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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 930541, 7 pages
http://dx.doi.org/10.1155/2014/930541
Research Article

Bifurcation Analysis of an SIR Epidemic Model with the Contact Transmission Function

1School of Science, North University of China, Taiyuan, Shanxi 030051, China
2Xinjiang Agriculture Second Division Korla Hospital, Korla, Xinjiang 841000, China

Received 8 December 2013; Accepted 23 December 2013; Published 21 January 2014

Academic Editor: Kaifa Wang

Copyright © 2014 Guihua Li and Gaofeng 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 consider an SIR endemic model in which the contact transmission function is related to the number of infected population. By theoretical analysis, it is shown that the model exhibits the bistability and undergoes saddle-node bifurcation, the Hopf bifurcation, and the Bogdanov-Takens bifurcation. Furthermore, we find that the threshold value of disease spreading will be increased, when the half-saturation coefficient is more than zero, which means that it is an effective intervention policy adopted for disease spreading. However, when the endemic equilibria exist, we find that the disease can be controlled as long as we let the initial values lie in the certain range by intervention policy. This will provide a theoretical basis for the prevention and control of disease.