I_n^\ast}{\sum_{i=1}^n \beta_i I_i^\ast}+m_{1n}I_{n}^\ast & m_{2n}I_{n}^\ast& m_{3n}I_{n}^\ast & \hdots&\hdots &-\diamond_{nn}
\end{pmatrix}$$ where $$\diamond_{11}=\left(\nu_hE_h^*\frac{\sum_{i=2}^n \beta_i I_i^\ast }{\sum_{i=1}^n \beta_i I_i^\ast }+\sum_{j=1}^nm_{1j}I_{j}^*\right),\quad\textrm{
and for   } k\geq 2,\quad
 \diamond_{kk}=\sum_{j=1}^{n}m_{kj}I_j^\ast.$$ The matrix $B$ is irreducible. Indeed, since $\mathbf{I}_h^*\gg0$, we notice that all elements of the second upper diagonal of $B$ are all non zero, as $m_{i+1,i}=\gamma_{i,i+1}$, and thus represent the incremental transition between infectious classes. This, along with the first column, makes the matrix $B$ irreducible. Hence, it could be shown that $\dim(\ker(B))= 1$; and by the Kirchhoff’s matrix tree theorem[@bollobas2013modern; @moon1970counting], $c_i=-C_{ii}\gg0$ where $C_{ii}$ is the cofactor of the $i^{th}$ diagonal of $B$. Hence, it exists $c=(c_1,c_2,\dots,c_n)^T\gg0$ such that $Bc=0$. Moreover, this implies that, in Equation (\[CombinedLyap3\]), we have: $$\begin{aligned}
0&=&  -  c_1\left(\nu_hE_h^*+\sum_{j=1}^nm_{1j}I_{j}^*\right)\frac{I_1}{I_1^*}+c_1\sum_{j=1}^nm_{1j}I_{j}      +c_1\sum_{j=1}^n m_{1j}I_{j}^*
 \nonumber\\
&&+\sum_{i=2}^n\left[
 c_i\sum_{j=1}^nm_{ij}I_j^\ast-c_i\left(\sum_{j=1}^nm_{ij}I_j^\ast\right)\frac{I_i}{I_i^*} 
      + c_i\sum_{j=1}^n m_{ij}I_j  
\right]\nonumber\\
&&+c_1\frac{\nu_hE_h^\ast}{\sum_{l=1}^n \beta_l I_l^\ast}\left(\sum_{i=1}^n  \beta_i I_i\right)
   \end{aligned}$$ Thus, (\[CombinedLyap3\]) yields to:

$$\begin{aligned}
\label{CombinedLyap4}
\dot{\mathcal V} 
 &= c_0\mu_h S_h^\ast\left(2-\frac{S_h}{S_h^\ast}-\frac{S_h^\ast}{S_h}\right)+c_{v,1}(\mu_v+\delta_v)S_v^*\left(2-\dfrac{S_v^*}{S_v}-\dfrac{S_v}{S_v^*}\right)\nonumber\\
 & +c_0p\pi\left(2-\frac{E_h^\ast}{E_{h}}-\frac{E_{h}}{E_h^*}  \right)+\sum_{i=1}^nc_ip_i\pi_h\left(2-\frac{I_i^\ast}{I_i} -\frac{I_i}{I_i^*}\right) \nonumber\\
        & +c_{v,1}\sum_{i=1}^naS_v^\ast\frac{\beta_i I_i^\ast}{N_i}\underbrace{\left(6 -\frac{S_h^\ast}{S_h}-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}  -\frac{E_h}{E_h^*}\frac{I_1^\ast}{I_1}  -\frac{S_v^\ast}{S_v}-\frac{S_v}{S_v^\ast} \frac{I_i}{I_i^\ast} \frac{E_v^\ast}{E_{v}}-\frac{E_vI_v^\ast}{E_v^\ast I_v} \right)}_{\mathcal W_i}\nonumber\\
 & +\sum_{i=1}^n\sum_{j=1}^nc_im_{ij}I_j^\ast \left(
1
       -\frac{I_i^\ast}{I_i} \frac{I_j^\ast}{I_j}  
\right).
   \end{aligned}$$

The first three terms of (\[CombinedLyap4\]) are definite-positive. Now, we will break down the last two terms in (\[CombinedLyap4\]) into definite-negative terms. Indeed, following [@guo2012global], we transform each theses expressions 