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June  2020, 10(2): 207-225. doi: 10.3934/naco.2019048

## Optimal control of an HIV model with CTL cells and latently infected cells

 Laboratory of Mathematics and Applications, Faculty of Sciences and, Techniques, Hassan II University of Casablanca, PO Box 146, Mohammedia, Morocco

Received  February 2019 Revised  July 2019 Published  September 2019

This paper deals with an optimal control problem for an human immunodeficiency virus (HIV) infection model with cytotoxic T-lymphocytes (CTL) immune response and latently infected cells. The model under consideration describes the interaction between the uninfected cells, the latently infected cells, the productively infected cells, the free viruses and the CTL cells. The two treatments represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. Existence of the optimal control pair is established and the Pontryagin's minimum principle is used to characterize these two optimal controls. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to show the role of optimal therapy in controlling the infection severity.

Citation: Jaouad Danane, Karam Allali. Optimal control of an HIV model with CTL cells and latently infected cells. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 207-225. doi: 10.3934/naco.2019048
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Surface plot of $R_0$ (left) and contour plot of $R_0$ (right)
The uninfected cells as function of time
The latently infected cells as function of time
The infected cells as function of time
The HIV virus as function of time
The CTL response as function of time
The optimal control $u_1$ (left) and the optimal control $u_2$ (right) versus time
The optimal control $u_1$ (left) and the optimal control $u_2$ (right) versus time
The behavior of the infection dynamics
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