is $$\begin{aligned}
\label{CombinedLyapGeneFinal}
 \dot {\mathcal V}
 &= c_0\mu_h S_h^\ast\left(2-\frac{S_h}{S_h^\ast}-\frac{S_h^\ast}{S_h}\right)+c_{v,1}(\mu_v+\delta_v)S_v^*\left(2-\dfrac{S_v^*}{S_v}-\dfrac{S_v}{S_v^*}\right) \nonumber\\
 &+c_{v,1}aS_v^\ast\sum_{i=1}^n \frac{\beta_i I_i^\ast}{N_h}\left[
 \left(1-\frac{S_h^\ast}{S_h}+\ln\frac{S_h^\ast}{S_h}\right)+\left(1-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}+\ln\frac{S_hI_vE_h^\ast}{S_h^\ast I_v^\ast E_{h}} \right)\right.\nonumber\\
 & \left.+\left(1-\frac{E_vI_v^\ast}{E_v^\ast I_v} +\ln\frac{E_vI_v^\ast}{E_v^\ast I_v} \right)
 + \left(1 -\frac{E_h}{E_h^\ast}\frac{I_1^\ast}{I_1} +\ln\frac{E_hI_1^\ast}{E_h^\ast I_1}   \right)
 +\left(1-\frac{S_vI_i E_v^\ast}{S_v^\ast I_i^\ast E_{v}} +\ln\frac{S_vI_i E_v^\ast}{S_v^\ast I_i^\ast E_{v}} \right)
 \right.\nonumber\\
  & \left.+\left(1 
  -\frac{S_v^\ast}{S_v}+\ln\frac{S_v^\ast }{S_v}\right)\right]+c_0p\pi\left(2-\frac{E_h^\ast}{E_{h}}-\frac{E_{h}}{E_h^*}  \right)
  +\sum_{i=1}^nc_ip_i\pi_h\left(2-\frac{I_i^\ast}{I_i}-\frac{I_i}{I_i^\ast}\right) \nonumber\\
  &  \sum_{i=1}^nc_i \sum_{j=1}^{n}m_{ij}I_j^\ast\left(1- \frac{I_i^\ast}{I_i} \frac{I_j}{I_j^\ast}+\ln\frac{I_i^\ast I_j}{I_iI_j^\ast}\right),
      \end{aligned}$$ which is definite-negative. Therefore, by Lyapunov’s stability theorem, the unique endemic equilibrium is GAS.

\[WEEGAS\] Let $({S}_h^\diamond, 0,\mathbf{I}_h^\diamond,{S}_v^0,0,0)$ be a weakly endemic equilibrium of Model (\[ModelAD2Compact\]). This equilibrium is GAS.

Let $({S}_h^\diamond, 0,\mathbf{I}_h^\diamond,{S}_v^0,0,0)$ be a weakly endemic equilibrium of Model (\[ModelAD2Compact\]).\
$\bullet$ If $\Beta=0$, we remark from the vector’s equations in Model (\[ModelAD2Compact\]) that $S_v\to S_v^0:=\frac{\pi_v}{\mu_v+\delta_v}$, $E_v\to0$ and $I_v\to0$ as $t\to\infty$. So, by the theory of asymptotically autonomous systems for triangular systems [@CasThi95; @0478.93044], Model (\[ModelAD2Compact\]) is equivalent to $$\left\{\begin{array}{llll}\label{WeakEE}
\displaystyle\dot S_{h}=\pi_h\left( 1-p-\sum_{i=1}^{n+1}p_i \right)-\mu_h S_h\\
\displaystyle\dot E_{h}=p\pi_h-(\mu_h+\nu_h+\eta)E_h\\
\displaystyle\dot{\mathbf{I}}_h=\pi_h\mathbf{p}+\nu_hE_he_1-(\textrm{diag}(\alpha)-M)\mathbf{I}_h
\end{array}\right.$$ System (\[WeakEE\]) is triangular and linear, and its solutions converge toward $(\bar S_h,\bar E_h,\bar{\mathbf{I}}_h)$, where $\bar S_h:=\frac{\Lambda_h}{\mu_h}$, $\bar E_h:=\frac{p\pi_h}{\mu_h+\nu_h+\eta}$ and $\bar{\mathbf{I}}_h:=(\textrm{diag}(\alpha)-M)^{-1}\left( \pi_h\mathbf{p}+\frac{\nu_hp\pi_h}{\mu_h+\nu_h+\eta}e_1 \right)$. Thus, it follows that the weak endemic equilibrium $(\bar S_h,\bar E_h,\bar{\mathbf{I}}_h,S_v^0,0,0)$ of Model (\[ModelAD2Compact\]) is GAS.\
Before we start the proof of the next case, let us define the order relation for the 