Exploring the role of the potential surface in the behaviour of early warning signals

The theory of critical slowing down states that a system displays increasing relaxation times as it approaches a critical transition. These changes can be seen in statistics generated from timeseries data, which can be used as early warning signals of a transition. Such early warning signals would be of value for emerging infectious diseases or to understand when an endemic disease is close to elimination. However, in applications to a variety of epidemiological models there is frequent disagreement with the general theory of critical slowing down, with some indicators performing well on prevalence data but not when applied to incidence data. Furthermore, the alternative theory of critical speeding up predicts contradictory behaviour of early warning signals prior to some stochastic transitions. To investigate the possibility of observing critical speeding up in epidemiological models we characterise the behaviour of common early warning signals in terms of a system's potential surface and noise around a quasi-steady state. We then describe a method to obtain these key features from timeseries data, taking as a case study a version of the SIS model, adapted to demonstrate either critical slowing down or critical speeding up. We show this method accurately reproduces the analytic potential surface and diffusion function, and that these results can be used to determine the behaviour of early warning signals and correctly identify signs of both critical slowing down and critical speeding up.