Epidemic Spreading on Complex Networks as Front Propagation into an Unstable State

We study epidemic arrival times in meta-population disease models through the lens of front propagation into unstable states. We demonstrate that several features of invasion fronts in the PDE context are also relevant to the network case. We show that the susceptible-infected-recovered model on a network is linearly determined in the sense that the arrival times in the nonlinear system are approximated by the arrival times of the instability in the system linearized near the disease-free state. Arrival time predictions are extended to general compartmental models with a susceptible-exposed-infected-recovered model as the primary example. We then study a recent model of social epidemics where higher-order interactions lead to faster invasion speeds. For these pushed fronts, we compute corrections to the estimated arrival time in this case. Finally, we show how inhomogeneities in local infection rates lead to faster average arrival times.