B.T. Grenfell, O.N. Bjørnstad & B.F. Finkenstadt: Endemic and epidemic dynamics of measles. II. Scaling noise, determinism and predictability with the time series SIR model

Ecological Monographs (2002) 72: 185-202.


Two key, linked questions in population dynamics are the relative importance of noise vs density dependent nonlinearities and the limits on temporal predictability of population abundance. We propose that childhood microparasitic infections -- notably measles -- provide an unusually suitable empirical and theoretical test bed for addressing these issues. We base our analysis on a new mechanistic time series model for measles, the TSIR model, which captures the mechanistic essence of epidemic dynamics. The model, and parameter estimates based on short-term fits to pre-vaccination measles time series for 60 towns and cities in England and Wales, is introduced in the previous paper (Bjørnstad et al. 2001). Here, we explore how well the model predicts the long-terms dynamics of measles and the balance between noise and determinism, as a function of population size. The TSIR model captures the basic dynamical features of the long-term pattern of measles epidemics in large cities remarkably well (based on time and frequency domain analyses). In particular the model illustrates the impact of secular increases in birth rates, which cause a transition from biennial to annual dynamics. The model also captures the observed increase in epidemic irregularity with decreasing population size and the onset of local extinction below a Critical Community Size. Decreased host population size is shown to be associated with an increased impact of demographic stochasticity. The interaction between nonlinearity and noise is explored using local Lyapunov exponents (LLE). These testify to the high level of stability of the biennial attractor in large cities. Irregularities are due to the limit cycle evolving with changing human birth rates and not due to complex dynamics. The geometry of the dynamics (sign and magnitude of the LLEs across phase-space) is similar in the cities and the smaller urban areas. The qualitative difference in dynamics between small and large host communities is that demographic and extinction-recolonization stochasticities are much more influential in the former. The regional dynamics can therefore only by understood in terms of a core-satellite metapopulation structure for this host-enemy system. We also make a preliminary exploration of the model's ability to predict the dynamic consequences of measles vaccination.