4. 1 | Useful Simplification
Let us begin by noticing that, our system \((2)\) can be written in the
triangular form in \(\mathbb{R}^{n+2}\):
\begin{equation}
\left(\text{TS}\right)\left\{\begin{matrix}\dot{X}=F(X),\\
\dot{Y}=G\left(X,Y\right)\\
\end{matrix}\right.\ \nonumber \\
\end{equation}with\(X=S=(S_{1},S_{2},\ldots{,S}_{n-1})\in\mathbb{R}^{n-1}\),\(F\left(S\right)=MS\)
So,
\(\dot{S}=F\left(S\right)=MS\) (8)
Where M is the triangular matrix
\(\par
\begin{pmatrix}\left(2a_{1}-1\right)p_{1}-\ \ \mu_{1}&0&0&\ldots&0\\
2\left(1-a_{1}\right)p_{1}&\left(2a_{2}-1\right)p_{2}-\ \ \mu_{2}&0&\ldots&0\\
0&2\left(1-a_{2}\right)p_{2}&\left(2a_{3}-1\right)p_{3}-\ \ \mu_{3}&\cdots&0\\
:&:&\vdots&\ddots&\vdots\\
0&0&0&\cdots&\left({2a}_{n-1}-1\right)p_{n-1}-\ \ \mu_{n-1}\\
\end{pmatrix}\) (9)
and
\(Y=(\ T,T_{i},V\)),
\(G\left(\ S,T,T_{i},V\right)=\par
\begin{pmatrix}\lambda-(d+\ \mu_{n})T-kTV+2\left(1-a_{n-1}\right)p_{n-1}S_{n-1}\\
kTV-\rho T_{i}\\
\sigma T_{i}-\text{cV}\\
\end{pmatrix}\),
\begin{equation}
G\left(0,T,T_{i},V\right)=\begin{pmatrix}\lambda-(d+\ \mu_{n})T-k\text{TV}\\
kTV-\rho T_{i}\\
\sigma T_{i}-\text{cV}\\
\end{pmatrix}\nonumber \\
\end{equation}\begin{equation}
\dot{Y}=G\left(0,T,T_{i},V\right)\Leftrightarrow system\ \left(1\right),\ with\ "d"\ replaced\ by\ "d^{\prime}=(d+\ \mu_{n})"\nonumber \\
\end{equation}So, we need to study the global stability of the system (1)
Let us begin by proving the boundedness of the solutions of system (2)