# American Institute of Mathematical Sciences

March  2022, 15(3): 501-518. doi: 10.3934/dcdss.2021148

## Optimal control of an HIV model with a trilinear antibody growth function

 1 Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies, University Hassan II of Casablanca, P. O. Box 146, Mohammedia, Morocco 2 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

* Corresponding author: Delfim F. M. Torres

Received  June 2019 Revised  May 2021 Published  March 2022 Early access  November 2021

We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.

Citation: Karam Allali, Sanaa Harroudi, Delfim F. M. Torres. Optimal control of an HIV model with a trilinear antibody growth function. Discrete & Continuous Dynamical Systems - S, 2022, 15 (3) : 501-518. doi: 10.3934/dcdss.2021148
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##### References:
The evolution of the uninfected cells during time
The evolution of the infected cells during time
The evolution of the HIV virus during time
The evolution of the CTL cells during time
The evolution of the antibodies during time
The behaviour of the two optimal controls
Parameters, their symbols and meaning, and default values used in HIV literature
 Parameters Meaning Value References $\lambda$ source rate of CD4+ T cells $1$-$10$ cells $\mu l^{-1}$ days $^{-1}$ [4] $d$ decay rate of healthy cells $0.007$-$0.1$ days $^{-1}$ [4] $\beta$ rate at which CD4+ T cells become infected $0.00025$-$0.5$ $\mu l$ virion $^{-1}$ days $^{-1}$ [4] $a$ death rate of infected CD4+ T cells, not by CTL $0.2$-$0.3$ days $^{-1}$ [4] $\mu$ clearance rate of virus $2.06$-$3.81$ days $^{-1}$ [18] $N$ number of virions produced by infected CD4+ T-cells $6.25$-$23599.9$ virion $^{-1}$ [3,29] $p$ clearance rate of infection $1$-$4.048 \times 10^{-4}$ ml virion days $^{-1}$ [3,16] $c$ activation rate of CTL cells $0.0051$-$3.912$ days $^{-1}$ [3] $h$ death rate of CTL cells $0.004$-$8.087$ days $^{-1}$ [3] $q$ Neutralization rate of virions $0.12$ days $^{-1}$ Assumed $g$ activation rate of antibodies $0.00013$ days $^{-1}$ Assumed $\alpha$ death rate of antibodies $0.12$ days $^{-1}$ Assumed
 Parameters Meaning Value References $\lambda$ source rate of CD4+ T cells $1$-$10$ cells $\mu l^{-1}$ days $^{-1}$ [4] $d$ decay rate of healthy cells $0.007$-$0.1$ days $^{-1}$ [4] $\beta$ rate at which CD4+ T cells become infected $0.00025$-$0.5$ $\mu l$ virion $^{-1}$ days $^{-1}$ [4] $a$ death rate of infected CD4+ T cells, not by CTL $0.2$-$0.3$ days $^{-1}$ [4] $\mu$ clearance rate of virus $2.06$-$3.81$ days $^{-1}$ [18] $N$ number of virions produced by infected CD4+ T-cells $6.25$-$23599.9$ virion $^{-1}$ [3,29] $p$ clearance rate of infection $1$-$4.048 \times 10^{-4}$ ml virion days $^{-1}$ [3,16] $c$ activation rate of CTL cells $0.0051$-$3.912$ days $^{-1}$ [3] $h$ death rate of CTL cells $0.004$-$8.087$ days $^{-1}$ [3] $q$ Neutralization rate of virions $0.12$ days $^{-1}$ Assumed $g$ activation rate of antibodies $0.00013$ days $^{-1}$ Assumed $\alpha$ death rate of antibodies $0.12$ days $^{-1}$ Assumed
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