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Deterministic and Stochastic Models for Recurrent Epidemics

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A quantitative theory of epidemics in any complete sense is still a very long way off. The well-known complexity of most epidemiological phenomena is hardly surprising, for not only does it depend on the interactions between hosts and infecting organisms, each individual interaction itself usually a complicated and fluctuating biological process, but it is also, and this is a further point to be stressed, a struggle between opposing populations, the size of which may play a vital role. This last aspect is essentially one that can only be discussed in terms of statistical concepts. One contrast of my own approach with that of many earlier studies is that complete probabilistic or stochastic formulations have always been in mind. This enables the status of previous deterministic formulations, as approximations valid to some extent in the case of large numbers, to be examined. It will be shown that in some respects, as i epidemics always begin with only one or two infected persons, ii local units even of a large population are still small, the neglect of the random or chance factor can be quite misleading. I have been interested in particular in possible mechanisms for recurrent epidemics, when the susceptible population is in one way or other replenished. Measles, with children continually growing up into the critical age period, has been the explicit infectious disease usually in mind. The phenomenon of extinction or fade-out of infection largely decides whether the deterministic approximation has any relevance or not to the quasiperiodicity of recurrent epidemics. This chance of extinction, which alters with the characteristics of the model, is high for models appropriate to measles in comparatively small communities. This implies that the two-year cycle sometimes claimed as an inherent feature of measles, and an observed fact for many large urban areas, will be replaced by a longer average interval between epidemics for smaller communities.

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