The Pre-Epidemic Model
Consider a very large population in which nobody has smallpox. For the sake of simplicity it is assumed that the total size of the population is constant. Each day a number of individuals travel to and from areas of the world in which smallpox is endemic. Let α be the probability that on any given day at least one of the arriving individuals is infected with smallpox. (We assume that an individual arriving with the disease manifests no visible symptoms -- if the disease were apparent, the individual would be isolated immediately, and no outbreak of the disease could occur.) Note that the probability of disease importation depends on factors which are external to the population under study. Specifically, the probability that an infected person will arrive depends on the worldwide incidence of smallpox and the level of international travel to regions where the disease is endemic. These rates (and hence α) are determined outside the present mathematical model and are treated here as constants.3
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