The case for chaos in childhood epidemics. II. Predicting historical epidemics from mathematical models

Abstract

The case for chaos in childhood epidemics rests on two observations. The first is that historical epidemics show various 'fieldmarks' of chaos, such as positive Lyapunov exponents. Second, phase portraits reconstructed from real-world epidemiological time series bear a striking resemblance to chaotic solutions obtained from certain epidemiological models. Both lines of evidence are subject to dispute: the algorithms used to look for the fieldmarks can be fooled by short, noisy time series, and the same fieldmarks can be generated by stochastic models in which there is demonstrably no chaos at all. In the present paper, we compare the predictive abilities of stochastic models with those of mechanistic scenarios that admit to chaotic solutions. The main results are as follows: (i) the mechanistic models outperform their stochastic counterparts; (ii) forecasting efficacy of the deterministic models is maximized by positing parameter values that induce chaotic behaviour; (iii) simple mechanistic models are equal if not superior to more detailed schemes that include age structure; and (iv) prediction accuracy for monthly notifications declines rapidly with time, so that, from a practical standpoint, the results are of little value. By way of contrast, next amplitude maps can successfully forecast successive changes in maximum incidence one or more years into the future.