Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs

Kirui Wesley; Rotich Kiplimo Titus; Bitok Jacob; Lagat Cheruiyot Robert

doi:doi:10.11648/j.sjams.20150304.17

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Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs

Kirui Wesley, Rotich Kiplimo Titus, Bitok Jacob, Lagat Cheruiyot Robert

Published in Science Journal of Applied Mathematics and Statistics (Volume 3, Issue 4)

Received: 29 July 2015 Accepted: 8 August 2015 Published: 14 August 2015

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Abstract

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 (V_I) and noninfectious virus (V_N). 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 ε_c=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.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 3, Issue 4)
DOI	10.11648/j.sjams.20150304.17
Page(s)	204-213
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Equilibrium, Basic Reproductive Number, Delay, Stability, Bifurcation

References

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[6]	W. P. Nelson and A. S. Perelson, "Mathematical analysis of HIV-1 dynamics in vivo," SIAM Rev., vol. 41, p. 3- 44, 1999.
[7]	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.
[8]	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.
[9]	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
[10]	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.
[11]	T. R. Malthus, An essay on the principal of population: Penguin Books. Originally Published in 1798, 1970.
[12]	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.
[13]	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.
[14]	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.
[15]	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.
[16]	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.

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APA Style

Kirui Wesley, Rotich Kiplimo Titus, Bitok Jacob, Lagat Cheruiyot Robert. (2015). Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs. Science Journal of Applied Mathematics and Statistics, 3(4), 204-213. https://doi.org/10.11648/j.sjams.20150304.17

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ACS Style

Kirui Wesley; Rotich Kiplimo Titus; Bitok Jacob; Lagat Cheruiyot Robert. Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs. Sci. J. Appl. Math. Stat. 2015, 3(4), 204-213. doi: 10.11648/j.sjams.20150304.17

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AMA Style

Kirui Wesley, Rotich Kiplimo Titus, Bitok Jacob, Lagat Cheruiyot Robert. Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs. Sci J Appl Math Stat. 2015;3(4):204-213. doi: 10.11648/j.sjams.20150304.17

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@article{10.11648/j.sjams.20150304.17,
  author = {Kirui Wesley and Rotich Kiplimo Titus and Bitok Jacob and Lagat Cheruiyot Robert},
  title = {Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {3},
  number = {4},
  pages = {204-213},
  doi = {10.11648/j.sjams.20150304.17},
  url = {https://doi.org/10.11648/j.sjams.20150304.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20150304.17},
  abstract = {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 forτ1. 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.},
 year = {2015}
}

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TY  - JOUR
T1  - Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs
AU  - Kirui Wesley
AU  - Rotich Kiplimo Titus
AU  - Bitok Jacob
AU  - Lagat Cheruiyot Robert
Y1  - 2015/08/14
PY  - 2015
N1  - https://doi.org/10.11648/j.sjams.20150304.17
DO  - 10.11648/j.sjams.20150304.17
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 204
EP  - 213
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20150304.17
AB  - 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 forτ1. 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.
VL  - 3
IS  - 4
ER  -

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Author Information

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

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APA Style

Kirui Wesley, Rotich Kiplimo Titus, Bitok Jacob, Lagat Cheruiyot Robert. (2015). Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs. Science Journal of Applied Mathematics and Statistics, 3(4), 204-213. https://doi.org/10.11648/j.sjams.20150304.17

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ACS Style

Kirui Wesley; Rotich Kiplimo Titus; Bitok Jacob; Lagat Cheruiyot Robert. Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs. Sci. J. Appl. Math. Stat. 2015, 3(4), 204-213. doi: 10.11648/j.sjams.20150304.17

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AMA Style

Kirui Wesley, Rotich Kiplimo Titus, Bitok Jacob, Lagat Cheruiyot Robert. Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs. Sci J Appl Math Stat. 2015;3(4):204-213. doi: 10.11648/j.sjams.20150304.17

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@article{10.11648/j.sjams.20150304.17,
  author = {Kirui Wesley and Rotich Kiplimo Titus and Bitok Jacob and Lagat Cheruiyot Robert},
  title = {Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {3},
  number = {4},
  pages = {204-213},
  doi = {10.11648/j.sjams.20150304.17},
  url = {https://doi.org/10.11648/j.sjams.20150304.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20150304.17},
  abstract = {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 forτ1. 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.},
 year = {2015}
}

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TY  - JOUR
T1  - Modeling the Effects of Time Delay on HIV-1 in Vivo Dynamics in the Presence of ARVs
AU  - Kirui Wesley
AU  - Rotich Kiplimo Titus
AU  - Bitok Jacob
AU  - Lagat Cheruiyot Robert
Y1  - 2015/08/14
PY  - 2015
N1  - https://doi.org/10.11648/j.sjams.20150304.17
DO  - 10.11648/j.sjams.20150304.17
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 204
EP  - 213
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20150304.17
AB  - 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 forτ1. 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.
VL  - 3
IS  - 4
ER  -

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