Front propagation in a spatial system of weakly interacting networks

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 equation is solved for early times to obtain the local growth exponent. Next, the reaction diffusion equation is derived from the effective master equation and the effective diffusion coefficient and the overall proliferation rate are determined. Finally, the fourth order derivative in the reaction diffusion equation is 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.

Automated summary

This article investigates the spread of epidemics in a spatial metapopulation system composed of weakly interacting patches. Stochastic particle simulations using the SIR model show that the spatial spread of epidemics exhibits a propagating front after a short transient period. Theoretical analysis reveals that the speed of front propagation is determined by an effective diffusion coefficient and the local proliferation rate, similar to fronts described by the Fisher-Kolmogorov equation. To determine the speed of front propagation, analytical computations are conducted for the early-time dynamics in a local patch, assuming a constant disease duration. The resulting delay differential equation is solved to obtain the local growth exponent, and the reaction diffusion equation is derived from the effective master equation to determine the effective diffusion coefficient and the overall proliferation rate. A correction to the front propagation speed is obtained by taking into account the fourth order derivative in the reaction diffusion equation. The analytical results are in good agreement with stochastic particle simulations.