$\bullet$ If the conditions of , and are not satisfied; that is, if $\Beta\neq0$, $p=0$ and $\mathbf{p}$ is such that $\bigl\langle
      \Beta   
 \, \mbox{\Large $\mid$} \, (\textrm{diag}(\alpha)-M)^{-1}\mathbf{p}\bigr\rangle=0$. In this case, $C=0$ and $B$ could be written as: $$\begin{aligned}
  B   &=(\mu_v+\delta_v)\mu_hN_h \\
&-\frac{a^2\beta_{vh}\nu_v\nu_h}{(\mu_v+\nu_v+\delta_v)(\mu_h+\nu_h+\eta)}\frac{N_v}{N_h}\Lambda_h\bigl\langle
      \Beta   \, \mbox{\Large $\mid$} \,(\textrm{diag}(\alpha)-M)^{-1}e_1\bigr\rangle  \\
                           &=(\mu_v+\delta_v)\mu_hN_h \left[1
-\frac{a^2\beta_{vh}\nu_v\nu_h}{(\mu_v+\nu_v+\delta_v)(\mu_v+\delta_v)(\mu_h+\nu_h+\eta)\mu_h}\frac{N_v}{N_h}\frac{\Lambda_h}{N_h}\bigl\langle
      \Beta   \, \mbox{\Large $\mid$} \,(\textrm{diag}(\alpha)-M)^{-1}e_1\bigr\rangle   \right]\\
      &=(\mu_v+\delta_v)\mu_hN_h\left(1-\mathcal N_0^2(p,\mathbf{p},p_{n+1}) \right).\end{aligned}$$ Thus, it follows that if $\mathcal N_0^2(p,\mathbf{p},p_{n+1})\leq1$, then $B\geq0$, leading to $I_v^*=0$ and $\mathbf{I}_h^\ast=\pi_h(\textrm{diag}(\alpha)-M)^{-1}\mathbf{p}>\mathbf{0}_{\R^n}$. If $\mathcal N_0^2(p,\mathbf{p},p_{n+1})>1$, then $B<0$ and therefore $I_v^*>0$, leading to a strongly positive equilibrium.

The following two theorems establish the global stability analysis for the two types of endemic equilibria exhibited in . This gives a complete description of the global asymptotic behavior of System (\[ModelAD2Compact\]) whenever there is an influx of infected or infectious individuals into the population.

\[GASEEAD\]\
Let $({S}_h^*, {E}_h^*,\mathbf{I}_h^*,S_v^*,E_v^*,I_v^*)$ be a strongly endemic equilibrium of Model (\[ModelAD2Compact\]). This equilibrium is globally asymptotically stable whenever it exists.

\
Let consider the following Lyapunov function $  \mathcal V=  \mathcal V_h+  \mathcal V_v$, where $$\begin{aligned}
  \mathcal V_h&=&c_0\int_{S_h^\ast}^{S_h}\left(1-\frac{S_h^\ast}{x}\right)dx+c_0\int_{E_{h}^\ast}^{E_h}\left(1-\frac{E_h^\ast}{x}\right)dx+\sum_{i=1}^nc_{i}\int_{I_{i}^\ast}^{I_{i}}\left(1-\frac{I_{i}^\ast}{x}\right)dx,
  \end{aligned}$$ and, $$\mathcal V_{v}=c_{v}\int_{S_{v}^\ast}^{S_{v}}\left(1-\frac{S_{v}^\ast}{x}\right)dx +c_{v}\int_{E_{v}^\ast}^{E_{v}}\left(1-\frac{E_{v}^\ast}{x}\right)dx+\frac{\mu_v+\nu_v+\delta_v}{\nu_v}c_{v}\int_{I_{v}^\ast}^{I_{v}}\left(1-\frac{I_{v}^\ast}{x}\right)dx.$$ The coefficients $c=(c_1,c_2,\dots,c_n)^T$ are positive to be determined later. The coefficient $c_0$ and $c_{v}$ are related to $c_1$ as follows: $$\label{c0cvc1}c_0a\beta_{vh}\,S_{h}^*\dfrac{I_{v}^*}{N_{h}}=c_1\nu_hE_h^*\quad\textrm{and}\quad   c_{v} aS_v^\ast\frac{1}{N_h}=c_1\frac{\nu_hE_h^\ast}{\sum_{i=1}^n 