Abstract

Recurrent outbreaks of measles, chickenpox and mumps in cities are studied with a mathematical model of ordinary differential delay equations. For each calendar month a mean contact rate (fraction of susceptibles contacted per day by an infective) is estimated from the monthly reported cases over a 30- to 35-year period. For each disease the mean monthly contact rate is 1.7 to 2 times higher in the winter months than in the summer months; the seasonal variation is attributed primarily to the gathering of children in school. Computer simulations that use the seasonally varying contact rates reproduce the observed pattern of undamped recurrent outbreaks: annual outbreaks of chickenpox and mumps and biennial outbreaks of measles. The two-year period of measles outbreaks is the signature of an endemic infectious disease that would exhaust itself and become nonendemic if there were a minor increase in infectivity or a decrease in the length of the incubation period. For populations in which most members are vaccinated, simulations show that the persistence of the biennial pattern of measles outbreaks implies that the vaccine is not being used uniformly throughout the population.