models proposed in the literature in the following five ways:

-   If $D=\mathbf{0}_{n,n}$ and $\gamma_{ij}=0$ for all $i,j$, except when $j=i+1$, Model (\[ModelAD2Compact\]) consists of a class of staged progression vector-borne diseases models with an influx of infected individuals of each class into the considered population. In this case,

    -   Model (\[ModelAD2Compact\]) extents the existing stage progression vector-borne models to incorporate a differential proportions of the overall recruitment in all infected classes. This allows us to gauge the impact of imported cases on the dynamics of vector-borne infections. When $\mathbf{p}=\mathbf{0}$, $p=0$, in Model (\[ModelAD2Compact\]), we obtain the model proposed and studied in [@BicharaIggidrSmith2017]. Moreover, our model extends also [@palmer2018dynamics], for which $\beta_i=\beta$, for all $i$ and the recruitment constitutes of susceptible individuals only. Our model generalizes also [@cruz2012control], which considers Chagas disease model with two stages, namely acute and chronic phases.

    -   Model (\[ModelAD2Compact\]) generalizes existing models that investigate staged-progression for directly transmitted infections for which influx of infected individuals are considered [@guo2012impacts; @1056.92052] and [@BraVdd01], where no stages are considered in the latter.

    -   Model (\[ModelAD2Compact\]) extends also the model in [@tumwiine2010host], where the authors considered a host-vector model $SIR-SI$ with infectious immigrants in investigating the effects of the 