Research ArticleOpen AccessGoogle Scholar indexed
Analysis of a Delayed SIR Model with Exponential Birth and Saturated Incidence Rate
Department of Mathematics, North University of China, Taiyuan, China
Department of Mathematics, North University of China, Taiyuan, China
Department of Mathematics, North University of China, Taiyuan, China
- 1 Department of Mathematics, North University of China, Taiyuan, China
- 2 Department of Mathematics, North University of China, Taiyuan, China
- 3 Department of Mathematics, North University of China, Taiyuan, China
Applied Mathematics·Volume 04 (2013)·Pages 60–67·Published 30 September 2013·DOI10.4236/am.2013.410A2006
Copy link · social · email
Abstract
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.
KeywordsExponential BirthSIR ModelTime DelayHurwitz CriterionHopf Bifurcation
- R. M. Anderson and R. M. May, “Population Biology of Infectious Diseases I,” Nature, Vol. 280, 1979, pp. 361367. http://dx.doi.org/10.1038/280361a0
- R. M. May and R. M. Anderson, “Population Biology of Infectious Diseases II,” Nature, Vol. 280, 1979, pp. 455461. http://dx.doi.org/10.1038/280455a0
- H. W. Hethcote and P. van den Driessche, “An SIS EpiDemic Model with Variable Population Size and a Delay,” Journal of Mathematical Biology, Vol. 34, No. 2, 1995, pp. 177-194. http://dx.doi.org/10.1007/BF00178772
- F. Chamchod and N. F. Britton, “Analysis of a VectorBias Model on Malaria Transmission,” Bulletin of Mathematical Biology, Vol. 73, No. 3, 2011, pp. 639-657. http://dx.doi.org/10.1007/s11538-010-9545-0
- K. L. Cooke and P. van den Driessche, “Analysis of an SEIRS Epidemic Model with Two Delays,” Journal of Mathematical Biology, Vol. 35, No. 2, 1996, pp. 240-260. http://dx.doi.org/10.1007/s002850050051
- Y. Takeuchi, W. Ma and E. Beretta, “Global Asymptotic Properties of a Delay SIR Epidemic Model with Finite Incubation Times,” Nonlinear Analysis, Vol. 42, No. 6, 2000, pp. 931-947. http://dx.doi.org/10.1016/S0362-546X(99)00138-8
- J. Mena-Lorca and H. W. Hetheote, “Dynamic Models of Infectious Diseases as Regulators of Population Sizes,” Journal of Mathematical Biology, Vol. 30, No. 7, 1992, pp. 693-716.
- B. K. Mishra and D. K. Saini, “SEIRS Epidemic Model with Delay for Transmission of Malicious Objects in Computer Network,” Applied Mathematics and Computation, Vol. 188, No. 2, 2007, pp. 1476-1482. http://dx.doi.org/10.1016/j.amc.2006.11.012
- M. Y. Li, J. R. Graef, L. Wang and J. Karsai, “Global Dynamics of a SEIR Model with Varying Total Population Size,” Mathematical Biosciences, Vol. 160, No. 2, 1999, pp. 191-213. http://dx.doi.org/10.1016/S0025-5564(99)00030-9
- M. Gabriela, M. Gomes, L. J. White and G. F. Medley, “The Reinfection Threshold,” Journal of Theoretical Biology, Vol. 236, No. 1, 2005, pp. 111-113. http://dx.doi.org/10.1016/j.jtbi.2005.03.001
- Z. Jiang and J. Wei, “Stability and Bifurcation Analysis in a Delayed SIR Model,” Chaos, Solitons & Fractals, Vol. 35, No. 3, 2008, pp. 609-619. http://dx.doi.org/10.1016/j.chaos.2006.05.045
- T. Zhang and Z. Teng, “Global Behavior and Permanence of SIRS Epidemic Model with Time Delay,” Nonlinear Analysis: Real World Applications, Vol. 9, No. 4, 2008, pp. 1409-1424. http://dx.doi.org/10.1016/j.nonrwa.2007.03.010