Migratory fluxes of humans and of insects of various species have favoured the spreading of diseases world-wise. It is important to stress that those epidemics can have strong social and economical impacts if not seriously controlled. Only in 2010 in Brazil, one million infected individual of which 80,000 where hospitalised. I shall present the SIR-Network model, introduced in Stolerman et al (2015), and revisite the SIR model with birth and death terms and time-varying infectivity parameter β(t). In the particular case of a sinusoidal parameter, we show that the average Basic Reproduction Number Ro, already introduced in Bacaer et al. (2006) is not the only relevant parameter and we emphasise the role played by the initial phase, the amplitude and the period. For a quite general slowly varying β(t) (not necessarily periodic) infectivity parameter all the trajectories of the system are proven to be attracted into a tubular region around a suitable curve, which is then an approximation of the underlying attractor. Numerical simulations are given and comparison with real data from Dengue epidemics in Rio de Janeiro allow us to estimate the infectivity rate and make predictions about what are the periods more at risk of infection.