Browse > Article
http://dx.doi.org/10.4134/JKMS.2012.49.4.779

GLOBAL STABILITY OF HIV INFECTION MODELS WITH INTRACELLULAR DELAYS  

Elaiw, Ahmed (Department of Mathematics Faculty of Science King Abdulaziz University, Department of Mathematics Faculty of Science Al-Azhar University)
Hassanien, Ismail (Department of Mathematics Faculty of Science Assiut University)
Azoz, Shimaa (Department of Mathematics Faculty of Science Assiut University)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.4, 2012 , pp. 779-794 More about this Journal
Abstract
In this paper, we study the global stability of two mathematical models for human immunodeficiency virus (HIV) infection with intra-cellular delays. The first model is a 5-dimensional nonlinear delay ODEs that describes the interaction of the HIV with two classes of target cells, $CD4^+$ T cells and macrophages taking into account the saturation infection rate. The second model generalizes the first one by assuming that the infection rate is given by Beddington-DeAngelis functional response. Two time delays are used to describe the time periods between viral entry the two classes of target cells and the production of new virus particles. Lyapunov functionals are constructed and LaSalle-type theorem for delay differential equation is used to establish the global asymptotic stability of the uninfected and infected steady states of the HIV infection models. We have proven that if the basic reproduction number $R_0$ is less than unity, then the uninfected steady state is globally asymptotically stable, and if the infected steady state exists, then it is globally asymptotically stable for all time delays.
Keywords
global stability; HIV dynamics; time delay; direct Lyapunov method;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Y. Wang, Y. Zhou, J. Heffernan, and J. Wu, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci. 219 (2009), no. 2, 104-112.   DOI   ScienceOn
2 Q. Xie, D. Huang, S. Zhang, and J. Cao, Analysis of a viral infection model with delayed immune response, Appl. Math. Model. 34 (2010), no. 9, 2388-2395.   DOI   ScienceOn
3 R. Xu, Global stability of an HIV-1 infection model with saturation infection and intra- cellular delay, J. Math. Anal. Appl. 375 (2011), no. 1, 75-81.   DOI   ScienceOn
4 H. Zhu, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math. Medic. Biol. 25 (2008), 99-112.   DOI   ScienceOn
5 Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl. 375 (2011), 14-27.   DOI   ScienceOn
6 P. W. Nelson, J. Murray, and A. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci. 163 (2000), no. 2, 201-215.   DOI   ScienceOn
7 P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci. 179 (2002), no. 1, 73-94.   DOI   ScienceOn
8 A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41 (1999), no. 1, 3-44.   DOI   ScienceOn
9 M. A. Nowak and R. M. May, Virus Ddynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., Oxford, 2000.
10 A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz, and D. D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature 387 (1997), 188-191.   DOI   ScienceOn
11 X. Shi, X. Zhou, and X. Song, Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear Anal. Real World Appl. 11 (2010), no. 3, 1795-1809.   DOI   ScienceOn
12 X. Song, S. Wang, and J. Dong, Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, J. Math. Anal. Appl. 373 (2011), no. 2, 345-355.   DOI   ScienceOn
13 X. Song, X. Zhou, and X. Zhao, Properties of stability and Hopf bifurcation for a HIV infection model with time delay, Appl. Math. Model. 34 (2010), no. 6, 1511-1523.   DOI   ScienceOn
14 X. Wang, Y. Tao, and X. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response, Nonlinear Dynam. 62 (2010), no. 1-2, 67-72.   DOI
15 A. M. Elaiw and X. Xia, HIV dynamics: Analysis and robust multirate MPC-based treatment schedules, J. Math. Anal. Appl. 356 (2009), no. 1, 285-301.
16 J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
17 A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May, and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 90 (1996), 7247-7251.
18 G. Huang, Y. Takeuchi, and W. Ma, Lyapunov functionals for delay differential equations model of viral infection, SIAM J. Appl. Math. 70 (2010), no. 7, 2693-2708.   DOI   ScienceOn
19 Z. Hu, X. Liu, H. Wang, and W. Ma, Analysis of the dynamics of a delayed HIV pathogenesis model, J. Comput. Appl. Math. 234 (2010), no. 2, 461-475.   DOI   ScienceOn
20 G. Huang, W. Ma, and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett. 24 (2011), no. 7, 1199-1203.   DOI   ScienceOn
21 D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl. 335 (2007), no. 1, 683-691.   DOI   ScienceOn
22 M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol. 72 (2010), no. 6, 1492-1505.   DOI   ScienceOn
23 J. Mittler, B. Sulzer, A. Neumann, and A. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients, Math. Biosci. 152 (1998), 143-163.   DOI   ScienceOn
24 Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis functional response, Nonlinear Anal. 74 (2011), no. 9, 2929-2940.   DOI   ScienceOn
25 A. M. Elaiw, I. A. Hassanien, and S. A. Azoz, Global properties of a class of HIV models with Beddington-DeAngelis functional response, (submitted).
26 D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol. 64 (2002), 29-64.   DOI   ScienceOn
27 R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^+$ T-cells, Math. Biosci. 165 (2000), 27-39.   DOI   ScienceOn
28 A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2253-2263.   DOI   ScienceOn