Multiple-step contact tracing is not always logistically feasible. It is therefore important to determine the benefit provided by it. For both network types (figure 8), there are considerable differences between the final epidemic sizes for single-and multiple-step contact tracing at high tracing rates, particularly for assortatively mixed networks (figure 8b). In multiple-step tracing, a triggering node can generate other triggering nodes, potentially creating a cascade of triggering nodes throughout the infected portion of the network. This leads to tracing a higher number of infectious nodes and reducing the number of links that successfully transmit the disease. For high tracing rates, the epidemics are short-lived, and, on assortatively mixed networks, as a result of the faster initial epidemic growth rate, the extra proportion of untraced infectious nodes in the case of single-step tracing generates a higher number of infections and hence there is a marked difference between single-and multistep tracing. In figure 8 , the faster decrease in the proportion of traced nodes with increasing tracing probability indicates earlier and more effective control in the case of multiple-step contact tracing compared with the single-step case.