With iterative tracing, isolation of a single infected results in quarantine of a cluster of infecteds, all mutually linked by traceable contacts ( Figure 1B) . By recognizing these clusters of infectives, an epidemic of infected individuals can be regarded as an epidemic of traceable clusters. Each untraceable contact in the transmission tree results in infection of a new cluster index case, so the average number of untraceable contact infections caused by a single cluster, the cluster reproduction ratio R c , determines the effectiveness of iterative contact tracing. If Y(p c ) denotes the expected cumulative infectiousness of a cluster at the time of cluster quarantine, then