where the first equality uses the definition of the auto-convolution, the second is a re-ordering of integrals, the third involves changing variables to u = t * -s, the fourth is a factorisation and the fifth arises by induction. The sixth uses the definition of the individual reproduction number R (r ) one obtains ignoring household structure from equation (12) . The household reproduction number can be expressed in terms of the individual reproduction number R (r ) as