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Nonlinear dynamics and chaos in a fractional-order HIV model. (English) Zbl 1181.37124
Summary: We introduce fractional order into an HIV model. We consider the effect of viral diversity on the human immune system with frequency dependent rate of proliferation of cytotoxic T-lymphocytes (CTLs) and rate of elimination of infected cells by CTLs, based on a fractional-order differential equation model. For the one-virus model, our analysis shows that the interior equilibrium which is unstable in the classical integer-order model can become asymptotically stable in our fractional-order model and numerical simulations confirm this. We also present simulation results of the chaotic behaviors produced from the fractional-order HIV model with viral diversity by using an Adams-type predictor-corrector method.

MSC:
37N25Dynamical systems in biology
37D45Strange attractors, chaotic dynamics
92C37Cell biology
26A33Fractional derivatives and integrals (real functions)
28A80Fractals
WorldCat.org
Full Text: DOI EuDML
References:
[1] M. A. Nowak, R. M. May, and K. Sigmund, “Immune responses against multiple epitopes,” Journal of Theoretical Biology, vol. 175, no. 3, pp. 325-353, 1995. · doi:10.1006/jtbi.1995.0146
[2] M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune responses to persistent viruses,” Science, vol. 272, no. 5258, pp. 74-79, 1996. · doi:10.1126/science.272.5258.74
[3] M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, UK, 2000. · Zbl 1101.92028
[4] S. Iwami, S. Nakaoka, and Y. Takeuchi, “Frequency dependence and viral diversity imply chaos in an HIV model,” Physica D, vol. 223, no. 2, pp. 222-228, 2006. · Zbl 1126.34032 · doi:10.1016/j.physd.2006.09.011
[5] E. Ahmed and A. S. Elgazzar, “On fractional order differential equations model for nonlocal epidemics,” Physica A, vol. 379, no. 2, pp. 607-614, 2007. · doi:10.1016/j.physa.2007.01.010
[6] E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 542-553, 2007. · Zbl 1105.65122 · doi:10.1016/j.jmaa.2006.01.087
[7] M. P. Lazarević, “Finite time stability analysis of PD\alpha fractional control of robotic time-delay systems,” Mechanics Research Communications, vol. 33, no. 2, pp. 269-279, 2006. · Zbl 1192.70008 · doi:10.1016/j.mechrescom.2005.08.010
[8] T. J. Anastasio, “The fractional-order dynamics of brainstem vestibulo-oculomotor neurons,” Biological Cybernetics, vol. 72, no. 1, pp. 69-79, 1994. · doi:10.1007/BF00206239
[9] T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua’s system,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485-490, 1995. · doi:10.1109/81.404062
[10] B. Bonilla, M. Rivero, L. Rodríguez-Germá, and J. J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equations,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 79-88, 2007. · Zbl 1120.34323 · doi:10.1016/j.amc.2006.08.105
[11] A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “On the fractional-order logistic equation,” Applied Mathematics Letters, vol. 20, no. 7, pp. 817-823, 2007. · Zbl 1140.34302 · doi:10.1016/j.aml.2006.08.013
[12] R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. · Zbl 0998.26002
[13] K. S. Cole, “Electric conductance of biological systems,” in Proceedings of the Cold Spring Harbor Symposia on Quantitative Biology, pp. 107-116, Cold Spring Harbor, NY, USA, January 1993.
[14] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[15] Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor’s formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286-293, 2007. · Zbl 1122.26006 · doi:10.1016/j.amc.2006.07.102
[16] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[17] D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in Systems Applications, vol. 2, pp. 963-968, IMACS IEEE-SMC, Lille, France, 1996.
[18] J. Velasco-Hernández, J. García, and D. Kirschner, “Remarks on modeling host-viral dynamics and treatment,” in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods, and Theory, vol. 125 of The IMA Volumes in Mathematics and Its Applications, pp. 287-308, Springer, New York, NY, USA, 2002. · Zbl 1021.92017
[19] R. R. Regoes, D. Wodarz, and M. A. Nowak, “Virus dynamics: the effect of target cell limitation and immune responses on virus evolution,” Journal of Theoretical Biology, vol. 191, no. 4, pp. 451-462, 1998. · doi:10.1006/jtbi.1997.0617
[20] 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
[21] A. S. Perelson, D. E. Kirschner, and R. De Boer, “Dynamics of HIV infection of CD4+ T cells,” Mathematical Biosciences, vol. 114, no. 1, pp. 81-125, 1993. · Zbl 0796.92016 · doi:10.1016/0025-5564(93)90043-A
[22] K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1-4, pp. 3-22, 2002. · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[23] K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31-52, 2004. · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be
[24] Y. Iwasa, F. Michor, and M. Nowak, “Some basic properties of immune selection,” Journal of Theoretical Biology, vol. 229, no. 2, pp. 179-188, 2004. · doi:10.1016/j.jtbi.2004.03.013