Influence of Perfect Vaccination of Highly Connected Individuals on the Median Smallpox Outbreak Size. Each point was obtained from 1,000 simulations. Initial value: 100 index cases in a population of 100,000. Vaccine efficacy = 100%. Targeted vaccination: (a) a random sample of 10% of the population is asked to name casual contacts each (number of contacts asked is given next to the curves); (b) these contacts are sorted by frequency; (c) vaccination starts with the most frequently named person and progresses towards less frequently named ones. The horizontal axis shows what percentage of the population is vaccinated.