Stability of a Delayed SIQRS Model with Temporary Immunity

Affiliation(s)

Department of Mathematics, Faculty of Exact Sciences, University Constantine 1, Algeria.

Department of Mathematics, Faculty of Exact Sciences, University Constantine 1, Algeria.

ABSTRACT

This paper addresses a time-delayed SIQRS model with a linear incidence rate. Immunity gained by experiencing the disease is temporary; whenever infected, the disease individuals will return to the susceptible class after a fixed period of time. First, the local and global stabilities of the infection-free equilibrium are analyzed, respectively. Second, the endemic equilibrium is formulated in terms of the incidence rate, and locally asymptotic stability. Finally we use the adomian decomposition method is applied to the system epidemiologic. This method yields an analytical solution in terms of convergent infinite power series.

KEYWORDS

Adomian Method; Epidemiology; Mathematical Model; The Epidemic Model; The Equilibrium Points

Adomian Method; Epidemiology; Mathematical Model; The Epidemic Model; The Equilibrium Points

Cite this paper

L. Chahrazed and R. Lazhar, "Stability of a Delayed SIQRS Model with Temporary Immunity,"*Advances in Pure Mathematics*, Vol. 3 No. 2, 2013, pp. 240-245. doi: 10.4236/apm.2013.32034.

L. Chahrazed and R. Lazhar, "Stability of a Delayed SIQRS Model with Temporary Immunity,"

References

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[6] B. Batiha, M. S. M. Noorani and I. Hashim, “Numerical Solutions of the Nonlinear Integro-Differential Equations,” International Journal of Open Problems in Computer Science, Vol. 1, No. 1, 2008, pp. 34-42.

[7] D. J. Evansa and K. R. Raslan, “The Adomian Decompositio Methode for Solving Delay Differential Equation,” International Journal of Computer Mathematics, Vol. 00, No. 0, 2004, pp.1-6.

[8] H. A. Zedan and Al-A. Eman, “Numerical Solutions for a Generalized Ito System by Using Adomian Decomposition Method,” International Journal of Mathematics and Computation, Vol. 4, No. S09. 2009, pp. 9-19.

[9] D. Kaya and Inc, “On the Solution of the Nonlinear Wave Equation by the Decomposition Method,” Bull. Malaysian Math. Soc. (Second Series) 22. 1999, p. 151-155.

[10] K. R. Raslan, “The Decomposition Methode for a Hirota-Satsuma Coupled KdV Equation and a Coupled MKdV Equation,” International Journal of Computer Mathematics, Vol. 81, No. 12, 2004, pp. 1497-1505. doi:10.1080/0020716042000261405

[11] S. Pamuk, “An Application for Linear and Nonlinear Heat Equations by Adomian’s Decomposition Method,” Applied Mathematics and Computation, Vol. 163, No. 1, 2005, pp. 89-96. doi:10.1016/j.amc.2003.10.051

[12] T. M.-D. Syed, “On Numerical Solutions of Two-Dimensional Boussinesq Equations by Using Adomian Decomposition and He’s Homotopy Perturbation Method,” Applications and Applied Mathematics. An International Journal, No. 1, 2010, pp. 1-11.

[13] V. Makarov and D. Dragunov, “A Numeric-Analytical Method for Solving the Cauchy Problem for Ordinary Diferential Equations,” Applied Mathematics and Computation, 2010, pp. 1-26.

[14] L. Wu, F.-D. Zong and J.-F. Zhang, “Adomian Decomposition Method for Nonlinear Differential-Difference Equation,” Communications in Theoretical Physics, Vol. 48, No. 6, 2007, pp. 983-986. doi:10.1088/0253-6102/48/6/004

[15] K. Wang, W. Wang, X. Liu, “Viral Infection Model with Periodic Lytic Immune Response,” Chaos, Solitons & Fractals, Vol. 28, No. 1, 2006, pp. 90-99. doi:10.1016/j.chaos.2005.05.003

[16] W. Wang, “Global Behavior of an SEIRS Epidemic Model with Time Delays,” Applied Mathematics Letters, Vol. 15, No. 4, 2002, pp. 423-428. doi:10.1016/S0893-9659(01)00153-7

[17] D. Greenhalgh, Q. J. A. Khanand and F. I. Lewis, “Recurrent Epidemic Cycles in an Infectious Disease Model with a Time Delay in Loss of Vaccine Immunity,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 63, No. 5-7, 2005, pp. 779-788. doi:10.1016/j.na.2004.12.018

[18] G. Li and J. Zhen, “Global Stability of a SEIR Epidemic Model with Infectious Force in Latent, Infected and Immune Period,” Chaos, Solitons & Fractals, Vol. 25, No. 5, 2005, pp. 1177-1184. doi:10.1016/j.chaos.2004.11.062

[19] Y. N. Kyrychko and K. B. Nlyuss, “Global Properties of a Delayed SIR Model with Temporary Immunity and Nonlinear Incidence Rate,” Nonlinear Analysis: Real World Applications, Vol. 6, No. 3, 2005, pp. 495-507. doi:10.1016/j.nonrwa.2004.10.001

[1] R. M. Anderson, et al., “A Preliminary Study of the Transmission Dynamics of the Human Immunodeficiency Virus (HIV), the Causative Agent of AIDS,” Mathematical Medicine and Biology, Vol. 3, No. 4, 1986, p. 229-263. doi:10.1093/imammb/3.4.229

[2] Y. Asif and K. Dogan, “A Numerical Comparison for Coupled Boussines Equations by Using the ADM,” Proceedings of Dynamical Systems and Applications, 5-10 July 2004, Antalya, pp. 730-736.

[3] N. T. J. Bailley, “Some Stochastic Models for Small Epidemics in Large Population,” Applied Statistics, Vol. 13, No. 1, 1964, pp. 9-19. doi:10.2307/2985218

[4] N. T. J. Bailley, “The Mathematical Theory of Infection Diseases and Its Application,” Applied Statistics, Vol. 26, No. 1, 1977, pp. 85-87. doi:10.2307/2346882

[5] M. S. Bartlett, “An Introduction to Stochastic Processes,” 3rd Edition, Cambridge University Press, Cambridge, 1978.

[6] B. Batiha, M. S. M. Noorani and I. Hashim, “Numerical Solutions of the Nonlinear Integro-Differential Equations,” International Journal of Open Problems in Computer Science, Vol. 1, No. 1, 2008, pp. 34-42.

[7] D. J. Evansa and K. R. Raslan, “The Adomian Decompositio Methode for Solving Delay Differential Equation,” International Journal of Computer Mathematics, Vol. 00, No. 0, 2004, pp.1-6.

[8] H. A. Zedan and Al-A. Eman, “Numerical Solutions for a Generalized Ito System by Using Adomian Decomposition Method,” International Journal of Mathematics and Computation, Vol. 4, No. S09. 2009, pp. 9-19.

[9] D. Kaya and Inc, “On the Solution of the Nonlinear Wave Equation by the Decomposition Method,” Bull. Malaysian Math. Soc. (Second Series) 22. 1999, p. 151-155.

[10] K. R. Raslan, “The Decomposition Methode for a Hirota-Satsuma Coupled KdV Equation and a Coupled MKdV Equation,” International Journal of Computer Mathematics, Vol. 81, No. 12, 2004, pp. 1497-1505. doi:10.1080/0020716042000261405

[11] S. Pamuk, “An Application for Linear and Nonlinear Heat Equations by Adomian’s Decomposition Method,” Applied Mathematics and Computation, Vol. 163, No. 1, 2005, pp. 89-96. doi:10.1016/j.amc.2003.10.051

[12] T. M.-D. Syed, “On Numerical Solutions of Two-Dimensional Boussinesq Equations by Using Adomian Decomposition and He’s Homotopy Perturbation Method,” Applications and Applied Mathematics. An International Journal, No. 1, 2010, pp. 1-11.

[13] V. Makarov and D. Dragunov, “A Numeric-Analytical Method for Solving the Cauchy Problem for Ordinary Diferential Equations,” Applied Mathematics and Computation, 2010, pp. 1-26.

[14] L. Wu, F.-D. Zong and J.-F. Zhang, “Adomian Decomposition Method for Nonlinear Differential-Difference Equation,” Communications in Theoretical Physics, Vol. 48, No. 6, 2007, pp. 983-986. doi:10.1088/0253-6102/48/6/004

[15] K. Wang, W. Wang, X. Liu, “Viral Infection Model with Periodic Lytic Immune Response,” Chaos, Solitons & Fractals, Vol. 28, No. 1, 2006, pp. 90-99. doi:10.1016/j.chaos.2005.05.003

[16] W. Wang, “Global Behavior of an SEIRS Epidemic Model with Time Delays,” Applied Mathematics Letters, Vol. 15, No. 4, 2002, pp. 423-428. doi:10.1016/S0893-9659(01)00153-7

[17] D. Greenhalgh, Q. J. A. Khanand and F. I. Lewis, “Recurrent Epidemic Cycles in an Infectious Disease Model with a Time Delay in Loss of Vaccine Immunity,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 63, No. 5-7, 2005, pp. 779-788. doi:10.1016/j.na.2004.12.018

[18] G. Li and J. Zhen, “Global Stability of a SEIR Epidemic Model with Infectious Force in Latent, Infected and Immune Period,” Chaos, Solitons & Fractals, Vol. 25, No. 5, 2005, pp. 1177-1184. doi:10.1016/j.chaos.2004.11.062

[19] Y. N. Kyrychko and K. B. Nlyuss, “Global Properties of a Delayed SIR Model with Temporary Immunity and Nonlinear Incidence Rate,” Nonlinear Analysis: Real World Applications, Vol. 6, No. 3, 2005, pp. 495-507. doi:10.1016/j.nonrwa.2004.10.001