For full understanding of contact tracing in our model, we analyzed four special cases regarding the infectious period and the detection time distribution. The infectious period was assumed to be either very short (all transmission occurs instantaneously, so t inf = t lat ) or very long (of infinite duration, so t inf = '). (In our model, infinite duration can be assumed because each infected will stop spreading the infection after detection. If some infecteds would never be detected, some large t inf ,' should be taken.) The detection time was assumed to be either fixed (a = ') or highly variable (a = 1, i.e. exponentially distributed). As a control we analyzed intermediate cases (results not shown) and four examples of real infections (influenza, SARS, smallpox, and foot-and-mouth disease), of which the parameter values are listed in Table 2 . These parameter values were obtained from literature [2, 3, [12] [13] [14] , assuming that the time to detection consists of the incubation period (time to symptom onset) plus a symptom-to-detection delay, which we assumed to be distributed as observed in the SARS epidemic (average 3.67 days, [2] ).