Outbreak statistics and scaling laws for externally driven epidemics

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 $P\left(n\right)\sim n^{-3/2}$ 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 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 $O\left(N^{1/3}\right)$ and $O\left(N\right)$ 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 $O\left(N^{\xi }\right)$, where $\xi \in (0,1]\setminus \{2/3\}$ and $O\left({(N/lnN)}^{2/3}\right)$ at the multicritical point.

Figure 1

(a) A realization of the externally driven SIR process with three outbreaks. (b) The decomposition of an outbreak into its constituent micro-outbreaks. Each micro-outbreak is associated with a single imported infection. (c) The $M/G/\infty$ queue where a micro-outbreak is analogous to a customer being serviced at a station. In this realization, the busy period starts when the first customer enters service and ends when the second customer leaves service. The number of customers served during a busy period corresponds to the number of micro-outbreaks that constitute an outbreak. The statistics of the composite outbreaks depends strongly on the driving rate $\nu$, which is our primary focus here.

Figure 2

The probability of having an outbreak of size $n$ empirically calculated from ${10}^9$ realizations of the process for different values of $N$ and $\alpha =1,\nu =0.2$. The dashed line shows the slope of the analytical scaling predicted from theory [Eq. (21)], ignoring any constant prefactors. Inset: The collapse of outbreak sizes when scaled by $N^{2/3}$. The bump near the exponential cutoff represents the probability mass associated with outbreaks that would have continued along the power law in an infinite-size system but are clustered due to finite-size effects.

Figure 3

The probability of having an outbreak of size $n$ for $\alpha =1,\nu =1$. Dashed line plotted at an offset represents the analytical scaling [Eq. (23)]. Inset: The collapse of outbreak sizes when scaled by $N^{2/3}$.

Figure 4

The probability of having an outbreak of size $n$ for $\alpha =1,\nu =3$. Dashed line represents the analytical scaling [Eq. (27)] at an offset. The finite-size bump is more pronounced because it has accumulated outbreaks of two distinct scales. The bump starts at the scale of $N^{2/3}$ consistent with what is observed for $\nu \le 1$. However, the exponential cutoff which marks the end of the bump occurs at a different scale as evident from the lack of scaling collapse for large $n$ in the inset. The presence of two scales in the bump is discussed in detail in Sec. 3a and illustrated in Fig. 7.

Figure 5

Finite-size scaling for average outbreak size ($\nu =\{0.2,1,2,3\}$). The statistics of the power-law regime and the infinite regime were calculated separately using $N^{2/3}$ as the separation boundary. Note that the scaling for $\nu =3$ is purely logarithmic in the power-law regime. Dashed lines represent the scaling laws predicted from theory [Eq. (44)].

Figure 6

Scaling collapse at $M\sim {(N^{2}lnN)}^{1/3}$ for $\nu =1$. The $y$ axis is scaled by the theoretical scaling law of Eq. (23). Note that the scaling collapse is distinct from the one shown in inset (done at the scale of $N^{2/3}$). While the power laws collapse on top of each other at $N^{2/3}$, the exponential cutoffs collapse at ${(N^{2}lnN)}^{1/3}$. This separation of scales is more pronounced for $\nu >1$ (see Fig. 7).

Figure 7

Scaling of the outbreak size distribution by $M\sim N^{(\nu +1)/(\nu +2)}$ for $\nu =3$. The $y$ axis is scaled by the theoretical scaling law of Eq. (27). Similarly to Fig. 6, the scaling collapse is distinct from the one shown in inset (outbreaks scaled by $N^{2/3}$). The power-law regime exhibits a scaling collapse at $N^{2/3}$, while the exponential cutoffs collapse at $N^{4/5}$.

Figure 8

Finite-size scaling for average outbreak duration ($\nu =0.2,1,1.5,2$). The plots are split into two figures for clarity. The statistics of the power-law regime and the infinite regime were calculated separately using $N^{2/3}$ as the separation boundary. Inset shows the scaling behavior for $\nu =2$ on a log-linear plot. Dashed lines represent scaling laws predicted from theory [Eq. (54)].