Scaling Behavior of Threshold Epidemics

TL;DR

This paper analyzes the behavior of the SIR epidemic model at the critical threshold, deriving scaling laws for outbreak sizes and durations, supported by analytical and numerical methods.

Contribution

It provides new analytical scaling laws for epidemic sizes and durations at the threshold, validated by numerical simulations.

Findings

Scaling laws for outbreak size and duration at the epidemic threshold

Analytical expressions for time-dependent distribution functions

Numerical verification of the scaling predictions

Abstract

We study the classic Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease. In this stochastic process, there are two competing mechanism: infection and recovery. Susceptible individuals may contract the disease from infected individuals, while infected ones recover from the disease at a constant rate and are never infected again. Our focus is the behavior at the epidemic threshold where the rates of the infection and recovery processes balance. In the infinite population limit, we establish analytically scaling rules for the time-dependent distribution functions that characterize the sizes of the infected and the recovered sub-populations. Using heuristic arguments, we also obtain scaling laws for the size and duration of the epidemic outbreaks as a function of the total population. We perform numerical simulations to verify the scaling predictions and discuss the consequences of these scaling laws for near-threshold epidemic outbreaks.