Approximate aggregation of a two time scales periodic multi-strain SIS epidemic model: A patchy environment with fast migrations

In this work we consider a spatially distributed periodic multi strain SIS epidemic model. We let susceptible and infected individuals migrate between patches, with periodic migration rates. Considering that migrations are much faster than the epidemic process, we build up a less dimensional (aggregated) system that allows to study some features of the asymptotic behavior of the original model. In particular, we are able to define global reproduction numbers in the non-spatialized aggregated system that serve to decide the eradication or endemicity of the epidemic in the initial spatially distributed nonautonomous model. Comparing these global reproductive numbers with those corresponding to isolated patches we show that adequate periodic fast migrations can in many cases reverse local endemicity and get global eradication of the epidemic.