# zbMATH — the first resource for mathematics

The fractional SIRC model and influenza A. (English) Zbl 1235.92033
Summary: This paper deals with the fractional-order SIRC model associated with the evolution of influenza A disease in human populations. The qualitative dynamics of the model is determined by the basic reproduction number, $$R_0$$. We give a detailed analysis for the asymptotic stability of disease-free and positive fixed points. Nonstandard finite difference methods are used to solve and simulate the system of differential equations.

##### MSC:
 92C60 Medical epidemiology 37N25 Dynamical systems in biology 35Q92 PDEs in connection with biology, chemistry and other natural sciences 65C20 Probabilistic models, generic numerical methods in probability and statistics
Full Text:
##### References:
 [1] P. Palese and J. F. Young, “Variation of influenza A, B, and C viruses,” Science, vol. 215, no. 4539, pp. 1468-1474, 1982. [2] R. Anderson and R. May, Infectious Disease of Humans, Dynamics and Control, Oxford University Press, Oxford, UK, 1995. [3] R. G. Webster, W. J. Bean, O. T. Gorman, T. M. Chambers, and Y. Kawaoka, “Evolution and ecology of influenza A viruses,” Microbiological Reviews, vol. 56, no. 1, pp. 152-179, 1992. [4] W. O. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics,” Proceedings of Royal Society of London, vol. A 115, pp. 700-721, 1927. · JFM 53.0517.01 [5] R. Casagrandi, L. Bolzoni, S. A. Levin, and V. Andreasen, “The SIRC model and influenza A,” Mathematical Biosciences, vol. 200, no. 2, pp. 152-169, 2006. · Zbl 1089.92043 · doi:10.1016/j.mbs.2005.12.029 [6] G. P. Samanta, “Global dynamics of a nonautonomous SIRC model for influenza A with distributed time delay,” Differential Equations and Dynamical Systems, vol. 18, no. 4, pp. 341-362, 2010. · Zbl 1228.34130 · doi:10.1007/s12591-010-0066-y [7] L. Jódar, R. J. Villanueva, A. J. Arenas, and G. C. González, “Nonstandard numerical methods for a mathematical model for influenza disease,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 622-633, 2008. · Zbl 1151.92018 · doi:10.1016/j.matcom.2008.04.008 [8] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 [9] I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. · Zbl 0924.34008 [10] E. H. Elbasha, C. N. Podder, and A. B. Gumel, “Analyzing the dynamics of an SIRS vaccination model with waning natural and vaccine-induced immunity,” Nonlinear Analysis, vol. 12, no. 5, pp. 2692-2705, 2011. · Zbl 1225.37104 · doi:10.1016/j.nonrwa.2011.03.015 [11] R. Gorenflo, J. Loutchko, and Y. Luchko, “Computation of the Mittag-Leffler function E\alpha ,\beta (z) and its derivative,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 491-518, 2002. · Zbl 1027.33016 [12] E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems,” Physics Letters A, vol. 358, no. 1, pp. 1-4, 2006. · Zbl 1142.30303 · doi:10.1016/j.physleta.2006.04.087 [13] D. Matignon, “Stability results for fractional differential equations with applications to control processing,” Computational Engineering in Systems Applications, vol. 2, p. 963, 1996. [14] Y. Ding and H. Ye, “A fractional-order differential equation model of HIV infection of CD4+T -cells,” Mathematical and Computer Modelling, vol. 50, no. 3-4, pp. 386-392, 2009. · Zbl 1185.34005 · doi:10.1016/j.mcm.2009.04.019 [15] N. Özalp and E. Demiörciö, “A fractional order SEIR model with vertical transmission,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 1-6, 2011. · Zbl 1225.34011 · doi:10.1016/j.mcm.2010.12.051 [16] H. Ye and Y. Ding, “Nonlinear dynamics and chaos in a fractional-order HIV model,” Mathematical Problems in Engineering, vol. 2009, Article ID 378614, 12 pages, 2009. · Zbl 1181.37124 · doi:10.1155/2009/378614 · eudml:45837 [17] R. Anguelov and J. M.-S. Lubuma, “Nonstandard finite difference method by nonlocal approximation,” Mathematics and Computers in Simulation, vol. 61, no. 3-6, pp. 465-475, 2003. · Zbl 1015.65034 · doi:10.1016/S0378-4754(02)00106-4 [18] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 2005. · Zbl 1073.65552
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.