Although less efficient than ring vaccination in the sense that more vaccinations must be delivered to eliminate infection, comprehensive mass vaccination following the introduction of smallpox is sufficient to eliminate the infection. In Figure 10 , we show the probability of achieving containment (defined to be fewer than 500 total cases resulting from 10 index cases) for different levels of ring vaccination (0, 5, 10, and 20 vaccinations per day) and mass vaccination (0, 0.5%, 1%, and 2%; compare with the 10%-20% per day many jurisdictions in the United States are planning to vaccinate). Specifically, for each level of ring vaccination and mass vaccination, we used the same 1000 parameter sets used in Figure 5 , and performed 100 simulated epidemics for each parameter set. On the vertical axis, we plot the fraction of the 1000 scenarios for which each of the 100 simulated epidemics was contained. We further computed the fraction of scenarios for which none of the 100 simulated epidemics was contained; this is indicated by the colored segment in the small pie chart at each symbol. When the mass vaccination rate was 2% per day, the mean number of deaths (averaging over all scenarios and all simulations within each scenario) was 47.7, 33.7, 26.4, and 20.1 for a ring vaccination level of 0, 5, 10, and 20 per day (respectively) out of a population of 10000. Moreover, when we increased the mass vaccination level to 3%, an average of 28.9 deaths occurred when no ring vaccination was used, but this fell to 22.3 deaths when only 5 ring vaccinations per day were available (again assuming a population of 10000, and 10 index cases). With a mass vaccination level of 5% per day, an average of 18.8 deaths occurred without ring vaccination, and 15.8 deaths occurred when only 5 ring vaccinations per day were possible. (At a mass vaccination rate of 3% per day, containment as defined above was achieved in all 100 replications for 95% of the scenarios even without ring vaccination; at a mass vaccination rate of 5% per day, containment was achieved in all replications for all scenarios.) These results show that over a wide range of simulated epidemics, even seemingly small levels of ring vaccination (coupled with follow-up) may have a substantial effect in preventing epidemic spread and reducing deaths from smallpox, even during a mass vaccination campaign. Note that many jurisdictions in the United States are planning mass vaccination campaigns which could reach 10%-20% of the population per day, far greater than the mass vaccination levels examined here; it is interesting to note that mass vaccination cam- Figure 10 Mass and ring vaccination together. Low-level mass vaccination programs are improved substantially by the addition of ring vaccination. The shaded pie segments represent the fraction of 1000 scenarios for which containment (as defined in the text) was never realized; the vertical position of the pie chart represents the fraction of the 1000 "calibrated" scenarios for which containment was always achieved. As the fraction of the population mass vaccinated increases or the ring vaccination capacity increases, the probability of containment increases. paigns may be effective in preventing a widespread epidemic even at much lower levels than are being planned for. Where feasible, such rapid mass vaccination rapidly eliminates smallpox transmission in our model; vaccination of contacts is still beneficial, since we are assuming that earlier vaccination yields a greater probability of preventing or ameliorating infection (results not shown).