Optimal intervention strategies of a SI-HIV models with differential infectivity and time delays

Mboya Ba; P. Tchinda Mouofo; Mountaga Lam; Jean-Jules Tewa

doi:10.3934/dcdsb.2020035

Article Contents

2020, Volume 25, Issue 6: 2293-2306. Doi: 10.3934/dcdsb.2020035

This issue Previous Article Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions Next Article Dynamics of a metapopulation epidemic model with localized culling

Optimal intervention strategies of a SI-HIV models with differential infectivity and time delays

a.
University of Cheikh Anta Diop, Department of Mathematics and Computer SciencesFaculty of Science and Technic, P.O. Box 5005 Dakar, Senegal
b.
University of Yaounde I, Department of MathematicsFaculty of Science, Yaounde, Cameroon
c.
University of Yaounde I, Department of Mathematics and PhysicsNational Advanced School of Engineering, Yaounde, Cameroon

^* Corresponding author: Mboya Ba
^* Corresponding author: Mboya Ba

Received: December 31, 2018

Revised: July 31, 2019

Early access: February 2020

Published: June 2020

The first author is supported by NSF grant

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Abstract

HIV infection is divided into stages of infection which are determined by the CD4 cells count progression. Through each stage, the time delay for the progression is important because the duration of HIV infection varies according to the infectious. Retarded optimal control theory is applied to a system of delays ordinary differential equations modeling the evolution of HIV with differential infectivity. Seeking to reduce the population of the infective individuals with low CD$ 4 $ cells, we use the ARV drug to control the fraction of infective individuals that is identified and will be put under treatment. We use optimal control theory to study our proposed system. Numerical simulations are provided to illustrate the effect of the Antiretroviral treatment (ART) taking into account the delays.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Impact of time delay on dynamics of model system (1)

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Figure 2. Simulation results of the delayed HIV model (5) with control (red line) and without control (blue line)

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Figure 3. State variable with control and time delay (continuous curves) versus with control and without delay (dashed curves)

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Table 2. Sensitivity indices of $ \mathcal R_{0} $

Parameter	Sensitivity index	Value
$ \Lambda $	0	$ 10^4 $
$ c $	$ 1 $	$ +0.3 $
$ \beta_1 $	$ +0.7557 $	0.03
$ \beta_2 $	$ +0.2033 $	0.0384
$ \beta_3 $	$ +0.0388 $	0.03
$ \beta_4 $	+0.0021	0.02
$ \beta_5 $	$ +3.095\times10^{-8} $	0.01
$ k_1 $	$ +0.033 $	$ 3\times 10^{-5} $
$ k_2 $	$ -0.010 $	$ 4\times 10^{-5} $
$ k_3 $	$ -0.0012 $	$ 10^{-5} $
$ k_4 $	$ -1.7060\times 10^{-4} $	$ 10^{-5} $
$ d_5 $	$ -2.99\times 10^{-5} $	$ 3.3\times 10^{-3} $

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