Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Stochastic dynamics of SIRS epidemic models withrandom perturbation

1. School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024
2. Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH

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.
  Figure/Table
  Supplementary
  Article Metrics

Keywords extinction; Ergodic property; positive recurrence.

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

References

  • 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.

 

This article has been cited by

  • 1. R. Rajaji, M. Pitchaimani, Analysis of Stochastic Viral Infection Model with Immune Impairment, International Journal of Applied and Computational Mathematics, 2017, 3, 4, 3561, 10.1007/s40819-017-0314-8
  • 2. Tingting Tang, Zhidong Teng, Zhiming Li, Threshold Behavior in a Class of Stochastic SIRS Epidemic Models With Nonlinear Incidence, Stochastic Analysis and Applications, 2015, 33, 6, 994, 10.1080/07362994.2015.1065750
  • 3. Nguyen Huu Du, Nguyen Thanh Dieu, Long-time behavior of an SIR model with perturbed disease transmission coefficient, Discrete and Continuous Dynamical Systems - Series B, 2016, 21, 10, 3429, 10.3934/dcdsb.2016105
  • 4. Lei Wang, Zhidong Teng, Tingting Tang, Zhiming Li, Threshold Dynamics in Stochastic SIRS Epidemic Models with Nonlinear Incidence and Vaccination, Computational and Mathematical Methods in Medicine, 2017, 2017, 1, 10.1155/2017/7294761
  • 5. Ramziya Rifhat, Lei Wang, Zhidong Teng, Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients, Physica A: Statistical Mechanics and its Applications, 2017, 481, 176, 10.1016/j.physa.2017.04.016
  • 6. Jinhu Xu, Yicang Zhou, Global stability of a multi-group model with vaccination age, distributed delay and random perturbation, Mathematical Biosciences and Engineering, 2015, 12, 5, 1083, 10.3934/mbe.2015.12.1083
  • 7. Dan Li, Jing’an Cui, Meng Liu, Shengqiang Liu, The Evolutionary Dynamics of Stochastic Epidemic Model with Nonlinear Incidence Rate, Bulletin of Mathematical Biology, 2015, 77, 9, 1705, 10.1007/s11538-015-0101-9
  • 8. Zhidong Teng, Lei Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physica A: Statistical Mechanics and its Applications, 2016, 451, 507, 10.1016/j.physa.2016.01.084
  • 9. Ramziya Rifhat, Qing Ge, Zhidong Teng, The Dynamical Behaviors in a Stochastic SIS Epidemic Model with Nonlinear Incidence, Computational and Mathematical Methods in Medicine, 2016, 2016, 1, 10.1155/2016/5218163
  • 10. Can Chen, Yanmei Kang, The asymptotic behavior of a stochastic vaccination model with backward bifurcation, Applied Mathematical Modelling, 2016, 40, 11-12, 6051, 10.1016/j.apm.2016.01.045
  • 11. Yongli Cai, Yun Kang, Weiming Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Applied Mathematics and Computation, 2017, 305, 221, 10.1016/j.amc.2017.02.003
  • 12. Tayebeh Waezizadeh, Adel Mehrpooya, Maryam Rezaeizadeh, Shantia Yarahmadian, Mathematical models for the effects of hypertension and stress on kidney and their uncertainty, Mathematical Biosciences, 2018, 10.1016/j.mbs.2018.08.013
  • 13. M. Pitchaimani, M. Brasanna Devi, Effects of randomness on viral infection model with application, IFAC Journal of Systems and Control, 2018, 10.1016/j.ifacsc.2018.09.001
  • 14. S.P. Rajasekar, M. Pitchaimani, Qualitative analysis of stochastically perturbed SIRS epidemic model with two viruses, Chaos, Solitons & Fractals, 2019, 118, 207, 10.1016/j.chaos.2018.11.023
  • 15. R. Rajaji, , Advances in Algebra and Analysis, 2018, Chapter 46, 415, 10.1007/978-3-030-01120-8_46
  • 16. Guoqiang Wang, Jinchen Ji, Jin Zhou, Stochastic distribution synchronization and pinning control for complex heterogeneous dynamical networks, Asian Journal of Control, 2019, 10.1002/asjc.2044
  • 17. Xiaoming Fu, On invariant measures and the asymptotic behavior of a stochastic delayed SIRS epidemic model, Physica A: Statistical Mechanics and its Applications, 2019, 10.1016/j.physa.2019.04.181
  • 18. Mohamed El Fatini, Mohamed El Khalifi, Richard Gerlach, Aziz Laaribi, Regragui Taki, Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence, Physica A: Statistical Mechanics and its Applications, 2019, 120696, 10.1016/j.physa.2019.03.061
  • 19. Ramziya Rifhat, Ahmadjan Muhammadhaji, Zhidong Teng, Asymptotic properties of a stochastic SIRS epidemic model with nonlinear incidence and varying population sizes, Dynamical Systems, 2019, 1, 10.1080/14689367.2019.1620689

Reader Comments

your name: *   your email: *  

Copyright Info: 2014, Qingshan Yang, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved