1 | INTRODUCTION
HIV-1 is a retrovirus discovered in 1984 Montagnier 1, that targets the helper CD+T cells of the immune system, making people more vulnerable to diseases and infections. According to the World Health Organization’s guidelines, there is no viable cure or vaccine for HIV, but treatments can improve patients’ quality of life. Understanding the dynamics of HIV-1 by mathematical modeling plays an important role in medicine. The basic mathematical model that describes the interaction of the immune system with HIV considers three populations: uninfected T cells, infected T cells, and free virus 2-6
\begin{equation} \begin{matrix}\left\{\begin{matrix}\frac{\text{dT}}{\text{dt}}=\lambda-dT-kTV\\ \frac{dT_{i}}{\text{dt}}=kTV-\rho T_{i}\\ \frac{\text{dV}}{\text{dt}}=\sigma T_{i}-\text{cV}\\ \end{matrix}\right.&\ \ \ \ \ \ \ \ \ \ (1)\\ \end{matrix}\nonumber \\ \end{equation}
Inside any organism, damaged cells need to be regenerated by new ones. The new cells need a source, which requires a continuous role to replace the older cells. In each organ stem cells are planning this important role to replace the older cells. Hence, thinking about their transplantation in the patient can be a good idea to replace injured cells and help in restoring the function of damaged cells. Recent studies have shown the efficacy of this technique to regenerate some body organs 7. The author and coworkers recently proposed a new mathematical model to treat HIV-1 patients by engraftment of the type of cells able to transform to T-cells (CD4+T)8,9 :
\begin{equation} \begin{matrix}\left\{\begin{matrix}\frac{dS_{1}}{\text{dt}}=\left(2a_{1}-1\right)p_{1}S_{1}-\ \ \mu_{1}S_{1}=F_{1}(S)\\ \frac{dS_{2}}{\text{dt}}=\left(2a_{2}-1\right)p_{2}S_{2}+2\left(1-a_{1}\right)p_{1}S_{1}-\ \ \mu_{2}S_{2}{=F}_{2}(S)\\ \vdots\\ \frac{dS_{n-1}}{\text{dt}}=\left({2a}_{n-1}-1\right)p_{n-1}S_{n-1}+2\left(1-a_{n-2}\right)p_{n-2}S_{n-2}-\ \ \mu_{n-1}S_{n-1}=F_{n-1}(S)\\ \frac{\text{dT}}{\text{dt}}=\lambda-\left(d+\ \mu_{n}\right)T-kTV+2\left(1-a_{n-1}\right)p_{n-1}S_{n-1}=G_{1}(S,T,T_{i},T_{i})\\ \frac{dT_{i}}{\text{dt}}=kTV-\rho T_{i}=G_{2}(S,T,T_{i},T_{i})\\ \frac{\text{dV}}{\text{dt}}=\sigma T_{i}-cV=G_{3}(S,T,T_{i},T_{i})\\ \end{matrix}\right.&\ \ \ (2)\\ \end{matrix}\nonumber \\ \end{equation}
Where \(S_{i}\) denotes the density of stem cells at the\(i\)th stage of differentiation, for\(i\ =1,\ 2,\ .\ .\ ,\ n-\ 1\), \(p_{i}\) denotes the proliferation rate, \(a_{i}\) denotes the fraction of self-renewal, and \(\mu_{i}\)denotes the death rate. \(S_{n-1}\) transforms to mature T-cells. These T-cells, The T cells transformed from \(S_{n-1}\) cells, die at a rate \(\mu_{n}\). The T cells produced by the thymus are generated at rate \(\lambda\) and die at a rate \(d\). T cells become infected by free virus at rate \(k\). Infected cells, \(T_{i}\), die at rate \(\rho\). Virus particles V are produced at rate \(\sigma\) and are cleared at rate \(c\). S is the vector with components\(S_{1}\),…, \(S_{n-1}\)
Different numbers of compartment \(n\) for the lineage with different systems were chosen in previous stem cell models 10- 15 .
All the constants in system (2) are non negative. We also suppose the following biologically relevant assumptions 16-17 :
\begin{equation} \begin{matrix}\left\{\begin{matrix}S_{i}\left(0\right)\geq 0,\ for\ i=1,\ldots,n\\ \mu_{n}>0,\ \mu_{i}\geq 0,\ for\ i=1,\ldots,n-1\\ c>0,\rho>0,d>0,p_{i}>0,\ for\ i=1,\ldots,n-1\\ a_{i}\in\left[\left.\ 0,\frac{1}{2}\right),\ \right.\ \ for\ i=1,\ldots,n-1\\ \left(2a_{i}-1\right)p_{i}<\mu_{i}\ ,\ \ \ for\ i=1,\ldots,n-1\\ \end{matrix}\right.&\ \ \ \ \ \ \ \ (3)\\ \end{matrix}\nonumber \\ \end{equation}
For simplicity, let
\(d+\mu_{n}=d^{\prime}\) (4)
\(R_{0}=\frac{k\text{λσ}}{\text{cρ}d^{\prime}}\) (5)
We will study, in this article, the global dynamics of (2). In section 2, we shall state the fundamental results concerning the stability of the system. In section 3, we present some lemmas, useful for the study of the global stability that we prove in section 4. We then conclude in the last section and discuss the biological significance of our results