GLOBAL STABILITY OF HIV INFECTION MODELS WITH INTRACELLULAR DELAYS

Title & Authors
GLOBAL STABILITY OF HIV INFECTION MODELS WITH INTRACELLULAR DELAYS
Elaiw, Ahmed; Hassanien, Ismail; Azoz, Shimaa;

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, $\small{CD4^+}$ 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 $\small{R_0}$ 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;
Language
English
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27.
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References
1.
D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol. 64 (2002), 29-64.

2.
R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of \$CD4^+\$ T-cells, Math. Biosci. 165 (2000), 27-39.

3.
A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2253-2263.

4.
A. M. Elaiw, I. A. Hassanien, and S. A. Azoz, Global properties of a class of HIV models with Beddington-DeAngelis functional response, (submitted).

5.
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.

6.
J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.

7.
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.

8.
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.

9.
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.

10.
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.

11.
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.

12.
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.

13.
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.

14.
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.

15.
Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl. 375 (2011), 14-27.

16.
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.

17.
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.

18.
M. A. Nowak and R. M. May, Virus Ddynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., Oxford, 2000.

19.
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.

20.
A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41 (1999), no. 1, 3-44.

21.
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.

22.
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.

23.
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.

24.
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.

25.
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.

26.
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.

27.
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.

28.
H. Zhu, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math. Medic. Biol. 25 (2008), 99-112.