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