\beta_i I_i^\ast}.$$ This function is definite positive. We want to prove that its derivative along the trajectories of System (\[ModelAD2Compact\]) is definite-negative. Throughout the proof, we will be using the component-wise endemic relations (\[ModelAD2Endemic1\]). That is, $$\label{ModelAD2Endemic}
\left\{\begin{array}{llll} 
\displaystyle \pi_h\left( 1-p-\sum_{i=1}^{n+1}p_i \right)=a\,\beta_{vh}\,S_{h}^\ast\,\dfrac{I_{v}^\ast}{N_{h}}+\mu_h S_h^\ast\\
\displaystyle (\mu_h+\nu_h+\eta)E_h^\ast=p\pi_h+ \,a\beta_{vh}\,S_{h}^\ast\,\dfrac{I_{v}^\ast}{N_{h}}\\
\displaystyle \alpha_1I_1^\ast=p_1\pi_h+\nu_hE_h^\ast+\sum_{j=2}^nm_{1j}I_{j}^\ast\\
\displaystyle \alpha_2I_{2}^\ast=p_2\pi_h+\sum_{j=1}^nm_{2j}I_j^\ast \\ 
\vdots\\
\displaystyle \alpha_iI_i^\ast=p_i\pi_h+ \sum_{j=1}^nm_{ij}I_j^\ast\\ 
\displaystyle \alpha_nI_n^\ast=p_n\pi_h+\sum_{j=1}^{n}m_{nj}I_{j}^\ast\\
\displaystyle \Lambda_v=a S_{v}^\ast\sum_{i=1}^n\dfrac{\beta_i I_{i}^\ast}{N_{h}}+(\mu_v+\delta_v)S_{v}^\ast\\
\displaystyle (\mu_v+\nu_v+\delta_v)E_{v}^\ast=a S_{v}^\ast\sum_{i=1}^n\dfrac{\beta_i I_{i}^\ast}{N_{h}}\\
\displaystyle (\mu_v+\delta_v)I_{v}^\ast=\nu_vE_v^\ast
\end{array}\right.$$

The derivative of $\mathcal{V}_h$ along the trajectories of System (\[ModelAD2Compact\]) is: $$\begin{aligned}
\label{Lyapc1}
  \dot {\mathcal V_h} &=&c_0\left(1-\frac{S_h^\ast}{S_h}\right)\dot{S}_h+c_0\left(1-\frac{E_h^\ast}{E_{h}}\right)\dot E_{h}+\sum_{i=1}^nc_i\left(1-\frac{I_i^\ast}{I_i}\right)\dot I_i\nonumber\\
   &=&c_0\mu_h S_h^\ast\left(2-\frac{S_h}{S_h^\ast}-\frac{S_h^\ast}{S_h}\right)+c_0a\beta_{vh}S_{h}^*\,\dfrac{I_{v}^*}{N_{h}}\left(2 -\frac{S_h^\ast}{S_h}-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}\right)+c_0p\pi\left(2-\frac{E_h^\ast}{E_{h}}-\frac{E_{h}}{E_h^*}  \right)\nonumber\\
  &&-c_0\left(  \,\beta_{vh}\,S_{h}^*\dfrac{I_{v}^*}{N_{h}} \right)\frac{E_{h}}{E_h^*}+c_0a\,\beta_{vh}S_h^\ast\,\dfrac{I_{v}}{N_{h}}+\sum_{i=1}^nc_i\left(1-\frac{I_i^\ast}{I_i}\right)\dot I_i    \end{aligned}$$ Using the endemic relation $\displaystyle\alpha_1I_1^*=p_1\pi_h+\nu_hE_h^*+\sum_{j=1}^nm_{1j}I_{j}^*$, and the relationship between $c_0$ and $c_1$, Equation (\[Lyapc1\]) yields to $$\begin{aligned}
\label{Lyapc111}
  \dot{\mathcal V}_h
 &=&c_0\mu_h S_h^\ast\left(2-\frac{S_h}{S_h^\ast}-\frac{S_h^\ast}{S_h}\right)+c_0a\beta_{vh}S_{h}^*\,\dfrac{I_{v}^*}{N_{h}}\left(3 -\frac{S_h^\ast}{S_h}-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}  -\frac{E_h}{E_h^*}\frac{I_1^\ast}{I_1}\right)+c_0p\pi\left(2-\frac{E_h^\ast}{E_{h}}-\frac{E_{h}}{E_h^*}  \right)\nonumber\\
  &&+c_0a\,\beta_{vh}S_h^\ast\,\dfrac{I_{v}}{N_{h}}+ c_1p_1\pi_h\left(2-\frac{I_1^\ast}{I_1} -\frac{I_1}{I_1^\ast}\right)    -c_1\left(\nu_hE_h^*+\sum_{j=1}^nm_{1j}I_{j}^*\right)\frac{I_1}{I_1^*}+c_1\sum_{j=1}^nm_{1j}I_{j} \nonumber\\
     && +c_1\sum_{j=1}^n m_{1j}I_{j}^*
-c_1\frac{I_1^\ast}{I_1}(\sum_{j=1}^nm_{1j}I_{j}) +\sum_{i=2}^nc_i\left(1-\frac{I_i^\ast}{I_i}\right)\dot I_i 
   \end{aligned}$$ Noting that, from the endemic relations (\[ModelAD2Endemic\]), we have $\alpha_{i}I_i^\ast=p_i\pi_h+ \sum_{j=1}^nm_{ij}I_j^\ast$, and thus, the last term of Equation (\[Lyapc111\]) leads to $$\begin{aligned}
\label{ciIi2}
     c_i\left(1-\frac{I_i^\ast}{I_i}\right)\dot I_i 
       & =&c_ip_i\pi_h\left(2-\frac{I_i^\ast}{I_i} -\frac{I_i}{I_i^*}\right)+ c_i\sum_{j=1}^nm_{ij}I_j^\ast-c_i\left(\sum_{j=1}^nm_{ij}I_j^\ast\right)\frac{I_i}{I_i^*}\nonumber\\
       &&+ c_i\sum_{j=1}^n m_{ij}I_j -c_i\frac{I_i^\ast}{I_i}\left(\sum_{j=1}^n m_{ij}I_j \right)
        \end{aligned}$$

Moreover, we can check that the derivative of $\mathcal V_v$ along the trajectories of System (\[ModelAD2Compact\]) is: $$\label{VvDot1}
         \dot{\mathcal V_v}=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}  