O.N. Bjørnstad, B.F. Finkenstadt & B.T. Grenfell: Endemic and epidemic dynamics of measles. I. estimating transmission rates and their scaling using a time series SIR model

Ecological Monographs (2002) 72: 169-184.


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 around 300-500 thousand. We develop a model (the TSIR model = Time series Susceptible-Infected-Recovered) 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 pre-vaccination era (1944-1967). The 60 cities analyzed span a size range from London (3.3 million inhabitants) to Teignmouth (10 thousand). 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-term 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 appears to be constant (at about 3%). Through the epidemic cycle, the susceptible population is depleted from a maximum of around 5% to a minimum of around 1.5-2%. The disease is, thus, efficient in 'regulating' the susceptible population -- even in small cities that undergo recurrent epidemics with frequent disease extinction. 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.