SIR model with time dependent infectivity parameter : approximating the epidemic attractor and the importance of the initial phase.

We consider a 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 ¯ R o , introduced in [Bacaër & Guernaoui, 2006], is not the only relevant parameter and we emphasize the rôle played by the initial phase, the amplitude and the period. For a (general) periodic infectivity parameter β (t) a periodic orbit exists, as already proved in [Katriel, 2014]. In the case of a slowly varying β (t) an approximation of such a solution is given, which is shown to be asymptotically stable under an extra assumption on the slowness of β (t). For a non necessarily periodic β (t) , all the trajectories of the system are proved to be attracted into a tubular region around a suitable curve, which is then an approximation of the underlying attractor. Numerical simulations are given