3C -A severe smallpox epidemic is controlled by ring vaccination despite the large number of initial cases. The parameters are identical to Figure 3A , except that 1000 index cases inaugurate the attack in these scenarios (and ring vaccination capacity is much greater, as indicated). While not recommended, ring vaccination may ultimately halt epidemics beginning with many index cases if sufficient vaccination capacity were available, contact finding feasible, and follow-up sufficient. 3D -Tracing contacts of contacts (red) is beneficial when sufficient contact tracing/ring vaccination capacity exists (dotted lines). In these scenarios, all parameters are the same as in Figure 3A ; the number of contact tracings possible per day is either 20 or 40 per day. Contacts of contacts are traced in two scenarios; in the other two, only direct contacts of cases are traced. For low levels of ring vaccination (20 per day), tracing contacts of contacts is harmful; for high levels (40 per day) of ring vaccination, it is beneficial to trace contacts of contacts. When the contact tracing/ring vaccination capacity is too small to adequately cover contacts of the cases themselves, diversion of resources to contacts of contacts is harmful; however, provided that sufficient capacity exists, tracing contacts of contacts helps outrun the chain of transmission. Each line corresponds to the average of 100 realizations. The average number infected on each day is plotted in the Figure. The figure illustrates that when ring vaccination capacity is low, tracing contacts of contacts (as modeled) yields a more severe average epidemic; when ring vaccination capacity is large, tracing contacts of contacts results in a less severe average epidemic; if the contact tracing/ring vaccination capacity is too low to cover adequately the contacts of contacts in addition to the contacts of cases, extension of tracing to the contacts of contacts (the second ring) is harmful; however, if there is sufficient capacity to cover the contacts of contacts, then the tracing of contacts of contacts is beneficial.