compact positively invariant for System (\[ModelAD2Compact\]).

In the next section, we investigate the steady states solutions of Model (\[ModelAD2Compact\]) and their asymptotic behavior.

Global Stability Analysis {#sec:GSVBAP}
=========================

The next theorem establishes the existence of endemic equilibria of System (\[ModelAD2Compact\]) and provides conditions under which they may exist. Following Thieme [@MR1993355], we use the nomenclature *strongly* endemic equilibrium if all of its components are positive and *weakly* endemic equilibrium, if at least one of the infected component is positive. Naturally, we start with the assumption that $\mathbf{p}\neq\mathbf{0}_{\mathbb{R}^n}$. The case $\mathbf{p}=\mathbf{0}_{\mathbb{R}^n}$ is dealt in .

\[ExistenceEEAD\]

The equilibria of System (\[ModelAD2Compact\]) are as follows:

1.  \[i\] If $\Beta=0$, a unique weakly endemic equilibrium $(\bar{S}_h, \bar{E}_h,\bar{\mathbf{I}}_h,\bar{S}_v,0,0)$ exists.

2.  \[ii\] If $\Beta\neq0$ and $p\neq0$, it exists a unique strongly endemic equilibrium $({S}_h^*, {E}_h^*,\mathbf{I}_h^*,S_v^*,E_v^*,I_v^*)$.

3.  \[iii\] If $\Beta$ and $\mathbf{p}$ are such that $\bigl\langle
          \Beta   
     \, \mbox{\Large $\mid$} \, (\textrm{diag}(\alpha)-M)^{-1} \mathbf{p}  
    \bigr\rangle\neq0$, a unique strongly endemic equilibrium $({S}_h^\sharp, E_h^\sharp,\mathbf{I}_h^\sharp,{S}_v^\sharp,E_v^\sharp,I_v^\sharp)$ exists.

4.  \[iv\] If , and are not satisfied, then a threshold $\mathcal N_0^2(p,\mathbf{p},p_{n+1})$, defined by: $$\mathcal N_0^2(p,\mathbf{p},p_{n+1})=\frac{a^2\beta_{vh}\nu_v\nu_h}{\alpha_v(\mu_v+\delta_v)\alpha_h\mu_h}\frac{N_v}{N_h}\frac{\pi_h\left( 1-p-\mathds{1}^T\mathbf{p}-p_{n+1} \right)}{N_h}\bigl\langle
          \Beta   
     \, \mbox{\Large $\mid$} \, (\textrm{diag}(\alpha)-M)^{-1} e_1  
    \bigr\rangle,$$ exists and for which

    -   If $\mathcal N_0^2(p,\mathbf{p},p_{n+1})\leq1$, a unique weakly endemic equilibrium $({S}_h^\diamond, 0,\mathbf{I}_h^\diamond,{S}_v^0,0,0)$ where $\mathbf{I}_h^\diamond>0$ 