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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

,

### 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

on PubMed

Pages: 63-81

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### Abstract:

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 (ARVâ€™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.

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