$ i-k\geq 0$. We denote by $\delta_{ij}$ the *per capita* rate at which the host regresses from stage $i$ to stage $j$, where $j\leq i$. These generalizations are illustrated in .\
In concert, the overall dynamics of the Host-Vector infection is given by: $$\label{ModelAD}
\left\{\begin{array}{llll}\displaystyle\dot S_{h}=\pi_h\left( 1-p-\sum_{i=1}^{n+1}p_i \right)-a\,\beta_{vh}\,S_{h}\,\dfrac{I_{v}}{N_{h}}-\mu_h S_h\\
\displaystyle\dot E_{h}=p\pi_h+ \,a\beta_{vh}\,S_{h}\,\dfrac{I_{v}}{N_{h}}-(\mu_h+\nu_h+\eta)E_h\\
\displaystyle\dot I_{1}=p_1\pi_h+\nu_hE_h-(\mu_h+\eta_1)I_1-I_{1}\sum_{j=2}^n\gamma_{1j}+\sum_{j=2}^n\delta_{j1}I_{j}\\
\displaystyle\dot I_{2}=p_2\pi_h-(\mu_h+\eta_2)I_2-I_{2}\left(\sum_{j=1,j<2}^n\delta_{2j}+\sum_{j=1,j>2}^n\gamma_{2j}\right)+\left(\sum_{j=1,j>2}^n\delta_{j2}I_j+\sum_{j=1,j<2}^n\gamma_{j2}I_j\right)\\ 
\vdots\\
\displaystyle\dot I_{i}=p_i\pi_h-(\mu_h+\eta_i)I_i-I_{i}\left(\sum_{j=1,j<i}^n\delta_{ij}+\sum_{j=1,j>i}^n\gamma_{ij}\right)+\left(\sum_{j=1,j>i}^n\delta_{ji}I_j+\sum_{j=1,j<i}^n\gamma_{ji}I_j\right)\\ 
\displaystyle\dot I_{n}=p_i\pi_h-(\mu_h+\eta_n)I_n-I_{n}\sum_{j=1}^{n-1}\gamma_{nj}+\sum_{j=1}^{n-1}\delta_{jn}I_{j}\\
\displaystyle \dot R_h=p_{n+1}\pi_h+\sum_{i=1}^n\eta_iI_{i}-\mu_hR\\
\displaystyle\dot S_{v}=\pi_v-a S_{v}\sum_{i=1}^n\dfrac{\beta_i I_{i}}{N_{h}}-(\mu_v+\delta_v)S_{v}\\
\displaystyle\dot E_{v}=a S_{v}\sum_{i=1}^n\dfrac{\beta_i I_{i}}{N_{h}}-(\mu_v+\nu_v+\delta_v)E_{v}\\
\displaystyle\dot I_{v}=\nu_vE_v-(\mu_v+\delta_v)I_{v}
\end{array}\right.$$

To ease the notations, let us denote by $$m_{ij}=\left\{\begin{array}{lll}\delta_{ji}\quad\textrm{if}\quad i<j,\\
0\quad\textrm{if}\quad i=j,\\
\gamma_{ji}\quad\textrm{if}\quad i>j,
\end{array}\right.$$and $\alpha_v=\mu_v+\nu_v+\delta_v$, $\alpha_1=\mu_h+\eta_1 +\sum_{j=2}^n\gamma_{1j}=\mu_h+\eta_1 +\sum_{j=2}^nm_{j1}$ and for $i\geq2$, $$\begin{aligned}
\alpha_i&=&\mu_h+\eta_i +\left(\sum_{j=1,j<i}^n\delta_{ij}+\sum_{j=1,j>i}^n\gamma_{ij}\right)\nonumber\\
&=&\mu_h+\eta_i + \sum_{j=1}^nm_{ji}. \nonumber\end{aligned}$$ The matrix $M=(m_{ij})_{1\leq i,j\leq n}$ is the transfer matrix between Host’s infectious classes and the parameters $\alpha_i$ represent the rates at which infected of stage $i$ leave this stage. The total host population is asymptotically constant. Indeed, its evolution is given by $\dot N_h=\pi_h-\mu_hN_h$ and thus, it is straightforward to show that $\displaystyle\lim_{t\to\infty}N_h=\dfrac{\pi_h}{\mu_h}.$ Moreover, the subsystem describing the dynamics of the host is triangular, and hence we can disregard the dynamics of the recovered host $R_h$. Hence, by abusively denoting $\displaystyle\lim_{t\to\infty}N_h$ again by $N_h$ and using the theory of asymptotically autonomous systems for triangular systems [@CasThi95; 