Outbreak statistics and scaling laws for externally driven epidemics

Power-lawscalings are ubiquitous to physical phenomena undergoing a continuous phase transition. The classic susceptible-infectious-recovered (SIR) model of epidemics is one such example where the scaling behavior near a critical point has been studied extensively. In this system the distribution of outbreak sizes scales as P(n) similar to n(-3/2) at the critical point as the system size N becomes infinite. The finite-size scaling laws for the outbreak size and duration are also well understood and characterized. In this work, we report scaling laws for a model with SIR structure coupled with a constant force of infection per susceptible, akin to a "reservoir forcing". We find that the statistics of outbreaks in this system fundamentally differ from those in a simple SIR model. Instead of fixed exponents, all scaling laws exhibit tunable exponents parameterized by the dimensionless rate of external forcing. As the external driving rate approaches a critical value, the scale of the average outbreak size converges to that of the maximal size, and above the critical point, the scaling laws bifurcate into two regimes. Whereas a simple SIR process can only exhibit outbreaks of size O(N-1/3) and O(N) depending on whether the system is at or above the epidemic threshold, a driven SIR process can exhibit a richer spectrum of outbreak sizes that scale as O(N-xi), where xi is an element of (0,1]\{2/3} and O((N/lnN) 2/3) at the multicritical point.