The contact structure of the network is illustrated in Figure  2 . Observe that individuals b and c are household contacts of individual a, so that if individual a were a smallpox case, an attempt would be made to find and vaccinate individuals b and c as household contacts of a case. If individuals a and b were both cases, then two attempts could be made to find individual c. We have modeled the effect of multiple contact-finding attempts conservatively in the sense that if the first attempt to find an individual as a household contact (of a case or of a contact) is determined to fail, no further attempts will be made. This maintains the failure rate of contact tracing (looked at from the standpoint of finding individuals) even in large households. Similar considerations apply to workplace/social groups. Figure 2 Network structure shown for households (joined by thick lines) of size 3 and workplace/social groups of size 4 (joined by thin lines); a small portion of the network is shown. Individual a has two household contacts (b and c), and three workplace/social contacts (d, e, and f). If individual a were a smallpox case, the household contacts would be at highest risk for acquiring smallpox, followed by workplace/social contacts; all individuals in the population are at a low risk of casual transmission from individual a. Case investigation of individual a would identify the direct contacts b-f with probabilities that depend on whether the contact is household or workplace/social; if such individuals are identified, they will be vaccinated. If contacts of contacts are being traced, the investigation will subsequently identify individuals g-p. 