An hypothesis for the periodicity of smallpox epidemics as revealed by time series analysis.

Parish registers have been studied by time series analysis to detect smallpox epidemics in England during 1600-1800. Confirmatory evidence was provided by the seasonality of child mortality. A 5-year cycle in smallpox epidemics was detected in medium-sized, rural towns. Consideration of the mathematics of the dynamics of viral diseases suggests that the true interepidemic period where smallpox is endemic should be 2-3 years and it is concluded that, in the towns studied, the disease was not endemic but that the oscillations were established by 5-year cycles of periods of famine associated with high wheat prices. The cross-correlation function between the two cycles shows zero lag and the input-output function shows significant coherence. Another epidemic follows only when a sufficient density of susceptibles has been established by births, so that the cycles become phase-locked. It is predicted that smallpox (i) was endemic in London and other large cities, with 2-3 year epidemics, (ii) was epidemic with a 5-year oscillation in rural towns, (iii) did not reach epidemic proportions in scattered communities.