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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 916130, 11 pages
http://dx.doi.org/10.1155/2013/916130
Research Article

Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received 28 August 2013; Revised 23 September 2013; Accepted 23 September 2013

Academic Editor: Massimiliano Ferrara

Copyright © 2013 Yakui Xue and Tiantian Li. 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

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any , we show that the disease-free equilibrium is globally asymptotically stable; when , the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for any ; when , the disease will persist. However, for any , the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.