Noise and Nonlinearity in Measles Epidemics: Combining
Mechanistic and Statistical Approaches to Population Modeling

S. P. Ellner,1 B. A. Bailey,1 G. V. Bobashev,1 A. R. Gallant,2 B. T. Grenfell,3 and D. W. Nychka1

1. Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695-8203;
2. Department of Economics, University of North Carolina, Chapel Hill, North Carolina 27599-3305;
3. Department of Zoology, Cambridge University, Downing Street, Cambridge CB2 3EJ, United Kingdom

Submitted July 7, 1997; Accepted November 17, 1997

ABSTRACT: We present and evaluate an approach to analyzing population dynamics data using semimechanistic models. These models incorporate reliable information on population structure and underlying dynamic mechanisms but use nonparametric surface-fitting methods to avoid unsupported assumptions about the precise form of rate equations. Using historical data on measles epidemics as a case study, we show how this approach can lead to better forecasts, better characterizations of the dynamics, and a better understanding of the factors causing complex population dynamics relative to either mechanistic models or purely descriptive statistical time-series models. The semimechanistic models are found to have better forecasting accuracy than either of the model types used in previous analyses when tested on data not used to fit the models. The dynamics are characterized as being both nonlinear and noisy, and the global dynamics are clustered very tightly near the border of stability (dominant Lyapunov exponent 0). However, locally in state space the dynamics oscillate between strong short-term stability and strong short-term chaos (i.e., between negative and positive local Lyapunov exponents). There is statistically significant evidence for short-term chaos in all data sets examined. Thus the nonlinearity in these systems is characterized by the variance over state space in local measures of chaos versus stability rather than a single summary measure of the overall dynamics as either chaotic or nonchaotic.

Keywords: population dynamics, modeling, measles, time-series analysis, local Lyapunov exponents.