MODELING THE DIFFERENT SMALLPOX EPIDEMICS IN ENGLAND

Time series analysis has revealed two different patterns of smallpox e pidemics in Britain in the seventeenth and eighteenth centuries: in la rge conurbations (exemplified by London) the disease was endemic where as medium-sized rural towns (exemplified by Penrith, Cumbria) suffered from 5 year epidemics with no cases of smallpox in the inter-epidemic years. The oscillations (epidemics) persisted for over 150 years and it is suggested that both systems were pumped up by regular fluctuatio ns in susceptibility (delta beta). Modelling suggests that: (i) the na tural frequency of oscillations in large cities is two years and the s ystem is pumped up by a 1 year, seasonal input; (ii) it takes five yea rs to build up a pool of susceptibles in medium-sized towns by new bir ths and epidemics are then triggered by a 5 year input. The equations represent a system that has two components, a basic linear element wit h the remainder of the system being nonlinear; modelling a progressive increase in delta beta in London illustrates theoretically how a pred ominantly linear response changes to a nonlinear response and ultimate ly to chaos. A variation in susceptibility is a theoretical condition for inducing chaos; the undriven system cannot become chaotic. Modelli ng populations of progressively increasing size/density and applying a I year or 5 year sinusoidal oscillation in delta beta illustrates the fundamental distinction in the response of medium-sized rural towns a nd large cities.