Analytic models for SIR disease spread on random spatial networks

We study the propagation of susceptible-infectious-recovered (SIR) disease on random networks with spatial structure. We consider random spatial networks (RSNs) with heterogeneous degrees, where any two nodes are connected by an edge with a probability that depends on their spatial distance and expected degrees. This is a natural spatial extension of the well-known Erdos-Renyi and Chung-Lu random graphs. In the limit of high node density, we derive partial integro-differential equations governing the spread of SIR disease through an RSN. We demonstrate that these nonlocal evolution equations quantitatively predict stochastic simulations. We analytically show the existence of travelling wave solutions on the RSNs. If the distance kernel governing edge placement in the RSNs decays slower than exponential, the population-scale dynamics are dominated by long-range hops followed by local spread of travelling waves. This provides a theoretical modelling framework for disease propagation on networks with spatial structure, extending analytic SIR modelling capabilities from networks without spatial structure to the important real-life case of networks with spatial structure. Potential applications of this modelling framework include studying the interplay between long-range and short-range transmissions in designing polio vaccination strategies, wavelike spread of the plague in medieval Europe, and recent observations of how epidemics like Ebola evolve in modern connected societies.