Multi-Compartmental Staged Progression Endemic Models with Fast Transitions.

We present a model of infectious disease dynamics where individuals can transition between different compartments, which may have distinct epidemiological characteristics. Within each compartment, epidemic dynamics are represented by a staged progression epidemic model. Individual transitions between compartments occur on a faster time scale, allowing the initial model to be reduced for analysis. In the reduced model, disease eradication and endemicity are characterized by the basic reproduction number. The relationship between this basic reproduction number and those associated with each compartment is analyzed by considering each compartment in isolation. This allows the study of the role of transitions in epidemic dynamics. Endemicity is represented by uniform persistence relative to the total number of infected individuals. It is verified that, for a sufficiently large ratio between time scales, the initial model shares the uniform persistence of the reduced model. The influence of transitions on disease eradication/endemicity is illustrated by different results. In particular, the conditions for transition rates are determined so that endemicity (eradication) in each isolated compartment results in global eradication (endemicity). These results can provide some tools for managing epidemics in the context of individuals transiting between compartments with different epidemiological properties.