Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 636153, 16 pages
doi:10.1155/2008/636153
Research Article
Analysis of a Delayed SIR Model with Nonlinear Incidence Rate
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.
Abstract
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
ℜ0 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 ℜ0 and time
delay (i.e., incubation time length). If ℜ0 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.