Thus having averaged over all possible times of infection in the chain of transmission events in the household, infectious households are stratified by their size and by the number of cases in each generation. Using the notation defined earlier, I define the state vector X = {n, m 1 , …, m n } of variables which define the infectiousness and susceptibility of the infected household, where n is the household size and m i is the number of infected individuals in the i th generation of infection in the household. The infectiousness of a household with this state X towards other households, mediated by random mixing of individuals between households, is the sum of the infectiousness of all the individuals each given by equation (27) 