Conditions for Permanence and Ergodicity of Certain SIR Epidemic Models

  • Nguyen Huu Du
  • Nguyen Thanh Dieu
  • Nguyen Ngoc Nhu


In this paper, we study sufficient conditions for the permanence and ergodicity of a stochastic susceptible-infected-recovered (SIR) epidemic model with Beddington-DeAngelis incidence rate in both of non-degenerate and degenerate cases. The conditions obtained in fact are close to the necessary one. We also characterize the support of the invariant probability measure and prove the convergence in total variation norm of the transition probability to the invariant measure. Some of numerical examples are given to illustrate our results.


SIR model Extinction Permanence Stationary distribution Ergodicity 

Mathematics Subject Classification

34C12 60H10 92D25 



We gratefully thank the reviewers and the editor for their positive and helpful suggestions, which help to improve the presentation of the paper. Authors would like to express our gratitude to Nguyen Hai Dang for his valuable comments which helped to improve the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03-2017.23.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Nguyen Huu Du
    • 1
  • Nguyen Thanh Dieu
    • 2
  • Nguyen Ngoc Nhu
    • 1
  1. 1.Department of Mathematics, Mechanics and InformaticsHanoi National UniversityHanoiVietnam
  2. 2.Department of MathematicsVinh UniversityVinhVietnam

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