4. 4 | Proof of the principle results
Let us prove the free disease case\(P_{u}=(0,0,\ldots,0,\frac{\lambda}{d},0,0)\) is globally
asymptotically stable in \(\mathbb{R}_{+}^{n+2}\) if \(R_{0}\leq 1\)
If \(R_{0}\leq 1\) , we have just to apply lemma 3.3. Since
A3 is already done in subsection 4.2, we have to verify
A1 and A2
A1: The global stability of \(S=0\) for the liner
system \(\dot{S}=MS\), where \(M\) is the matrix given by (9), is
insured by the negativity of the eigenvalues of the matrix M.
A2: \(X_{o}=(\frac{\lambda}{d},0,0)\) is globally
asymptotically stable in \(\mathbb{R}_{+}^{3}\), for the equation
\begin{equation}
\dot{X}=G\left(X,0\right)\Longleftrightarrow\left(1\right),"d"\ replaced\ by\ "d^{\prime}=(d+\ \mu_{n})"\nonumber \\
\end{equation}This is already given by part (i) of theorem 4.3
If \(R_{0}>1,\) then, let us prove the endemic point \(P_{e}\) is
globally asymptotically stable for \((2)\) in the positive octant\(\mathbb{R}_{+}^{n+2}\). Like for the previous paragraph, we
have just to apply lemma 3.3, so,
we have just to verify A1, since A2 and
A3 have been already done.
A2 :\(X_{o}=(\frac{\lambda}{d}\frac{1}{R_{0}},\ \frac{dc}{\text{kσ}}\left(R_{0}-1\right),\ \frac{d}{k}\left(R_{0}-1\right)\))
is a globally asymptotically stable point for
\begin{equation}
\dot{X}=G\left(X,0\right)\Longleftrightarrow\left(1\right),"d"\ replaced\ by\ "d^{\prime}=(d+\ \mu_{n})"\nonumber \\
\end{equation}This is already given by part (ii) of theorem 4.3