Eur. Phys. J. B (2012) 85: 145
https://doi.org/10.1140/epjb/e2012-30117-0

Regular Article

Scaling behavior of threshold epidemics

¹ Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, 87545 NM, USA
² Department of Physics, Boston University, Boston, 02215 MA, USA

^a e-mail: ebn@lanl.gov

Received: 9 February 2012
Received in final form: 9 March 2012
Published online: 7 May 2012

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.