\right)+aS_v^\ast\sum_{i=1}^n \frac{\beta_i I_i}{N_h}-aS_v^\ast\sum_{i=1}^n \frac{\beta_i I_i^\ast}{N_h}\frac{I_v}{I_v^\ast}\right),$$ where $\displaystyle\mathcal A_v=(\mu_v+\delta_v)S_v^*\left(2-\dfrac{S_v^*}{S_v}-\dfrac{S_v}{S_v^*}\right).$ Combining equations (\[Lyapc111\]), (\[ciIi2\]), and (\[VvDot1\]), we obtain: $$\begin{aligned}
\label{CombinedLyap}
 \dot{\mathcal V} 
        &=&c_0\mu_h S_h^\ast\left(2-\frac{S_h}{S_h^\ast}-\frac{S_h^\ast}{S_h}\right)+c_0a\beta_{vh}S_{h}^*\,\dfrac{I_{v}^*}{N_{h}}\left(3 -\frac{S_h^\ast}{S_h}-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}  -\frac{E_h}{E_h^*}\frac{I_1^\ast}{I_1}\right)+c_0p\pi\left(2-\frac{E_h^\ast}{E_{h}}-\frac{E_{h}}{E_h^*}  \right)\nonumber\\
  &&+c_0a\,\beta_{vh}S_h^\ast\,\dfrac{I_{v}}{N_{h}}+\sum_{i=1}^nc_ip_i\pi_h\left(2-\frac{I_i^\ast}{I_i} -\frac{I_i}{I_i^*}\right)   -  c_1\left(\nu_hE_h^*+\sum_{j=1}^nm_{1j}I_{j}^*\right)\frac{I_1}{I_1^*}+c_1\sum_{j=1}^nm_{1j}I_{j} \nonumber\\
     && +c_1\sum_{j=1}^n m_{1j}I_{j}^*
-c_1\frac{I_1^\ast}{I_1}\left(\sum_{j=1}^nm_{1j}I_{j}\right) \nonumber\\
&&+\sum_{i=2}^n\left[
 c_i\sum_{j=1}^nm_{ij}I_j^\ast-c_i\left(\sum_{j=1}^nm_{ij}I_j^\ast\right)\frac{I_i}{I_i^*} 
      + c_i\sum_{j=1}^n m_{ij}I_j -c_i\frac{I_i^\ast}{I_i}\left(\sum_{j=1}^n m_{ij}I_j \right)
\right]\nonumber\\
&&+c_{v,1}\left( \mathcal A_v+\sum_{i=1}^naS_v^\ast\frac{\beta_i I_i^\ast}{N_i}\left(3  -\frac{S_v^\ast}{S_v}-\frac{S_v}{S_v^\ast} \frac{I_i}{I_i^\ast} \frac{E_v^\ast}{E_{v}}-\frac{E_vI_v^\ast}{E_v^\ast I_v}  \right)+aS_v^\ast\sum_{i=1}^n \frac{\beta_i I_i}{N_h}-aS_v^\ast\sum_{i=1}^n \frac{\beta_i I_i^\ast}{N_h}\frac{I_v}{I_v^\ast}\right)
   \end{aligned}$$ Given the relationship (\[c0cvc1\]), the linear terms in $I_v$ in Equation (\[CombinedLyap\]) cancel. Furthermore, by substituting $\mathcal A_v$ by its expression and $c_{v,1}$ by their expressions, Equation (\[CombinedLyap\]) leads to $$\begin{aligned}
\label{CombinedLyap3}
\dot{\mathcal V} 
        &=&c_0\mu_h S_h^\ast\left(2-\frac{S_h}{S_h^\ast}-\frac{S_h^\ast}{S_h}\right)+c_{v,1}(\mu_v+\delta_v)S_v^*\left(2-\dfrac{S_v^*}{S_v}-\dfrac{S_v}{S_v^*}\right)\nonumber\\
        &&+c_0p\pi\left(2-\frac{E_h^\ast}{E_{h}}-\frac{E_{h}}{E_h^*}  \right)+\sum_{i=1}^nc_ip_i\pi_h\left(2-\frac{I_i^\ast}{I_i} -\frac{I_i}{I_i^*}\right) \nonumber\\
        &&+c_{v,1}\sum_{i=1}^naS_v^\ast\frac{\beta_i I_i^\ast}{N_i}\left(6 -\frac{S_h^\ast}{S_h}-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}  -\frac{E_h}{E_h^*}\frac{I_1^\ast}{I_1}  -\frac{S_v^\ast}{S_v}-\frac{S_v}{S_v^\ast} \frac{I_i}{I_i^\ast} \frac{E_v^\ast}{E_{v}}-\frac{E_vI_v^\ast}{E_v^\ast I_v} \right)\nonumber\\
  &&  -  c_1\left(\nu_hE_h^*+\sum_{j=1}^nm_{1j}I_{j}^*\right)\frac{I_1}{I_1^*}+c_1\sum_{j=1}^nm_{1j}I_{j}      +c_1\sum_{j=1}^n m_{1j}I_{j}^*
-c_1\frac{I_1^\ast}{I_1}\left(\sum_{j=1}^nm_{1j}I_{j}\right) \nonumber\\
&&+\sum_{i=2}^n\left[
 c_i\sum_{j=1}^nm_{ij}I_j^\ast-c_i\left(\sum_{j=1}^nm_{ij}I_j^\ast\right)\frac{I_i}{I_i^*} 
      + c_i\sum_{j=1}^n m_{ij}I_j -c_i\frac{I_i^\ast}{I_i}\left(\sum_{j=1}^n m_{ij}I_j \right)
\right]\nonumber\\
&&+c_1\frac{\nu_hE_h^\ast}{\sum_{l=1}^n \beta_l I_l^\ast}\left(\sum_{i=1}^n  \beta_i I_i\right)
   \end{aligned}$$ We choose the vector $c=(c_1,c_2,\dots,c_n)^T$ to be the solution of the linear system $Bc=0$, where $$\label{B}
B=\begin{pmatrix}
 -\diamond_{11} & m_{21}I_1^\ast &  m_{31}I_{1}^\ast & \dots & \dots & m_{n1}I_{1}^\ast\\
\nu_hE_h^*\frac{\beta_2 I_2^\ast}{\sum_{i=1}^n \beta_i I_i^\ast}+ m_{12}I_{2}^\ast  & -\diamond_{22} & m_{32}I_2^\ast & \dots &\dots & m_{n2}I_{2}^\ast\\
\nu_hE_h^*\frac{\beta_3 I_3^\ast}{\sum_{i=1}^n \beta_i I_i^\ast} + m_{13}I_{3}^\ast & m_{23}I_{3}^\ast & -\diamond_{33} & \dots& \dots&  m_{n3}I_{n}^\ast\\
 \vdots& \vdots & \vdots& \ddots & \ddots& \vdots \\
\nu_hE_h^* \frac{\beta_n 