Processes and their Applications, 2002, 97(1):95-110.
[15] LI Dan, LIU Shengqiang, CUI Jingan. Threshold dynamics and ergodicity of an SIRS epidemic model with markovian switching[J]. Journal of Differential Equations, 2017, 263(12):8873-8915.
[1] ZHANG Dao-xiang, HU Wei, TAO Long, ZHOU Wen. Dynamics of a stochastic SIS epidemic model with different incidences and double epidemic hypothesis [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(5): 10-17.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd