Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs
Science Journal of Applied Mathematics and Statistics
Volume 3, Issue 4, August 2015, Pages: 204-213
Received: Jul. 29, 2015;
Accepted: Aug. 8, 2015;
Published: Aug. 14, 2015
Views 3868 Downloads 112
Kirui Wesley, Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya
Rotich Kiplimo Titus, Department of Center for Teacher Education, Moi University, Eldoret, Kenya
Bitok Jacob, Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya
Lagat Cheruiyot Robert, Department of Mathematics and Actuarial Science, South Eastern Kenya University, Kitui, Kenya
Mathematical models to describe in vivo and in vitro immunological response to infection in humans by HIV-1 have been of major concern due to the rich variety of parameters affecting its dynamics. In this paper, HIV-1 in vivo dynamics is studied to predict and describe its evolutions in presence of ARVs using delay differential equations. The delay is used to account for the latent period of time that elapsed between HIV – CD4+ T cell binding (infection) and production of infectious virus from this host cell. The model uses four variables: healthy CD4+T-cells (T), infected CD4+T-cells (T*), infectious virus (VI) and noninfectious virus (VN). Of importance is effect of time delay and drug efficacy on stability of disease free and endemic equilibrium points. Analytical results showed that DFE is stable for all τ>0. On the other hand, there is a critical value of delay τ1>0, such that for all τ>τ1, the EEP is stable but unstable forτ<τ1. The critical value of delayτ1 is the bifurcation value where the HIV-1 in vivo dynamics undergoes a Hopf-bifurcation. This stability in both equilibria is achieved only if the drug efficacy 0≤ε≤1 is above a threshold value of ε_c. Numerical simulations show that this stability is achieved at the drug efficacy of εc=0.59 and time delay of τ1=0.65 days. This ratifies the fact that if CD4+T cells remain inactive for long periods of time τ>τ1 the HIV-1 viral materials will not be reproduced, and the immune system together with treatment will have enough time to clear the viral materials in the blood and thus the EEP is maintain.
Rotich Kiplimo Titus,
Lagat Cheruiyot Robert,
Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs, Science Journal of Applied Mathematics and Statistics.
Vol. 3, No. 4,
2015, pp. 204-213.
M. T. L. Roos, J. M. A. Lange, and R. E. Y. Degoede, "Viral phenotype and immune response in primary HIV-1 infection," J. Infec. Dis., vol. 165, pp. 427-432, 1992.
N. J. Dimmock, A. J. Easton, and K. N. Leppard, Introduction to modern virology. Library of congress cataloging-in-publication data, 6th ed. UK: Blackwell Publishing, 2007.
A. M. Elaiw, “Global Dynamics of an HIV infection model with two classes of Target cells and distributed day”, Discrete Dynamics in Nature and society, vol 2012.
H. W. Hethcotes, “The mathematics of infectious diseases”, SIAM Review 42, No.4 599-653, 2000.
H. Khalid and Y. Noura, “Two optimal treatments of Hiv infection model”, world journal of modeling and simulation vol.8 No.1 27-35, 2012.
W. P. Nelson and A. S. Perelson, "Mathematical analysis of HIV-1 dynamics in vivo," SIAM Rev., vol. 41, p. 3- 44, 1999.
R. V. Culshaw and R. Shigui, “A delay-differential equation models of HIV infection of CD4+T-cells, Mathematical Biosciences vol.165, pp. 27-39, 2000.
K. T. Rotich and C. R. Lagat, “The bounds of time lag and chemotherapeutic efficacy in the control of HIV/AIDS”, IJMR, PAK Publishing Group, 3(6):63-81, 2014.
C. Shanshan, S. Jumping and W. Junjie. “Time-delay induced instabilities and Hopf Bifurcations in general reaction-diffusion systems”. Journal of nonlinear science vol.23 (1-23). 2013
S. Xinyu and C. Shuhan. “A Delay- Differential model of HIV infection of CD4+T-cells”. Journal of Korean Mathematical Society.42 No.5 (1071-1086), 2005.
T. R. Malthus, An essay on the principal of population: Penguin Books. Originally Published in 1798, 1970.
M. Markowitz, Y. Cao, and A. Hurley, "Triple therapy with AZT, 3TC and ritonavir in 12 subjects newly infected with HIV-1," presented at the XI International Conference on AIDS, Vancouver, 11 July 1996, Abstr. Th.B.933, 1996.
R. M. Gulick, J. Mellors, and D. Havlir, "Potent and sustainable antiretroviral activity of indinavir (IDV), zidovudine (ZDV) and lamiduvive (3TC)," presented at the XI International Conference on AIDS, Vancouver, 11 July 1996, Abstr. Th.B.931, 1996.
Y. M. Li and S. Hongying, "Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-1 infection," Bull Math Biol. Springer Verlag, vol. 10, 2010.
N. Radde, "The impact of time delays on the robustness of biological oscillators and the effect of bifurcations on the inverse problem," EURASIP Journal of Bio informatics and Systems Biology. Hindawi Publishing Corporation, vol. 9, 2009.
D. E. Kirschner and G. F. Webb, "A model for treatment strategy in the chemotherapy of AIDS," Bulletin of Mathematical Biology, Elsevier Science Inc, vol. 58, pp. 367 -390, 1996.