A controlled treatment strategy applied to HIV immunology model

Shohel Ahmed; Abdul Alim; Sumaiya Rahman

doi:10.3934/naco.2018019

Article Contents

2018, Volume 8, Issue 3: 299-314. Doi: 10.3934/naco.2018019

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A controlled treatment strategy applied to HIV immunology model

1.
Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh
2.
Department of Statistics, University of Dhaka, Dhaka-1000, Bangladesh

* Corresponding author: SHOHEL AHMED, E-mail: shohel2443@math.buet.ac.bd
* Corresponding author: SHOHEL AHMED, E-mail: shohel2443@math.buet.ac.bd

Received: April 2017

Revised: November 2017

Published: June 2018

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Abstract

Optimal control can be helpful to test and compare different vaccination strategies of a certain disease. This study investigates a mathematical model of HIV infections in terms of a system of nonlinear ordinary differential equations (ODEs) which describes the interactions between the human immune systems and the HIV virus. We introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The aim is to obtain a new optimal chemotherapeutic strategy where an isoperimetric constraint on the chemotherapy supply plays a crucial role. We outline the steps in formulating an optimal control problem, derive optimality conditions and demonstrate numerical results of an optimal control for the model. Numerical results illustrate how such a constraint alters the optimal vaccination schedule and its effect on cell-virus interactions.

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Mathematics Subject Classification: Primary: 37N35, 49J15, 58F17; Secondary: 58E25.

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Figure 1. The battle between HIV and the immune system begins in the earnest after the virus replicates in the infected cells and the new viral particles escape.[26]

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Figure 2. Particles of HIV (green spheres), the virus that causes AIDS, bud from an infected white blood cell before moving on to infect other cells. [27]

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Figure 3. Global number of AIDS-related deaths, new HIV-infections and people living with HIV (1990-2015)

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Figure 4. HIV immunology model with chemotherapy function $u(t)$.

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Figure 5. HIV model with control and without control

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Figure 6. HIV model with different percentage of chemotherapy

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Figure 7. HIV model with different value of weight parameter $a$

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Figure 8. HIV model with different value of k

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Figure 9. HIV model with and without constraints.

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Figure 10. Total Chemotherapy amount during treatment period.

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Figure 11. HIV model for different chemotherapy amount

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Figure 12. Total Chemotherapy amount during treatment period

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Table 1. Description of parameter and values of the HIV model [15]

Parameters	Description	Value

$m_1$	Death rate of Uninfected CD4$^+$T cell population	$ 0.02/d$
$m_2$	Death rate of infected CD4$^+$T cell population	$ 0.5/d$
$m_3$	Death rate of free virus	$ 4.4/d$
$k$	Rate of CD4$^+$T cell become infected by free virus	$ 2..4\times10^{-5}mm^{3} /d$
$r$	Rate of growth for the CD4$^+$T cell population	$ 0.03/d$
$N$	Number of free virus produced by $T_i$ cells	300
$T_{max}$	Maximum CD4$^+$T cell population level	$ 1.5\times10^{3}/mm^{3}$
$s$	Source term for Uninfected CD4$^+$T cells	$ 10 d^{-1} mm^{-3}$
$a$	Weight parameter	0.05

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