model when there is no influx of infected individuals into the population, which surprisingly has not been done (). In this case, the model exhibits the threshold phenomenon – the basic reproduction number $\mathcal R_0^2$ determines the outcome of the disease both in host and vector population. It happens that $\mathcal R_0^2:=\mathcal N^2(0,\mathbf{0},0)$.\
$\bullet$ Illustrations of the results and numerical simulations are carried out in .

Formulation of the model {#sec:SPHostFlux}
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We consider a disease whose evolution is captured by a host-vector interaction for which the host population is composed of susceptible, exposed, recovered and infectious of stage $i$ ($1\leq i\leq n$). These subpopulations are denoted respectively by $S_h$, $E_h$, $R_h$ and $I_i$. The total host population is therefore $N_h=S_h+E_h+\sum_{i=1}^nI_i+R_h$. The vector population $N_v$ is composed of susceptible, exposed, and infectious arthropods; denoted by $S_v$, $E_v$, and $I_v$, respectively.\
The total host population is replenished through a constant recruitment, $\pi_h$, that includes birth and migratory influx of individuals. Of this constant recruitment, a proportion $p$, $p_i$ ($i=1,\dots,n$) and $p_{n+1}$ is latent, infectious at stage $i$ and recovered, respectively. Thus, the total recruitment in the susceptible class is $\pi_h\left( 1-p-\sum_{i=1}^{n+1}p_i \right)$. Naturally, we assume that, $0\leq p\leq1$, $0\leq p_i\leq1$ and 