We evaluated iterative tracing with and without delays for the same cases as single-step tracing, resulting in Figures similar to Figures 2 and 3 (see Supporting Information). As expected, the universal effect of iterative tracing is to lower p c *, although the lower limits 121/R 0 pre as observed in three of the four special cases remain intact. The largest difference between single-step and iterative tracing is observed when single-step tracing is on the brink of being effective, so that is the only situation where iterative tracing will make a difference (see also Figure 4 ). Without delays this difference is rather intuitive: single-step tracing is not effective if the latent period is small, but iterative tracing will always be effective, because if all contacts are traceable, the first detection is immediately followed by quarantine of every infected. This is the case with influenza, although the improvement will probably not be sufficient to making tracing effective for influenza control. On the other hand, when the latent period is large and single-step tracing is effective, iterative tracing only improves on this if there are significant delays, providing one or two more days to reach the required p c * (smallpox, SARS, and FMD).