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

Jaouad Danane; Karam Allali

doi:10.3934/naco.2019048

Article Contents

2020, Volume 10, Issue 2: 207-225. Doi: 10.3934/naco.2019048

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

Jaouad Danane^, and
Karam Allali

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

^* Corresponding author: Jaouad Danane. Email: jaouaddanane@gmail.com
^* Corresponding author: Jaouad Danane. Email: jaouaddanane@gmail.com

Received: February 2019

Revised: July 2019

Published: April 2020

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Abstract

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.

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Figure 1. Surface plot of $ R_0 $ (left) and contour plot of $ R_0 $ (right)

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Figure 2. The uninfected cells as function of time

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Figure 3. The latently infected cells as function of time

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Figure 4. The infected cells as function of time

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Figure 5. The HIV virus as function of time

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Figure 6. The CTL response as function of time

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Figure 7. The optimal control $ u_1 $ (left) and the optimal control $ u_2 $ (right) versus time

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Figure 8. The optimal control $ u_1 $ (left) and the optimal control $ u_2 $ (right) versus time

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Figure 9. The behavior of the infection dynamics

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