Analysis of a differential equation model of HIV infection of CD4+ T-cells with saturated reverse function

In this paper, an ordinary differential equation model of HIV infection of CD4+ T-cells with saturated reverse function is studied. We prove that if the basic reproduction number R0

Anahtar Kelimeler:

HIV infection, Globally asymptotical stability, Periodic solution, Permanence

Analysis of a differential equation model of HIV infection of CD4+ T-cells with saturated reverse function

In this paper, an ordinary differential equation model of HIV infection of CD4+ T-cells with saturated reverse function is studied. We prove that if the basic reproduction number R0

Keywords:

HIV infection, Globally asymptotical stability, Periodic solution, Permanence,

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Brauer, F.: Backward bifurcations in simple vaccination models, J. Math. Anal. Appl. 298, 418–431 (2004).
De Boer, R.J., Perelson, A.S.: Target cell limited and immune control models of HIV infection: A comparison, J. Theoret. Biol. 190, 201–214 (1998).
Essunger, P., Perelson, A.S.: Modeling HIV infection of CD4 T-cell subpopulations, J. Theor. Biol. 170, 367–391 (1994).
Gantmacher, F.R.: The Theory of Matrices, Chelsea Publ. Co., New York, 1959.
Hirsch, M.W.: Systems of diﬀerential equations which are competitive or cooperative, IV, SIAM J. Math. Anal. 21, –1234 (1990).
Kepler, T.B., Perelson, A.S.: Cyclic re-entry of germinal center B cells and the eﬃciency of aﬃnity maturation, Immunol. Today 14, 412–415 (1993).
Kirschner D.: Using mathematics to understand HIV immune dynamics, AMS. Notices, 43, 191–202 (1996).
Levy, J.A.: HIV and the pathogenesis of AIDS, Washington DC: ASM Press, 1994.
Liu, W.M., Levin, S.A., Iwasa, Y.: Inﬂuence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol. 23, 187–204 (1986).
Martin R.H., Jr.: Logarithmic norms and projections applied to linear diﬀerential systems, J. Math. Anal. Appl. , 432–454 (1974).
McLean, A.R., Rosado, M.M., Agenes, F., Vasconcellos, R., Freitas, A.A.: Resource competition as a mechanism for B cell homeostasis, Proc. Natl. Acad. Sci. USA 94, 5792–5797 (1997).
Muldowney, J.S.: Dichotomies and asymptotic behaviour for linear diﬀerential systems, Trans. Amer. Math. Soc. , 465–584 (1984).
Muldowney, J. S.: Compound matrices and ordinary diﬀerential equations, Rocky Mountain J. Math. 20, 857–872 (1990).
Nowak, M.A., Bonhoeﬀer, S., Shaw, G.M., May, R.M.: Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations, J. Theor. Biol. 184, 203–217 (1997).
Ouifki, R., Witten, G.: A model of HIV-1 infection with HAART therapy and intracellular delay, Discrete Cont. Dyn-B 8, 229–240 (2007).
Percus, J.K., Percus, O.E., Perelson, A.S.: Predicting the size of the T-cell receptor and antibody combining region from consideration of eﬃcient self-nonself discrimination, Proc. Natl. Acad. Sci. USA 90, 1691–1695 (1993).
Perelson, A.S., Essunger, P., Cao, Y.: Decay characteristics of HIV-1-infected compartments during combination therapy, Nature 387, 188–191 (1997).
Perelson, A.S., Nelson P.W.: Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41, 3–44 (1999).
Smith, H.L.: Monotone dynamical systems: An Introduction to the theory of competitive and cooperative systems, Trans. Amer. Math. Soc. vol. 41, 1995.
Smith, H.L.: Systems of ordinary diﬀerential equations which generate an order preserving ﬂow, SIAM Rev. 30, –98 (1998).
Smith, H.L., Zhu, H.R.: Stable periodic orbits for a class of three-dimensional competitive systems, J. Diﬀerential Equations 110, 143–156 (1994).
Song, X.Y. and Neumann A.U.: Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl. (1), 281–297 (2007).
Srivastava P.K., Banerjee, M., Chandra, P.: Modeling the drug therapy for HIV infection, J. Biol. Syst. 17, 213–223 (2009).
Srivastava, P.K., Chandra, P.: Modeling the dynamics of HIV and CD4+T cells during primary infection, Nonlinear Anal. RWA 11, 612–618 (2010).
Thieme, H.R.: Persistence under relaxed point-dissipativity (with applications to an endemic model), SIAM J. Math. Anal. 24, 407–435 (1993).
Zack, J.A., Arrigo, S.J., Weitsman, S.R., Go, A.S., Haislip A., Chen, I.S.: HIV-1 entry into quiescent primary lymphocytes: molecular analysis reveals a labile latent viral structure, Cell 61, 213–222 (1990).
Zack, J.A., Haislip, A.M., Krogstad, P., Chen, I.S.: Incompletely reverse-transcribed human immunodeŞciency virus type 1 genomes in quiescent cells can function as intermediates in the retroviral cycle, J. Virol. 66, 1717–1725 (1992).
Zandrea Ambrose, John G. Julias, Paul L. Boyer, Vineet N. KewalRamani, Stephen H. Hughes, The level of reverse transcriptase (RT) in human immunodeŞciency virus type 1 particles aﬀects susceptibility to nonnucleoside RT inhibitors but not to Lamivudine, J. Virol. 80, 2578–2581 (2006).
Zhen, J., Haque M., Liu Q.X.: Pulse vaccination in the periodic infection rate SIR epidemic model, International Journal of Biomathematics 1, 409–432 (2008).
Zhou, X.Y., Song X.Y., Shi, X.Y.: A diﬀerential equation model of HIV infection of CD4+T-cells with cure rate, J. Math. Anal. Appl. 34, 1342–1355 (2008). Xiangyun SHI
College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Henan-P.R. CHINA Gang LI
School of Computer and Information Technology, Xinyang Normal University, Xinyang 464000, Henan-P.R. CHINA Xueyong ZHOU
College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Henan-P.R. CHINA e-mail: xueyongzhou@126.com Xinyu SONG
College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Henan-P.R. CHINA