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International Journal of Mathematical Research

December 2014, Volume 3, 6, pp 63-81

The Bounds of Time Lag and Chemotherapeutic Efficacy in the Control of HIV/AIDS

Rotich K. Titus


Lagat C. Robert

Rotich K. Titus 1

Lagat C. Robert 2

  1. Moi University, School of Education, Department of Center for Teacher Education, Eldoret, Kenya 1

  2. South Eastern Kenya University, Mathematics and Actuarial Science Department, Kitui, Kenya 2


The current use of Highly Active Anti-Retroviral Therapy (HAART) strategy to control Human Immunodeficiency Virus (HIV) and Acquired Immune Deficiency Syndrome (AIDS) is inefficient in eradicating HIV/AIDS due to inadequate understanding of the dynamics relating to interaction between the immune system components and HIV. As a result, a pool of potential transmitters is continuously created and thus HIV has remained a pandemic. In this paper, we formulate a mathematical model using differential equations to study the effects of time lag τ>0 due to cellular latency and pharmacological delays and chemotherapy on the control strategy of AIDS epidemic. Equilibrium points of the model are computed and used to determine the reproductive ratio〖 R〗_0. This important threshold parameter is then used to determine the critical bounds of time lag τ∈[τ_min,τ_min]  and therapeutic window C_p∈[MEC,MTC] that is, the bounds; above Minimum Effect Concentration (MEC) and below Minimum Toxic Concentration (MTC), where drug plasma concentration C_p should lie for effective maintenance of low levels of viral load and reduction of drug toxicity. The mathematical model gives qualitative understanding of HIV prognostic information which is a means of rejuvenating the existing Antiretroviral drugs (ARV’s). Numerical simulations show that a stable and persistent endemic equilibrium state of low viral load is achieved when these thresholds τ∈[0,25] and C_p∈[0.79,0.91] are satisfied. This persistent equilibrium state will lead to eventual eradication of HIV/AIDS.

Contribution/ Originality
This paper contributes the first logical analysis on the effect of intracellular delay of HIV viral infection on drug efficacy. The study originates new formula for finding the optimal therapeutic window [MEC,MTC]=[0.79,0.91] of HAART necessary to control the reproductive ratio R_0 of HIV to less than one and evade the risk of drug toxicity. 



  1. 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.
  2. AIDSinfo Health Information Specialist, Available: [Accessed 30/Oct/2012], 2012.
  3. 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.
  4. T. R. Malthus, An essay on the principal of population: Penguin Books. Originally Published in 1798, 1970.
  5. A. S. Perelson and P. W. Nelson, "Mathematical analysis of HIV-1 dynamics in vivo," SIAM Rev., vol. 41, p. 3, 1999.
  6. R. V. Culshaw, "Mathematical modeling of AIDS progression: Limitations, expectations, and future directions," Journal of American Physicians and Surgeons, vol. 11, pp. 101-104, 2006.
  7. B. Flugentius, L. S. Luboobi, and J. Y. T. Mugisha, "Periodicity of the HIV/AIDS epidemic in a mathematical model that incorporates complacency," American Journal of Infectious Diseases, vol. 1, pp. 55-60, 2005.
  8. 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.
  9. 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.
  10. L. Jiangtao, W. W. Hager, and W. Rongling, "A differential equation model for functional mapping of a virus-cell dynamic system," J. Math. Biol. Mathematical Biology, vol. 61, p. 115 2010.
  11. O. Arino, M. L. Hbid, and E. Ait Dads, "Delay differential equations and applications. Springer, Series II," Mathematics, Physics and Chemistry, vol. 205, 2006.
  12. P. Van Den Driessche and J. Watmough, "Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission," Math. Biosci., vol. 180, p. 2948, 2002.
  13. K. B. Blyuss and Y. N. Kyrychko, "Global properties of a delay SIR model with temporary immunity and nonlinear incidence rate," Nonlinear Anal RWA, vol. 6, p. 495507, 2005.
  14. Y. Kuang, Delay differential equations with applications in population dynamics. Boston: Academics Press, 1993.
  15. G. Huang and Y. Takeuchi, "Lyapunov functionals for delay differential equations model for viral infections," SIAM J Appl Math., vol. 70, p. 26932708, 2010.
  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.
  17. 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.
  18. 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.
  19. L. Xiaoling and R. J. Bhaskara, "Application of pharmacokinetics and pharmacodynamics in the design of controlled delivery systems," J. Control. Release, vol. 71, pp. 1-39, 2006.
  20. K. T. Rotich, "Effects of time Lag on stability and persistence of oscillating immune response to HIV infection," Unpublished PhD Thesis, Moi University, 2012.
  21. O. Dickmann, J. A. P. Heesterbeek, and J. A. J. Metz, "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases," J. Math. Biol., vol. 35, pp. 503-522, 1990.
  22. A. S. Perelson, D. E. Kirschner, and R. De Boer, "Dynamics of HIV infection of CD 4 + T-cells," Math. Biosci., vol. 114, p. 81, 1993.


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