vectors as follows: $u\leq v$ if $u_i\leq v_i$, for all $i$, where $u_i$ and $v_i$ are components of $u$ and $v$ respectively. Similarly, $u< v$ if $u\leq v$ and $u\neq v$. Also $u\gg v$ if $u_i>v_i$, for all $i$.\
$\bullet$ If of is satisfied with $\mathcal N_0^2(p,\mathbf{p},p_{n+1})\leq1$. That is, $\Beta\neq0$, $p=0$ and $\mathbf{p}$ and $M$ are such that $\bigl\langle
      \Beta   
 \, \mbox{\Large $\mid$} \, (\textrm{diag}(\alpha)-M)^{-1}\mathbf{p}\bigr\rangle=0$. These imply that, using the endemic relations, $$(\textrm{diag}(\alpha)-M)^{-1} \mathbf{p} >0.$$ Moreover, it follows that $\bigl\langle
      \Beta   
 \, \mbox{\Large $\mid$} \, \mathbf{I}_h^\diamond  
\bigr\rangle=0$. This implies that it exists a subset $\mathcal J$ of $\{1,2,\dots,n\}$ such that $I^\diamond_i=0$, for all $i\in\mathcal J$, $I_i^\diamond>0$ for $i\in\{1,2,\dots,n\}\setminus\mathcal J$; and $\beta_i^\diamond>0$ for $i\in\{1,2,\dots,n\}\setminus\mathcal J$, and $\beta_i^\diamond=0$ for $i\in\mathcal J$. WLOG, suppose that $\mathcal J=\{1,2,\dots,s-1\}$ with $s\geq2$. Hence, the endemic relation $\mathbf{I}_h^\diamond=\pi_h(\textrm{diag}(\alpha)-M)^{-1}\mathbf{p}$ and the condition $\bigl\langle
      \Beta   
 \, \mbox{\Large $\mid$} \, (\textrm{diag}(\alpha)-M)^{-1}\mathbf{p}\bigr\rangle=0$ imply that $M$ has the form $$M=\begin{pmatrix}
 M_{11} & \textbf{0}_{s-1,n-s+1}\\
 M_{21} & M_{22}
 \end{pmatrix},$$ where $M_{11}\in\mathcal M_{s-1,s-1},\;\; M_{21}\in\mathcal M_{n-s+1,s-1},\;\; M_{22}\in\mathcal M_{n-s+1,n-s+1}$. Similarly, $p_i=0$ for all $i\in\mathcal J$ and $\mathbf{I}_h^\diamond=(0,\dots,0, I_s^\diamond,\dots,I_n^\diamond )$ where $I_i^\diamond>0$ for $s\leq i\leq n$.\
Let $\mathbf{c}=(\mathbf{c}_1,\mathbf{c}_2)^T$ where $\mathbf{c}_1=(c_1,\dots,c_{s-1})^T$ and $\mathbf{c}_2=(c_s,\dots,c_{n})^T$. The vector $\mathbf{c}_2$ is the solution of $\tilde B\mathbf{c}_2=0$ where $$\tilde B=\begin{pmatrix}
 -\tilde{b}_{s,s} & m_{s+1,s}I_s^\diamond &  