$0\leq p+\sum_{i=1}^{n+1}p_i<1$.\
Susceptible hosts are infected at the rate $a\beta_{vh}\frac{I_v}{N_n}$, where $a$ is the biting/landing rate and $\beta_{vh}$ is the Host’s infectiousness by arthropod per biting/landing. Vector’s infection term is captured by $a S_{v}\frac{\sum_{i=1}^n\beta_iI_i}{N_n},$ where $\beta_i$ is the vector’s infectiousness by infected hosts of stage $i$. This accounts for the differential infectivity of vectors with respect to hosts’ infectious stages.\
Motivated by [@guo2012impacts; @guo2012global; @1008.92032], we incorporate incremental and non-incremental amelioration and recrudescence in the infectiosity at each stage of the host’s infection. For instance, for models in [@BicharaIggidrSmith2017; @cruz2012control; @palmer2018dynamics], the transitions between infection stages are incremental, that is, always from stage $i$ to $i+1$. However, with vector-borne diseases, a bite of infected arthropod to an already infected host, say at stage $i$, may increase this host’s parasitemia, thereby catapulting its infectious class from stage $i$ to any stage, say $i+j$, where $1\leq i+j\leq n$. To incorporate this phenomenon, we denote by $\gamma_{ij}$, the *per capita* rate at which the host progresses from stage $i$ to stage $j$. Similarly, the increase of treatments (which decrease the parasiteamia in the blood-stream) of vector-borne diseases could alleviate the host’s infection and therefore, its stage could change form $i$ to $i-k$, where 