Outbreak statistics and scaling laws for externally driven epidemics

Abstract

Power-law scalings 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 $P(n) \sim n^{-3/2}$ 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 are fundamentally different than 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 $\mathcal{O}(N^{1/3})$ and $\mathcal{O}(N)$ 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 $\xi \in (0,1] \backslash \{2/3\}$ and $\mathcal{O}((N/\log N)^{2/3})$ at the multi-critical point.