  Traditional studies about disease dynamics have focused on global stability
issues, due to their epidemiological importance. We study a classical SIR-SI
model for arboviruses in two different directions: we begin by describing an
alternative proof of previously known global stability results by using only a
Lyapunov approach. In the sequel, we take a different view and we argue that
vectors and hosts can have very distinctive intrinsic time-scales, and that
such distinctiveness extends to the disease dynamics. Under these hypothesis,
we show that two asymptotic regimes naturally appear: the fast host dynamics
and the fast vector dynamics. The former regime yields, at leading order, a SIR
model for the hosts, but with a rational incidence rate. In this case, the
vector disappears from the model, and the dynamics is similar to a directly
contagious disease. The latter yields a SI model for the vectors, with the
hosts disappearing from the model. Numerical results show the performance of
the approximation, and a rigorous proof validates the reduced models.
