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

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.

Cite this paper

W. Wang, M. Liu and J. Zhao, "Analysis of a Delayed SIR Model with Exponential Birth and Saturated Incidence Rate,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 60-67. doi: 10.4236/am.2013.410A2006.

W. Wang, M. Liu and J. Zhao, "Analysis of a Delayed SIR Model with Exponential Birth and Saturated Incidence Rate,"

References

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[17] A. Abta, A. Kaddar and H. T. Alaoui, “Global Stability for Delay SIR and SEIR Epidemic Models with Saturated Incidence Rates,” Journal of Differential Equations, Vol. 2012, No. 23, 2012, pp. 1-13.

[18] H. Wei, X. Li and M. Martcheva, “An Epidemic Model of a Vector-Borne Disease with Direct Transmission and Time Delay,” Journal of Mathematical Analysis and Applications, Vol. 342, No. 2, 2008, pp. 895-908. http://dx.doi.org/10.1016/j.jmaa.2007.12.058

[19] R. Xu and Z. Ma, “Global Stability of a Delayed SEIRS Epidemic Model with Saturation Incidence Rate,” Nonlinear Dynamics, Vol. 61, No. 1, 2010, pp. 229-239. http://dx.doi.org/10.1007/s11071-009-9644-3

[20] R. Xu, Z. Ma and Z. Wang, “Global Stability of a Delayed SIRS Epidemic Model with Saturation Incidence Rate and Temporary,” Computers & Mathematics with Applications, Vol. 59, No. 9, 2010, pp. 3211-3221. http://dx.doi.org/10.1016/j.camwa.2010.03.009

[21] H. Huo and Z. Ma, “Dynamics of a Delayed Epidemic Model with Non-Monotonic Incidence Rate,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 2, 2010, pp. 459-468.

[22] R. Xu and Z. Ma, “Stability of a Delayed SIRS Epidemic Model with a Nonlinear Incidence Rate,” Chaos, Solitons & Fractals, Vol. 41, No. 5, 2009, pp. 2319-2325. http://dx.doi.org/10.1016/j.chaos.2008.09.007

[1] 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

[2] 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

[3] 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

[4] 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

[5] 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

[6] 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

[7] 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.

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] V. Capasso and G. Serio, “A Generalization of the Kermack-Mckendrick Deterministic Epidemic Model,” Mathematical Biosciences, Vol. 42, No. 1-2, 1978, pp. 41-61. http://dx.doi.org/10.1016/0025-5564(78)90006-8

[14] A. Kaddar, “On the Dynamics of a Delayed SIR Epidemic Model with a Modified Saturated Incidence Rate,” Journal of Differential Equations, Vol. 2009, No. 133, 2009, pp. 1-7.

[15] R. Xu and Z. Ma, “Global Stability of a SIR Epidemic Model with Nonlinear Incidence Rate and Time Delay,” Nonlinear Analysis: Real World Applications, Vol. 10, No. 5, 2009, pp. 3175-3189. http://dx.doi.org/10.1016/j.nonrwa.2008.10.013

[16] A. Kaddar, A. Abta and H. T. Alaoui, “Stability Analysis in a Delayed SIR Epidemic Model with a Saturated Incidence Rate,” Nonlinear Analysis: Modelling and Control, Vol. 15, No. 3, 2010, pp. 299-306.

[17] A. Abta, A. Kaddar and H. T. Alaoui, “Global Stability for Delay SIR and SEIR Epidemic Models with Saturated Incidence Rates,” Journal of Differential Equations, Vol. 2012, No. 23, 2012, pp. 1-13.

[18] H. Wei, X. Li and M. Martcheva, “An Epidemic Model of a Vector-Borne Disease with Direct Transmission and Time Delay,” Journal of Mathematical Analysis and Applications, Vol. 342, No. 2, 2008, pp. 895-908. http://dx.doi.org/10.1016/j.jmaa.2007.12.058

[19] R. Xu and Z. Ma, “Global Stability of a Delayed SEIRS Epidemic Model with Saturation Incidence Rate,” Nonlinear Dynamics, Vol. 61, No. 1, 2010, pp. 229-239. http://dx.doi.org/10.1007/s11071-009-9644-3

[20] R. Xu, Z. Ma and Z. Wang, “Global Stability of a Delayed SIRS Epidemic Model with Saturation Incidence Rate and Temporary,” Computers & Mathematics with Applications, Vol. 59, No. 9, 2010, pp. 3211-3221. http://dx.doi.org/10.1016/j.camwa.2010.03.009

[21] H. Huo and Z. Ma, “Dynamics of a Delayed Epidemic Model with Non-Monotonic Incidence Rate,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 2, 2010, pp. 459-468.

[22] R. Xu and Z. Ma, “Stability of a Delayed SIRS Epidemic Model with a Nonlinear Incidence Rate,” Chaos, Solitons & Fractals, Vol. 41, No. 5, 2009, pp. 2319-2325. http://dx.doi.org/10.1016/j.chaos.2008.09.007