Prospects for detecting early warning signals in discrete event sequence data: application to epidemiological incidence data

Emma Southall; Michael J. Tildesley; Louise Dyson

Variance is one of the most intuitive statistical indicators. As a system approaches a 313 critical transition the time taken to recover from small perturbations increases, as 314 described by Critical Slowing Down theory. This can be observed in the fluctuations 315 about the steady state, which on the approach to a critical transition take longer to 316 return and consequently vary far more, defining the increasing nature of variance as an 317 early warning signal. 318 We evaluate analytical solutions of the variance in prevalence using the derived SDE 319 for each model (Model 1: Eqn.4, Model 2: Eqn.9, Model 3: Eqn.15). We compare this 320 to theoretical solutions of the variance in incidence, using the transition rates for each 321 model (Table 1) to compute the rate of the Poisson process λ. Figure 1 We observe that variance in prevalence simulations (Figure 1 a(ii) , b(ii) and c(ii)) 335 increases on the approach to the critical transition, as predicted by critical slowing 336 down. In comparison the variance in incidence decreases before the critical transition 337 for all disease elimination models (Model 1 Fig. 1 a(i) and Model 2 Fig. 1 b(i) ) and 338 increases similarly to prevalence for the disease emergence model (Model 4 Fig. 1 c(i) ). 339 This is contradictory to the theory of critical slowing down theory which predicts that 340 derivations from the steady state values return increasingly slowly on the approach to a 341 transition.