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

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 lambda approximately 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.

Disentangling Extrinsic from Intrinsic Factors in Disease Dynamics: A Nonlinear Time Series Approach with an Application to Cholera

“…The parametric part of the model allows for a portion of the dynamics to be defined mechanistically, while the nonparametric part of the model allows for flexibility in long-term or interannual changes in a parameter of interest. Although the usefulness of semiparametric models has been recently underscored in the ecological literature (Ellner et al 1998), their application to nonlinear dynamics that are driven by long-term or interannual variation has not been recognized (with the exception of a fisheries model; A. Solow, personal communication). For the relatively long time series of disease records, gradual changes in parameter values are inevitable.…”

“…We rely here on a similar semiparametric approach to develop a time series model for diseases with temporary immunity and unspecified variation in the transmission rate. Nonlinear time series models of this sort allow us to combine mechanistic representations of the processes we know with phenomenological representations of unknown processes (Ellner et al 1998). To our knowledge, this is the first attempt at using a statistical time series approach to understand retrospectively fluctuations in disease cycles as the result of both intrinsic and extrinsic factors, with the latter not limited to noise and seasonality.…”

Semiparametric Estimation of the Duration of Immunity from Infectious Disease Time Series: Influenza as a Case-Study

“…We call model a semiparametric mechanistic model. Similar approaches were introduced in Ellner et al (1998).…”

“…To use model , we must first reconstruct the number of susceptible individuals. Ellner et al (1998) and Finkenstädt and Grenfell (2000) considered the case of non‐recurring SIR diseases like measles. This is simpler than our current problem because, once a host is infected, under the SIR mechanism they will never be reinfected.…”

Estimating the Nationwide Transmission Risk of Measles in US Schools and Impacts of Vaccination and Supplemental Infection Control Strategies

“…Mathematical models have long been used to predict the transmission risk of airborne infectious diseases in the built environment. Epidemic modeling approaches such as susceptible-infector-susceptible (SIS) (16), susceptible-infectious-recovered (SIR) (17) competing-risks (18,19), neural network (20), and Reed-Frost (21,22) models are used to describe the progression of a disease in a population, although it is shown that these models alone cannot explain the spread of airborne infectious diseases such as measles in indoors environments (22). Therefore, epidemic models are usually combined with other mathematical approaches to predict the risks associated with indoor spaces such as airplanes, hospitals (3), schools (17), residences (23), and healthcare facilities (24).…”