Analysis of a Delayed SIR Model with Exponential Birth and Saturated Incidence Rate

In this paper, a delayed SIR model with exponential demographic structure and the saturated incidence rate is formulated. The stability of the equilibria is analyzed with delay: the endemic equilibrium is locally stable without delay; and the endemic equilibrium is stable if the delay is under some condition. Moreover the dynamical behaviors from stability to instability will change with an appropriate critical value. At last, some numerical simulations of the model are given to illustrate the main theoretical results.

[1]  R. May,et al.  Population biology of infectious diseases: Part II , 1979, Nature.

[2]  Zhidong Teng,et al.  Global behavior and permanence of SIRS epidemic model with time delay , 2008 .

[3]  Abdelilah Kaddar,et al.  On the dynamics of a delayed SIR epidemic model with a modified saturated incidence rate , 2009 .

[4]  Dinesh Kumar Saini,et al.  SEIRS epidemic model with delay for transmission of malicious objects in computer network , 2007, Appl. Math. Comput..

[5]  M. Li,et al.  Global dynamics of a SEIR model with varying total population size. , 1999, Mathematical Biosciences.

[6]  Herbert W. Hethcote,et al.  An SIS epidemic model with variable population size and a delay , 1995, Journal of mathematical biology.

[7]  Rui Xu,et al.  Global stability of a delayed SEIRS epidemic model with saturation incidence rate , 2010 .

[8]  Nicholas F Britton,et al.  Analysis of a Vector-Bias Model on Malaria Transmission , 2011, Bulletin of mathematical biology.

[9]  Rui Xu,et al.  Global stability of a SIR epidemic model with nonlinear incidence rate and time delay , 2009 .

[10]  Yasuhiro Takeuchi,et al.  Global asymptotic properties of a delay SIR epidemic model with finite incubation times , 2000 .

[11]  K. L. Cooke,et al.  Analysis of an SEIRS epidemic model with two delays , 1996, Journal of mathematical biology.

[12]  Zhan-Ping Ma,et al.  Dynamics of a delayed epidemic model with non-monotonic incidence rate , 2010 .

[13]  Abdelilah Kaddar,et al.  Stability analysis in a delayed SIR epidemic model with a saturated incidence rate , 2010 .

[14]  Junjie Wei,et al.  Stability and bifurcation analysis in a delayed SIR model , 2008 .

[15]  Rui Xu,et al.  Stability of a delayed SIRS epidemic model with a nonlinear incidence rate , 2009 .

[16]  Herbert W. Hethcote,et al.  Dynamic models of infectious diseases as regulators of population sizes , 1992, Journal of mathematical biology.

[17]  G. Serio,et al.  A generalization of the Kermack-McKendrick deterministic epidemic model☆ , 1978 .

[18]  Rui Xu,et al.  Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity , 2010, Comput. Math. Appl..

[19]  Xue-Zhi Li,et al.  An epidemic model of a vector-borne disease with direct transmission and time delay , 2008 .

[20]  Graham F Medley,et al.  The reinfection threshold. , 2005, Journal of theoretical biology.

[21]  R. May,et al.  Population Biology of Infectious Diseases , 1982, Dahlem Workshop Reports.