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Mathematical Problems in Engineering
Volume 2017, Article ID 8098475, 11 pages
Research Article

Global Analysis of a Novel Nonlinear Stochastic SIVS Epidemic System with Vaccination Control

1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Xinzhu Meng; nc.ude.tsuds@601127zxm

Received 2 April 2017; Accepted 3 May 2017; Published 28 May 2017

Academic Editor: Honglei Xu

Copyright © 2017 Xiaona Leng 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.


This paper proposes a stochastic SIVS epidemic system with nonlinear saturated infection rate under vaccination and investigates the dynamics predicted by the model. By using Itô’s formula and Lyapunov methods, we first study the existence and uniqueness of global positive solution. Then we investigate the stochastic dynamics of the system and obtain the thresholds which govern the extinction and the spread of the epidemic disease. Results show that large stochastic noises can lead to the extinction of epidemic diseases; that is, stochastic disturbances can suppress the outbreak of epidemic diseases. Finally, we carry out a series of numerical simulations to demonstrate the performance of our theoretical findings.