Dynamics of Measles Epidemics: Estimating Scaling of Transmission Rates Using a Time Series Sir Model

Before the development of mass-vaccination campaigns, measles exhibited persistent fluctuations (endemic dynamics) in large British cities, and recurrent outbreaks (episodic dynamics) in smaller communities. The critical community size separating the two regimes was ∼300 000–500 000. We develop a model, the TSIR (Time-series Susceptible–Infected–Recovered) model, that can capture both endemic cycles and episodic outbreaks in measles. The model includes the stochasticity inherent in the disease transmission (giving rise to a negative binomial conditional distribution) and random immigration. It is thus a doubly stochastic model for disease dynamics. It further includes seasonality in the transmission rates. All parameters of the model are estimated on the basis of time series data on reported cases and reconstructed susceptible numbers from a set of cities in England and Wales in the prevaccination era (1944–1966). The 60 cities analyzed span a size range from London (3.3 × 106 inhabitants) to Teignmouth (10 500 inhabitants). The dynamics of all cities fit the model well. Transmission rates scale with community size, as expected from dynamics adhering closely to frequency dependent transmission (“true mass action”). These rates are further found to reveal strong seasonal variation, corresponding to high transmission during school terms and lower transmission during the school holidays. The basic reproductive ratio, R0, is found to be invariant across the observed range of host community size, and the mean proportion of susceptible individuals also appears to be constant. Through the epidemic cycle, the susceptible population is kept within a 3% interval. The disease is, thus, efficient in “regulating” the susceptible population—even in small cities that undergo recurrent epidemics with frequent extinction of the disease agent. Recolonization is highly sensitive to the random immigration process. The initial phase of the epidemic is also stochastic (due to demographic stochasticity and random immigration). However, the epidemic is nearly “deterministic” through most of the growth and decline phase.