Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 636153, 16 pages
Research Article

Analysis of a Delayed SIR Model with Nonlinear Incidence Rate

Jin-Zhu Zhang,1,2,3 Zhen Jin,3 Quan-Xing Liu,3 and Zhi-Yu Zhang2

1School of Mechantronic Engineering, North University of China, Taiyuan 030051, China
2Department of Basic Science, Taiyuan Institute of Technology, Taiyuan 030008, China
3Department of Mathematics, North University of China, Taiyuan 030051, China

Received 8 September 2008; Accepted 6 October 2008

Academic Editor: Manuel de La Sen

Copyright © 2008 Jin-Zhu Zhang 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.


An SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. The threshold value 0 determining whether the disease dies out is found. The results obtained show that the global dynamics are completely determined by the values of the threshold value 0 and time delay (i.e., incubation time length). If 0 is less than one, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, while if it exceeds one there will be an endemic. By using the time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical results demonstrate that the system with time delay exhibits rich complex dynamics, such as quasiperiodic and chaotic patterns.