Vaccination with vaccinia virus provided substantial protection against infection. Dixon assessed the risk of infection for an individual successfully vaccinated 3 years prior to exposure to be 0.1% the infection risk of an unvaccinated individual [17] . However, smallpox vaccination did not always take when applied, and moreover, in many instances, individuals who experienced a repeated vaccination failure developed severe smallpox upon exposure. The probability of a successful take depended on the vaccination method used; we assume that the take rate is between 95% and 100% [22, 28] . In addition to protection against infection, vaccination could in many cases modify the course of infection and reduce the severity. Vaccine protection waned over time, but individuals vaccinated 20 years prior to exposure were believed to still have half the infection probability that an unvaccinated person had [17] , and to have some protection against the most severe manifestations of smallpox. Dixon [17] believed that vaccine protection had at least three components, which decayed at different rates; for the purpose of this paper, we will assume that the severity of smallpox in previously any (recently or otherwise) vaccinated individuals follows the same distribution as for the vaccinated subjects seen in the case series observed by Rao in Madras [26] , except that anywhere from 0 to 5% of vaccinated subjects develop smallpox too mild to diagnose without special surveillance or awareness. Observe that the vaccinated cases studied by Rao were vaccinated (at some point in their lives) before exposure, rather than after exposure to smallpox.