SIR Fronts in Complex Networks with Metapopulation Structure

SIR dynamics has been studied extensively on complex networks, yielding insight into the effects of heterogeneity in contact patterns on the spread of infectious diseases. Separately, metapopulations have provided a paradigm for modeling systems with extended and ``patchy'' organization. In this paper we demonstrate how multi-type networks can be used to combine these paradigms such that simple disease dynamics models can include heterogeneity in connectivity and multi-scale structure. We first present a multi-type generalization of the Volz-Miller mean-field approximation for SIR dynamics on multi-type random graphs. We then use this technique to study the propagation of epidemic fronts in a simple metapopulation model with population centers composed of configuration model networks coupled on a one-dimensional lattice. Using the formalism of front propagation into unstable states, we derive the effective transport coefficients of the linear spreading: asymptotic speed, characteristic perturbation size, and diffusion coefficient for the pulled fronts, and explore their dependence on the underlying graph structure. We also derive the average steady-state incidence, the equilibrium spectrum, and the threshold for invasion.