@0478.93044], System (\[ModelAD\]) could equivalently be written in a compact form as follows: $$\label{ModelAD2Compact}
\left\{\begin{array}{llll}\displaystyle\dot S_{h}=\pi_h\left( 1-p-\mathds{1}^T\mathbf{p}-p_{n+1} \right)-a\,\beta_{vh}\,S_{h}\,\dfrac{I_{v}}{N_{h}}-\mu_h S_h\\
\displaystyle\dot E_{h}=p\pi_h+ \,a\beta_{vh}\,S_{h}\,\dfrac{I_{v}}{N_{h}}-(\mu_h+\nu_h+\eta)E_h\\
\displaystyle\dot{\mathbf{I}}_h=\pi_h\mathbf{p}+\nu_hE_he_1-(\textrm{diag}(\alpha)-M)\mathbf{I}_h\\
\displaystyle\dot S_{v}=\pi_v
 -a\dfrac{ S_{v}}{N_h}\bigl\langle
      \Beta   
 \, \mbox{\Large $\mid$} \, \mathbf{I}_h   
\bigr\rangle
-(\mu_v+\delta_v)S_{v}\\
\displaystyle\dot E_{v}=a\dfrac{ S_{v}}{N_h}\bigl\langle
      \Beta    
 \, \mbox{\Large $\mid$} \, \mathbf{I}_h   
\bigr\rangle-(\mu_v+\nu_v+\delta_v)E_{v}\\
\displaystyle\dot I_{v}=\nu_vE_v-(\mu_v+\delta_v)I_{v},
\end{array}\right.$$ where $\mathbf{I}_h=(I_1,I_2,\dots,I_n)^T$, $\Beta=(\beta_1,\beta_2,\dots,\beta_n)^T$, $\mathbf{p}=(p_1,p_2,\dots,p_n)^T$, and $M=G^T+D^T$ with $G=(\gamma_{ij})$ representing the progression matrix, or forward flow transition matrix while $D=(\delta_{ij})$ represents the amelioration matrix, or backward transition flow matrix. More precisely,

$$\label{GD}G=\begin{pmatrix}
    0 &   \gamma_{12}&\gamma_{13} & \dots  & \gamma_{1n}\\
  0 &0 &\gamma_{23} & \dots  & \gamma_{2n}\\
    \vdots& \vdots &\ddots  & \ddots &  \vdots \\
    0 & 0 &0 & \dots &  \gamma_{n-1,n}\\ 
0 & 0 &0 & \dots  & 0 
    \end{pmatrix}
   \quad\textrm{and}\quad 
    D=\begin{pmatrix}
    0 &  0&  \dots & 0& 0\\
   \delta_{21} &0 &  \dots & 0& 0\\
    \vdots& \ddots &\ddots   & \vdots& \vdots \\
     \delta_{n-1,1} & \delta_{n-1,2}   & \dots &0& 0\\ 
 \delta_{n1} & \delta_{n2}   & \dots &\delta_{n,n-1}& 0 
    \end{pmatrix}.$$

The parameters of System (\[ModelAD\]) are described in Table \[TableHV1\]. The flow chart capturing the infection process is represented in Fig. (\[fig:FlowSEIR1HostAmeliorationDeterioration\]).\

\[TableHV1\]

    Parameters                              Description
  --------------- ----------------------------------------------------------------
      $\pi_h$                         Recruitment of the host
      $\pi_v$                          Recruitment of vectors
        $p$                       Proportion of latent immigrants
       $p_i$              Proportion of 