Interpreting time-series analyses for continuous-time biological models—measles as a case study

Abstract

An increasing number of recent studies involve the fitting of mechanistic models to ecological time-series. In some cases, it is necessary for these models to be discrete-time approximations of continuous-time processes. We test the validity of discretization in the case of measles, where time-series models have recently been developed to estimate ecological parameters directly from data. We find that a non-homogeneous contact function is necessary to capture the host–parasite interaction in a discrete-time model, even in the absence of heterogeneities due to spatial or age structure. We derive a mathematical relationship describing the expected departure from mass-action transmission in terms of the epidemiological parameters in the model, and identify conditions under which the discretization process may fail.

Introduction

Continuous-time models, both deterministic (Anderson and May 1979, Anderson and May 1991; Murray, 1989) and stochastic (Bartlett 1960, Bartlett 1961; Mollison et al., 1994) have a long history in the study of biological systems. Relatively simple sets of coupled equations may be used to model the mechanisms underlying a continuously changing system. In many real-world systems, it is difficult to measure (or even identify) all state variables, and instead measurements are made of a single variable at discrete time intervals. This time-series can then be fitted to a mechanistic discrete-time model, of which the Nicholson–Bailey host–parasite model (Nicholson and Bailey, 1935) is an early example. While discrete-time models are often better suited to the available data, not all simple discrete-time models successfully reproduce the dynamics of the standard continuous-time models (Mollison and Din, 1993). It is important to examine the effect that the discretization process has on model behaviour before conclusions can be drawn from the time-series analysis.

To address this question, we need to analyse systems for which rich time-series are available, and which are sufficiently well understood for realistic discrete- and continuous-time models to be made. Measles is a perfect case study for this investigation, with its extensive notification time-series (Grenfell and Harwood, 1997) and relatively simple natural history. A family of continuous-time models based on the susceptible–infective–recovered (SIR) paradigm successfully capture the recurrent pre-vaccination sequence of measles epidemics and the impact of vaccination (Anderson and May, 1991; Schenzle, 1984; Earn et al., 2000). Early attempts to construct discrete-time models of measles (Fine and Clarkson, 1982) have been found to display unsatisfactory dynamics (Mollison and Din, 1993), failing to capture the strong biennial cycle seen in numerous pre-vaccination data sets (e.g. England and Wales, New York, Baltimore). A recently developed discrete-time model allows for the estimation of epidemiological parameters (notably the seasonality in transmission rate) directly from disease time-series (Finkenstädt and Grenfell, 2000). This time-series SIR (TSIR) model successfully bridges the gap between mechanistic models and time-series data, capturing many of the salient features of pre-vaccination measles in large cities (Bjørnstad et al., 2002; Grenfell et al., 2002). In the light of the earlier difficulties with discrete-time models, a thorough investigation of the dynamical behaviour of the TSIR model over the range of parameter values seen for measles is crucial for interpreting the time-series analysis.

Of particular interest are the intriguing departures from homogeneous ‘mass-action’ transmission discovered in Finkenstädt and Grenfell (2000). These findings could represent important evidence of population manifestations of spatial heterogeneity in infection. However, we first need to examine whether the discretization of the dynamics inherent in this time-series model affects estimates of mass-action transmission.

This paper begins with a discussion of current measles models, and then provides a comprehensive comparison of the time-series model with the more established continuous-time model. It becomes apparent that the use of a non-homogeneous contact function in discrete time is crucial for modelling mass-action transmission in continuous time. We provide a mathematical analysis of the relationship between the contact functions of the two models, and consider the consequences of significant changes to the system such as that induced by measles vaccination. Our findings raise important issues for a large class of discrete-time ecological models.