Time-scale Separation on Networks for Multi-City Epidemics

This paper explores a general multiple time-scale approach to model reduction in dynamic network-of-network systems, with a specific treatment on multi-city epidemics. Extensive coupling between layers of networks often complicates and obfuscates the interaction dynamics. Through the use of the natural time-scale differences in the system, these dynamics can be individually studied under a separation principle, simplifying analysis. We describe asymptotic stability of the network-of-networks through a composite Lyapunov function, and provide an M-matrix condition which guarantees such stability. Conservative bounds on the time-scale parameters are given quantitatively such that this condition is satisfied. This theory is consequently applied to the study of multicity epidemics, where the interactions within each city may be assumed to behave at their own time-scale, and the movement of population between cities occurs at the slowest time-scale. We show, under certain time-scale conditions, the isolated dynamics of an individual city can approximate that of the full multi-city dynamics in the approach towards the disease-free equilibrium.