全部 标题 作者
关键词 摘要


Global Stability of HIV Infection of CD4+ T Cells and Macrophages with CTL Immune Response and Distributed Delays

DOI: 10.1155/2013/653204

Full-Text   Cite this paper   Add to My Lib

Abstract:

We study the global stability of a human immunodeficiency virus (HIV) infection model with Cytotoxic T Lymphocytes (CTL) immune response. The model describes the interaction of the HIV with two classes of target cells, CD4+ T cells and macrophages. Two types of distributed time delays are incorporated into the model to describe the time needed for infection of target cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is determined by two threshold numbers, the basic reproduction number and the immune response reproduction number . We have proven that, if , then the uninfected steady state is globally asymptotically stable (GAS), if , then the infected steady state without CTL immune response is GAS, and, if , then the infected steady state with CTL immune response is GAS. 1. Introduction One of the most diseases that have attracted the attention of many mathematicians is the acquired immunodeficiency syndrome (AIDS) caused by human immunodeficiency virus (HIV). HIV infects the CD4+ T cell which plays the central role in the immune system. Mathematical modeling and model analysis of HIV dynamics are important to discover the dynamical behaviors of the viral infection process and estimating key parameter values which leads to development of efficient antiviral drug therapies. Several mathematical models have been proposed to describe the HIV dynamics with CD4+ T cells [1–15]. In these papers, the Cytotoxic T Lymphocytes (CTL) immune response was not taken into account. The role of CTL is universal and necessary to eliminate or control the disease during viral infections. In particular, as a part of innate response, CTL plays a particularly important rate in antiviral defense by attacking infected cells. The basic HIV infection model which takes into consideration the CTL immune response has been proposed in [16] as The state variables describe the plasma concentrations of , the uninfected CD4+ T cells; , the infected CD4+ T cells; , the free virus particles; and , the CTL cells at time . Here, (1) describes the population dynamics of the uninfected CD4+ T cells, where represents the rate of new uninfected cells that are generated from sources within the body, is the death rate constant, and is the infection rate constant at which a target cell becomes infected via contacting with virus. Equation (2) describes the population dynamics of the infected CD4+ T cells and shows that they die with rate constant and are killed by the CTL immune response with rate constant . Equation (3)

References

[1]  A. S. Perelson and P. W. Nelson, “Mathematical analysis of HIV-1 dynamics in vivo,” SIAM Review, vol. 41, no. 1, pp. 3–44, 1999.
[2]  D. S. Callaway and A. S. Perelson, “HIV-1 infection and low steady state viral loads,” Bulletin of Mathematical Biology, vol. 64, no. 1, pp. 29–64, 2002.
[3]  R. V. Culshaw and S. Ruan, “A delay-differential equation model of HIV infection of CD4+ T-cells,” Mathematical Biosciences, vol. 165, no. 1, pp. 27–39, 2000.
[4]  N. M. Dixit and A. S. Perelson, “Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay,” Journal of Theoretical Biology, vol. 226, no. 1, pp. 95–109, 2004.
[5]  J. E. Mittler, B. Sulzer, A. U. Neumann, and A. S. Perelson, “Influence of delayed viral production on viral dynamics in HIV-1 infected patients,” Mathematical Biosciences, vol. 152, no. 2, pp. 143–163, 1998.
[6]  P. W. Nelson, J. D. Murray, and A. S. Perelson, “A model of HIV-1 pathogenesis that includes an intracellular delay,” Mathematical Biosciences, vol. 163, no. 2, pp. 201–215, 2000.
[7]  P. W. Nelson and A. S. Perelson, “Mathematical analysis of delay differential equation models of HIV-1 infection,” Mathematical Biosciences, vol. 179, pp. 73–94, 2002.
[8]  A. M. Elaiw, “Global properties of a class of HIV models,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2253–2263, 2010.
[9]  A. M. Elaiw and X. Xia, “HIV dynamics: analysis and robust multirate MPC-based treatment schedules,” Journal of Mathematical Analysis and Applications, vol. 359, no. 1, pp. 285–301, 2009.
[10]  A. M. Elaiw, “Global dynamics of an HIV infection model with two classes of target cells and distributed delays,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 253703, 13 pages, 2012.
[11]  A. M. Elaiw and S. A. Azoz, “Global properties of a class of HIV infection models with Beddington-DeAngelis functional response,” Mathematical Methods in the Applied Sciences, vol. 36, pp. 383–394, 2013.
[12]  A. M. Elaiw and A. S. Alsheri, “Global dynamics of HIV infection of CD4+ T cells and macrophages,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 264759, 8 pages, 2013.
[13]  A. M. Elaiw, I. A. Hassanien, and S. A. Azoz, “Global stability of HIV infection models with intracellular delays,” Journal of the Korean Mathematical Society, vol. 49, pp. 779–794, 2012.
[14]  A. M. Elaiw, “Global properties of a class of virus infection models with multitarget cells,” Nonlinear Dynamics, pp. 1–13, 2011.
[15]  A. M. Elaiw and M. A. Alghamdi, “Global properties of virus dynamics models with multitarget cells and discrete-time delays,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 201274, 19 pages, 2011.
[16]  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.
[17]  M. Y. Li and H. Shu, “Impact of intracellular delays and target-cell dynamics on in vivo viral infections,” SIAM Journal on Applied Mathematics, vol. 70, no. 7, pp. 2434–2448, 2010.
[18]  Y. Li, R. Xu, Z. Li, and S. Mao, “Global dynamics of a delayed HIV-1 infection model with CTL immune response,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 673843, 13 pages, 2011.
[19]  X. Shi, X. Zhou, and X. Song, “Dynamical behavior of a delay virus dynamics model with CTL immune response,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1795–1809, 2010.
[20]  Y. Nakata, “Global dynamics of a cell mediated immunity in viral infection models with distributed delays,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 14–27, 2011.
[21]  X. Song, S. Wang, and J. Dong, “Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 345–355, 2011.
[22]  Q. Xie, D. Huang, S. Zhang, and J. Cao, “Analysis of a viral infection model with delayed immune response,” Applied Mathematical Modelling, vol. 34, no. 9, pp. 2388–2395, 2010.
[23]  A. S. Perelson, P. Essunger, Y. Cao et al., “Decay characteristics of HIV-1-infected compartments during combination therapy,” Nature, vol. 387, no. 6629, pp. 188–191, 1997.
[24]  X. Wang, A. Elaiw, and X. Song, “Global properties of a delayed HIV infection model with CTL immune response,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9405–9414, 2012.
[25]  M. Elaiw and A. M. Shehata, “Stability and feedback stabilization of HIV infection model with two classes of target cells,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 963864, 20 pages, 2012.
[26]  J. K. Hale and S. V. Lunel, Introduction To Functional Differential Equations, Springer, New York, NY, USA, 1993.
[27]  C. C. McCluskey, “Complete global stability for an SIR epidemic model with delay—distributed or discrete,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 55–59, 2010.

Full-Text

comments powered by Disqus