The results of a time-series analysis of smallpox mortality in London are given in Chapter 9 and are shown in Chapter 10 to be consistent with linearised and non-linear models in which the epidemics are driven by an oscillation in susceptibility (δβ) produced by seasonal climatic conditions when the frequency of this is at, or close to, ωr. These mathematical models are, necessarily, simplified and are not integrated into the population dynamics. An extended model of a driven system is introduced in this chapter; it includes the mortality from smallpox, the rate of vaccination and changes in population size. These are potentially important factors where the disease is lethal and a substantial proportion of those infected die or where the population is increasing steadily (as in London in the 18th century). The objectives of this modelling of the smallpox epidemics in London are:

1. (1) To test the robustness of the model when these additional parameters are included and to confirm, in particular, that the system can be driven by an oscillation in δβ.

2. (2) To simulate the pattern and detailed characteristics of the smallpox epidemics in London and so verify the model.

3. (3) To study the effect of varying the parameters, for example δβ or the proportion of the infectives dying, on the behaviour of the model. From such studies it is possible to predict some details of the epidemiology and biology of the disease in London in the 18th century, before the major impact of inoculation and vaccination.

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