Front propagation in a spatial system of weakly interacting networks.

Abstract

We consider the spread of epidemic in a spatial metapopulation system consisting of weakly interacting patches. Each local patch is represented by a network with a certain node degree distribution and individuals can migrate between neighboring patches. Stochastic particle simulations of the SIR model show that after a short transient, the spatial spread of epidemic has a form of a propagating front. A theoretical analysis shows that the speed of front propagation depends on the effective diffusion coefficient and on the local proliferation rate similarly to fronts described by the Fisher-Kolmogorov equation. To determine the speed of front propagation, first, the early-time dynamics in a local patch is computed analytically by employing degree based approximation for the case of a constant disease duration. The resulting delay differential equationis solved for early times to obtain the local growth exponent. Next, the reaction diffusion equationis derived from the effective master equationand the effective diffusion coefficient and the overall proliferation rate are determined. Finally, the fourth order derivative in the reaction diffusion equationis taken into account to obtain the discrete correction to the front propagation speed. The analytical results are in a good agreement with the results of stochastic particle simulations.