Asymptotic Properties of a Stochastic SIR Epidemic Model with Beddington–DeAngelis Incidence Rate

  • Nguyen Thanh DieuEmail author


In this paper, the stochastic SIR epidemic model with Beddington–DeAngelis incidence rate is investigated. We classify the model by introducing a threshold value \(\lambda \). To be more specific, we show that if \(\lambda <0\) then the disease-free is globally asymptotic stable i.e., the disease will eventually disappear while the epidemic is persistence provided that \(\lambda >0\). In this case, we derive that the model under consideration has a unique invariant probability measure. We also depict the support of invariant probability measure and prove the convergence in total variation norm of transition probabilities to the invariant measure. Some of numerical examples are given to illustrate our results.


SIR epidemic model Extinction Permanence Stationary distribution Ergodicity 

Mathematics Subject Classification

34C12 60H10 92D25 



Author would like to thank Vietnam Institute for Advance Study in Mathematics (VIASM) for supporting and providing a fruitful research environment and hospitality. This research was supported by Foundation for Science and Technology Development of Vietnam’s Ministry of Education and Training No. B2015-27-15.


  1. 1.
    Anderson, R.M., May, R.M.: Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford (1991)Google Scholar
  2. 2.
    Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)CrossRefGoogle Scholar
  3. 3.
    Dieu, N.T., Nguyen, D.H., Du, N.H., Yin, G.: Classification of asymptotic behavior in A stochastic SIR model. SIAM J. Appl. Dyn. SystGoogle Scholar
  4. 4.
    Du, N.H., Nguyen, D.H., Yin, G.: Conditions for permanence and ergodicity of certain stochastic predator-prey models. J. Appl. Probab. 53(1), 187–202 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gray, A., Greenhalgh, D., Hu, L., Mao, X., Pan, J.: A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71(3), 876–902 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599–653 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ichihara, K., Kunita, H.: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrsch. Verw. Gebiete 30, 235–254 (1974). Corrections in 39(1977) 81-84MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Publishing Co., Amsterdam (1989)zbMATHGoogle Scholar
  9. 9.
    Kaddar, A.: Stability analysis in a delayed SIR epidemic model with a saturated incidence rate. Nonlinear Anal. Model. Control 15(3), 299–306 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics (part I). Proc. R. Soc. Lond. Ser. A 115, 700–721 (1927)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics (part II). Proc. R. Soc. Ser. A 138, 55–83 (1932)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kliemann, W.: Recurrence and invariant measures for degenerate diffusions. Ann. Probab. 15(2), 690–707 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kortchemski, I.: A predator-prey SIR type dynamics on large complete graphs with three phase transitions. Stoch. Process. Appl. 125(3), 886–917 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lahrouz, A., Omari, L., Kiouach, D.: Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model. Nonlinear Anal. Model. Control 16(1), 59–76 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lin, Y., Jiang, D.: Threshold behavior in a stochastic SIS epidemic model with standard incidence. J. Dyn. Differ. Equ. 26(4), 1079–1094 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mao, X.: Stochastic Differential Equations and Their Applications. Horwood Publishing, Chichester (1997)zbMATHGoogle Scholar
  17. 17.
    Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schurz, H., Tosun, K.: Stochastic asymptotic stability of SIR model with variable diffusion rates. J. Dyn. Differ. Equ. 27(1), 69–82 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Stettner, L.: On the Existence and Uniqueness of Invariant Measure for Continuous Time Markov Processes, LCDS Report No. 86-16. Brown University, Providence (1986)CrossRefGoogle Scholar
  20. 20.
    Tornatore, E., Buccellato, S.M., Vetro, P.: Stability of a stochastic SIR system. Phys. A 354, 111–126 (2005)CrossRefGoogle Scholar
  21. 21.
    Wang, W., Zhao, X.Q.: Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11(4), 1652–1673 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsVinh UniversityVinhVietnam

Personalised recommendations