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

  • Nguyen Thanh Dieu
Article

Abstract

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.

Keywords

SIR epidemic model Extinction Permanence Stationary distribution Ergodicity 

Mathematics Subject Classification

34C12 60H10 92D25 

Notes

Acknowledgements

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.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsVinh UniversityVinhVietnam

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