Journal of Applied Mathematics
Volume 2013 (2013), Article ID 154387, 15 pages
http://dx.doi.org/10.1155/2013/154387
Research Article

Dynamical Analysis of SIR Epidemic Models with Distributed Delay

Wencai Zhao,1 Tongqian Zhang,1 Zhengbo Chang,1 Xinzhu Meng,2 and Yulin Liu2

1College of Science, Shandong University of Science and Technology, Qingdao 266590, China
2College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China

Received 16 December 2012; Revised 18 June 2013; Accepted 23 June 2013

Academic Editor: Han H. Choi

Copyright © 2013 Wencai Zhao 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

SIR epidemic models with distributed delay are proposed. Firstly, the dynamical behaviors of the model without vaccination are studied. Using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed. The basic reproduction number is got. In order to study the important role of vaccination to prevent diseases, the model with distributed delay under impulsive vaccination is formulated. And the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model are obtained by using Floquet’s theorem, small-amplitude perturbation skills, and comparison theorem. Lastly, numerical simulation is presented to illustrate our main conclusions that vaccination has significant effects on the dynamical behaviors of the model. The results can provide effective tactic basis for the practical infectious disease prevention.