Deterministic and stochastic models for the seasonal variability of measles transmission.

Fine and Clarkson used a discrete-time epidemic model with variable transmission parameter to analyze measles data for England and Wales for 1950-1965, during the time of biennial epidemics. Their model seems to provide a convincing fit when its parameters are estimated from these data. In particular, they obtained nearly equal estimates for the variable transmission parameter from the widely different data for epidemic and nonepidemic years. They did not, however, study the dynamics of their model. In this paper we make such a study, and find that the model is unsatisfactory in some important respects. It has a unique equilibrium (to be exact, a solution showing only seasonal variation), and neutrally stable oscillations around this equilibrium are possible, but with period approximately 3 years, not 2. (The model also has an exact 3-year cycle, while for intermediate initial values it shows unstable "chaotic" behavior.) If we vary the amount of variation in the transmission parameter, from no variation to 2.5 times the observed amount, the main periodicity remains approximately 3 years, though the complexity of behavior changes, increasing as the amount of variation increases. A continuous-time version of the model also has an equilibrium and a 3-year cycle, the main difference being that these are stable rather than neutrally stable. A stochastic version shows that simulations starting close to either the equilibrium or the 3-year cycle have a high probability of remaining close to the corresponding deterministic solutions for at least 20-30 years. The oscillatory features of the model are thus reasonably robust against stochastic fluctuations. We conclude that the simple homogeneous mixing model cannot be adapted to provide an adequate fit to the data, especially in explaining the observed biennial cycles. It seems likely that the pattern of seasonal variation found by Fine and Clarkson is essentially correct, but for accurate modeling of measles we need also to take into account heterogeneities of mixing in the population, especially those due to age and space.