6 | CONCLUSION AND DISCUSSION
This paper presents a study on the global stability of a system of ODEs, recently introduced 8. The system represents a new model to study the influence of the treatment of HIV-1 infection with stem cell transplantation, with multistage stem cell lineage. The results show that the basic reproduction number of virus\(R_{0}=\frac{k\text{λσ}}{\text{cρ}(d+\mu_{n})}\) is a sharp number that decreases with increasing \(\mu_{n}\), a constant that depends on the flux to death of the final stage of stem cell lineage.
If \(R_{0}>1,\) then we found that the endemic point \(P_{e}\) is globally asymptotically stable in \(\mathbb{R}_{+}^{n+2}\) .
If \(R_{0}\leq 1,\) then we proved that the uninfected point \(P_{u}\)is globally asymptotically stable in \(\mathbb{R}_{+}^{n+2}\) .
So, a person with \(R_{0}=\frac{k\text{λσ}}{\text{cρ}d}<1\ \)(before stem cell injection, corresponding to equation (1)), do not need to be treated, since the uninfected point \({P_{u}}^{*}\) is globally asymptotically stable for the system (1). The ill person will be automatically healed after a certain time, without any need to the therapy.
Contrariwise, for a person with\(R_{0}=\frac{k\text{λσ}}{\text{cρ}d}>1\) (before stem cell injection), there are two possibilities:
If biologically, it is possible to inject stem cells to lower the reproduction number\(R_{0}=\frac{k\text{λσ}}{\text{cρ}(d+\mu_{n})}\) to make it smaller than 1, then, the patient will be healed after a certain time of therapy, since \(P_{u}\) is globally asymptotically stable for the system (2).
If medically, it is not possible to lower the reproduction number\(R_{0}=\frac{k\text{λσ}}{\text{cρ}(d+\mu_{n})}\), to make it smaller than 1, then, the patient will never be healed.
Moreover, since the endemic point
\begin{equation} P_{e}\left(S_{1},S_{2},\ldots{,S}_{n-1},T,T_{I},V\right)=\left(0,0,\ldots,0,\ \frac{\lambda}{d^{{}^{\prime}}}\frac{1}{R_{0}},\ \frac{d^{{}^{\prime}}c}{\text{kσ}}\left(R_{0}-1\right),\ \frac{d^{{}^{\prime}}}{k}\left(R_{0}-1\right)\right)\nonumber \\ \end{equation}
of our stem cell therapy \((2)\), in the case \(R_{0}>1\) , is globally asymptotically stable , and has all its first components\(S_{1},\ S_{2},\ldots{,S}_{n-1}\) (corresponding to stem cell stages) equal to zero, then, either in the case when stem cell therapy can not offer a cure to that infected person with very high reproduction number, if we repeat the transplantation of stem cells in a manner to prevent its exhaustion from the patient, we delay progression to the chronic stage, and can prevent AIDS.
Conflict of Interest Statement: no conflict