A SIR forced model with interplays with the external world and periodic internal contact interplays

We investigated the behavior of a Susceptible-Infected-Recovered seasonally forced model for endemic childhood infectious diseases in the case where the target population is not isolated and, moreover, fast weekly fluctuations of the social contacts occur. We considered some key scenarios of interplay of Susceptible subjects with the external world, leading to subharmonic resonances and chaos. Our simulations suggest that the above–mentioned fast oscillations of the contact rate can cause the suppression/reduction of chaos and of subharmonic resonances. Thus, far from being filtered, they have an important role. If one considers an opposition of phase of the pattern of external infections w.r.t. the pattern of internal transmission rate, they result remarkably different from a scenario of synchrony. In most scenarios, the chaotic behavior is not associated to the phenomenon of the ‘atom–infectious’, i.e. the proportion of infectious is small but not unrealistic for large populations.