Mathematical Model for an Effective Management of HIV Infection.

Oladotun Matthew Ogunlaran,Suares Clovis Oukouomi Noutchie

doi:10.1155/2016/4217548

Oladotun Matthew Ogunlaran, Suares Clovis Oukouomi Noutchie

Open Access

https://doi.org/10.1155/2016/4217548

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Abstract

Human immunodeficiency virus infection destroys the body immune system, increases the risk of certain pathologies, damages body organs such as the brain, kidney, and heart, and causes death. Unfortunately, this infectious disease currently has no cure; however, there are effective retroviral drugs for improving the patients' health conditions but excessive use of these drugs is not without harmful side effects. This study presents a mathematical model with two control variables, where the uninfected CD4+T cells follow the logistic growth function and the incidence term is saturated with free virions. We use the efficacy of drug therapies to block the infection of new cells and prevent the production of new free virions. Our aim is to apply optimal control approach to maximize the concentration of uninfected CD4+T cells in the body by using minimum drug therapies. We establish the existence of an optimal control pair and use Pontryagin's principle to characterize the optimal levels of the two controls. The resulting optimality system is solved numerically to obtain the optimal control pair. Finally, we discuss the numerical simulation results which confirm the effectiveness of the model.

Highlights

Acquired immunodeficiency syndrome (AIDS) is caused by a virus known as human immunodeficiency virus (HIV)
One of the major havocs wrought by the HIV is the destruction of CD4+T cells which play a significant role in the regulation of the body immune system
By assuming that the constant recruitment number of new uninfected cells and the number of death of uninfected cells have already been incorporated in the logistic growth function and that the rate of infection of CD4+T cells by free virions has been saturated probably because of overcrowding of free virions or as a result of protection measures being used by the HIV patient, and we obtain the variant model

Summary

Introduction

Acquired immunodeficiency syndrome (AIDS) is caused by a virus known as human immunodeficiency virus (HIV). One of the major havocs wrought by the HIV is the destruction of CD4+T cells which play a significant role in the regulation of the body immune system. Several mathematical models have been formulated to study the interactions between HIV and CD4+T cells [10,11,12,13,14]. HIV is not yet curable, there are antiretroviral drugs that help in boosting the immune system against cell infections. These antiretroviral drugs are categorized into two groups which are reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). RTIs disrupt the conversion of RNA of the virus to DNA so that new HIV infection of cells is prevented. We incorporate two controls into the model and find the optimal treatment strategy that will produce maximum uninfected cells and minimum viral load with a minimum dose of drug therapies to prevent harmful effects associated with excessive use of drugs in the body

Model Formulation

Optimal Control Problem

Conclusion

Recommendation

Proof of Theorem 1