The probability that a typical infective generates a local epidemic is computed by using a branching process approximation [12] for the initial stages of the epidemic, and equating 'epidemic' with the event that the branching process does not become extinct. This calculation is well known (e.g. [13] , page 473), but is modified here to allow for the fact that the process is initiated by a random number of infected arrivals and some of them have spent a random part of their infectious period before arriving in the at-risk country. The distribution for the random number of individuals infected by an infected individual when all their contacts are with susceptible individuals is needed for the calculation. The lack of data prevents a definitive conclusion for the most appropriate offspring distribution for influenza transmission [14] , and we use a Poisson distribution with a mean equal to R, discounted for individuals who spent only some of their infectious period mixing in the at-risk country. A Poisson offspring distribution is appropriate when the area under the infectiousness function is non-random (i.e. all individuals have the same infection 'potential'). We assume that R is the same in the source region and the at-risk country. For an undetected infected traveler and all their in-flight offspring to fail to initiate an epidemic on arrival, all of the chains of transmission they initiate must fail to become large epidemics (see Supporting Information).