Estimating the distance to an epidemic threshold

O’Dea, Eamon B.; Park, Andrew W.; Drake, John M.

The epidemic threshold of the susceptible-infected-recovered (SIR) model is a boundary separating parameters that can permit epidemics from those that cannot. This threshold corresponds to points where the stability of the system’s equilibrium reaches zero. Consequently, we use the average rate at which deviations from the equilibrium shrink to define a distance to this threshold. However, the vital dynamics of the host population may occur slowly even when transmission is far from threshold levels. Here we show analytically how such slow dynamics can prevent estimation of the distance to the threshold for some individual variables of the model. 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 behaviour 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. The estimation approach is general and may be applicable to zoonotic pathogens such as MERS-CoV as well as vaccine-preventable diseases such as measles.