Discrete epidemic models with two time scales.

The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviors, or others. To include a sufficiently general disease model, we first build up from first principles a discrete-time susceptible-exposed-infectious-recovered-susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number $R_0$ . Then, we propose a general full model that includes sequentially the two processes at different time scales and proceed to its analysis through a reduced model. The basic reproduction number ${\bar{R}}_0$ of the reduced system gives a good approximation of $R_0$ of the full model since it serves at analyzing its asymptotic behavior. As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches.