Stochastic dynamics of SIRS epidemic models withrandom perturbation

  • Received: 01 February 2013 Accepted: 29 June 2018 Published: 01 March 2014
  • MSC : 37H10, 37H15, 60J60.

  • In this paper, we consider a stochastic SIRS model with parameterperturbation, which is a standard technique in modeling populationdynamics. In our model, the disease transmission coefficient and theremoval rates are all affected by noise. We show that the stochasticmodel has a unique positive solution as is essential in anypopulation model. Then we establish conditions for extinction orpersistence of the infectious disease. When the infective part isforced to expire, the susceptible part converges weakly to aninverse-gamma distribution with explicit shape and scale parameters.In case of persistence, by new stochastic Lyapunov functions, weshow the ergodic property and positive recurrence of the stochasticmodel. We also derive the an estimate for the mean of the stationarydistribution. The analytical results are all verified by computersimulations, including examples based on experiments in laboratorypopulations of mice.

    Citation: Qingshan Yang, Xuerong Mao. Stochastic dynamics of SIRS epidemic models withrandom perturbation[J]. Mathematical Biosciences and Engineering, 2014, 11(4): 1003-1025. doi: 10.3934/mbe.2014.11.1003

    Related Papers:

  • In this paper, we consider a stochastic SIRS model with parameterperturbation, which is a standard technique in modeling populationdynamics. In our model, the disease transmission coefficient and theremoval rates are all affected by noise. We show that the stochasticmodel has a unique positive solution as is essential in anypopulation model. Then we establish conditions for extinction orpersistence of the infectious disease. When the infective part isforced to expire, the susceptible part converges weakly to aninverse-gamma distribution with explicit shape and scale parameters.In case of persistence, by new stochastic Lyapunov functions, weshow the ergodic property and positive recurrence of the stochasticmodel. We also derive the an estimate for the mean of the stationarydistribution. The analytical results are all verified by computersimulations, including examples based on experiments in laboratorypopulations of mice.
    加载中
    [1] Stoch Anal Appl., 26 (2008), 274-297.
    [2] Nature., 280 (1979), 361-367, doi: 10.1038/280361a0.
    [3] Second edition. Hafner Press [Macmillan Publishing Co., Inc.] New York, 1975.
    [4] Ann Probab., 20 (1992), 312-321.
    [5] J. Math. Biol., 63 (2011), 433-457.
    [6] Springer, Berlin, 1993.
    [7] Stoch Dynam., 9 (2009), 231-252.
    [8] Ann. Math. Statist., 36 (1965), 552-558.
    [9] SIAM J. Appl. Math., 71 (2011), 876-902.
    [10] Alphen aan den Rijn, The Netherlands, 1980.
    [11] J. Math. Biol., 9 (1980), 37-47.
    [12] J. Differ. Equations., 217 (2005), 26-53.
    [13] Appl. Math. Lett., 15 (2002), 955-960.
    [14] Springer, London, 2004.
    [15] J. Math. Biol., 23 (1986), 187-204.
    [16] J. Math. Biol., 25 (1987), 359-380.
    [17] Phys. A., 388 (2009), 3677-3686.
    [18] Appl. Math. Lett., 17 (2004), 1141-1145.
    [19] Longman Scientific & Technical Harlow, UK, 1991.
    [20] Marcel Dekker, New York, 1994.
    [21] 2nd ed., Horwood Publishing, Chichester, 1997.
    [22] Springer-Verlag, Berlin, 1989.
    [23] Math. Biosci., 179 (2002), 1-19.
    [24] Phys. A., 354 (2005), 111-126.
    [25] SIAM. J. Control. Optim., 46 (2007), 1155-1179.

    © 2014 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
  • Reader Comments
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(21) PDF downloads(528) Cited by(26)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog