Abstract
In this work, an optimal control approach is presented in order to propose an optimal therapy for the treatment HIV infection using a combination of two appropriate treatment strategies. The optimal treatment duration and the optimal medications amount are considered. The main objective of this study is to be able to maximize the benet based on number of healthy CD4+ T-cells and CTL immune cells and to minimize the infection level and the overall treatment cost while optimizing the duration of therapy. The free terminal time optimal control problem is formulated and the Pontryagin's maximum principle is employedto provide the explicit formulations of the optimal controls. The corresponding optimality system with the additional transversality condition for the terminal time is derived and solved numerically using an adapted iterative method with a Runge-Kutta fourth order scheme and a gradient method routine.
Highlights
Recent data from the World Health Organization [19] show that approximately 34 million people worldwide are infected with human immunodeficiency virus (HIV), more than 30 million people died of acquired immune deficiency syndrome (AIDS)-related causes since twenty years
Since the discovery of human immunodeficiency virus (HIV) and the assertion that it is the cause of the acquired immune deficiency syndrome (AIDS), many scientific studies have focused on the HIV infection [8, 9, 11, 12, 23, 31, 39] and various mathematical models have been developed in order to suggest possible optimal treatment strategies for HIV infection [6, 7, 13, 29, 30, 33, 49, 51]
Since this study is interested primarily in the possible biological changes resulting from the introduction of an appropriate treatment in the equilibrium state [50], the mathematical analysis shows that any state variable relating to the dynamics of HIV particles can be omitted [50], which explains the absence of any specific compartment that characterizes the evolution of HIV concentration in the studied model
Summary
Recent data from the World Health Organization [19] show that approximately 34 million people worldwide are infected with HIV, more than 30 million people died of AIDS-related causes since twenty years. The scientific research continues for the development of an effective drug therapy the interest of optimal control theory [33] which is presented as an indispensable tool for a better understanding of the dynamics of immune system and the evolution of HIV infection in order to propose an appropriate treatment strategy [6, 13, 20, 29, 30]. An optimal control approach with free terminal time is proposed for the treatment of HIV infection during an optimal therapeutic period. This approach is based on the introduction of two optimal controls characterizing a combination treatment using both HAART and IL-2 immunotherapy.
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