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Mathematical Problems in Engineering
Volume 2016, Article ID 2034136, 8 pages
Research Article

Stability and Hopf Bifurcation Analysis of an Epidemic Model by Using the Method of Multiple Scales

College of Science, Henan University of Engineering, Zhengzhou 451191, China

Received 7 May 2016; Accepted 2 August 2016

Academic Editor: Oleg V. Gendelman

Copyright © 2016 Wanyong Wang and Lijuan Chen. 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.


A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.