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Power-law scalings are ubiquitous to physical phenomena undergoing acontinuous phase transition. The classic Susceptible-Infectious-Recovered (SIR)model of epidemics is one such example where the scaling behavior near acritical point has been studied extensively. In this system the distribution ofoutbreak sizes scales as $P(n) \sim n^{-3/2}$ at the critical point as thesystem size $N$ becomes infinite. The finite-size scaling laws for the outbreaksize and duration are also well understood and characterized. In this work, wereport scaling laws for a model with SIR structure coupled with a constantforce of infection per susceptible, akin to a `reservoir forcing'. We find thatthe statistics of outbreaks in this system are fundamentally different thanthose in a simple SIR model. Instead of fixed exponents, all scaling lawsexhibit tunable exponents parameterized by the dimensionless rate of externalforcing. As the external driving rate approaches a critical value, the scale ofthe average outbreak size converges to that of the maximal size, and above thecritical point, the scaling laws bifurcate into two regimes. Whereas a simpleSIR 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 epidemicthreshold, a driven SIR process can exhibit a richer spectrum of outbreak sizesthat 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.

Outbreak statistics and scaling laws for externally driven epidemics