The ordering of infection events has no influence on the final number of individuals infected [33] , and therefore this approximation will produce exact results for the final number of people infected in each household. Because of the possibility of ''later'' generations preceding ''earlier'' ones, as noted in the case of households of size 3 above, and because of ignoring the saturation effect present in equation (22) in terms of the actual generation times within households, this approximation will overestimate the time taken for individuals to become infected in the household. Because of the general form of the relation between generation time and reproduction number seen in equation (12) , this will result in over-estimates of the household reproduction number R * (r). To provide a counter-balancing under-estimate of R * (r), I also consider an alternative approximation obtained by assuming the same total number of cases as predicted by this Reed-Frost model, but where all cases are assumed to be infected by the first index case. This is not a formal lower bound, since in the limit of infinite infectiousness within the household, all members of the household will be infected simultaneously upon introduction of the infection into the household. I find however that even for the example of highly infectious measles virus (below), the under-approximation is sufficient to provide a practical lower bound.