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Oscillatory dynamics of smallpox and the impact of vaccination.

The evolution of smallpox epidemics in London, 1647-1893, was studied by time series analysis of deaths from the disease in the Bills of Mortality. The interepidemic interval (T) evolved progressively from 4 years to 2 years at 1800. The dynamics of epidemics during 1647-1800 are explicable in terms of the transmission of viral diseases which shows that (i) T is determined by the product of population size (N) and susceptibility (beta), (ii) T determines the mean age of catching the disease, (iii) the system will settle at its steady-state, endemic level unless the epidemics are driven. It is suggested that (i) the progressive change in T was initially caused by a rise in N and later by an increased beta related to malnutrition and (ii) the epidemics were driven by an oscillation in delta beta associated with seasonal dry conditions. The effects of variolation and vaccination became apparent after 1800: the endemic level fell progressively, the epidemics were reduced in amplitude and they were not driven. The dynamics of the disease can now be described by an SEIR model: severe outbreaks of smallpox are followed by decaying epidemics. Endemic smallpox mortality also interacts with the dynamics of the population so that a long wavelength oscillation (associated with recovery after the plague) and a 5/6 year (associated with immigration) oscillation are generated.