Abstract
Predicting infectious disease emergence or eradication depends on monitoring the distance to the epidemic threshold. One approach to such monitoring, the early warning signals approach, is to quantify the slowing down of dynamics that is characteristic of an approach to a threshold. However, in the susceptible-infected-recovered (SIR) model, the vital dynamics of the host population may occur slowly even when transmission is far from threshold levels. The extent to which fluctuations of an individual variable can provide an estimate of the distance to the threshold, then, depends on the relative weighting of transmission and vital dynamics in the fluctuations. Here we show analytically how this weighting depends on the covariance of the perturbations to a system with two degrees of freedom. Although these results are exact only in the limit of long-term observation of a large system, we find that they still provide useful insight into the behavior of estimates from simulations with a range of population sizes, environmental noise, and observation schemes. Having established some guidelines about when estimates are accurate, we then illustrate how multiple distance estimates can be used to estimate the rate of approach to the threshold.