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Stability of a Delayed SIQRS Model with Temporary Immunity
Department of Mathematics, Faculty of Exact Sciences, University Constantine 1, Algeria
Department of Mathematics, Faculty of Exact Sciences, University Constantine 1, Algeria
- 1 Department of Mathematics, Faculty of Exact Sciences, University Constantine 1, Algeria
- 2 Department of Mathematics, Faculty of Exact Sciences, University Constantine 1, Algeria
Advances in Pure Mathematics·Volume 03 (2013)·Pages 240–245·Published 5 March 2013·DOI10.4236/apm.2013.32034
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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.
KeywordsAdomian MethodEpidemiologyMathematical ModelThe Epidemic ModelThe Equilibrium Points
- 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
- 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.
- 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
- 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
- M. S. Bartlett, “An Introduction to Stochastic Processes,” 3rd Edition, Cambridge University Press, Cambridge, 1978.
- 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.
- 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.
- 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.
- 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.
- 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
- 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
- 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.
- 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.