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Computational and Mathematical Methods in Medicine
Volume 2017, Article ID 7294761, 20 pages
https://doi.org/10.1155/2017/7294761
Research Article

Threshold Dynamics in Stochastic SIRS Epidemic Models with Nonlinear Incidence and Vaccination

1Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China
2College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Correspondence should be addressed to Zhidong Teng; moc.anis@gnet_gnodihz

Received 30 July 2016; Accepted 21 November 2016; Published 16 January 2017

Academic Editor: Giuseppe Pontrelli

Copyright © 2017 Lei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the dynamical behaviors for a stochastic SIRS epidemic model with nonlinear incidence and vaccination are investigated. In the models, the disease transmission coefficient and the removal rates are all affected by noise. Some new basic properties of the models are found. Applying these properties, we establish a series of new threshold conditions on the stochastically exponential extinction, stochastic persistence, and permanence in the mean of the disease with probability one for the models. Furthermore, we obtain a sufficient condition on the existence of unique stationary distribution for the model. Finally, a series of numerical examples are introduced to illustrate our main theoretical results and some conjectures are further proposed.