The model defined above assumes that the function b (t, t) describes the ''natural history'' of infection in each infected individual. Before specialising to the model of household transmission, it is first worth considering the case where different individuals experience different ''natural histories'', defined here by the susceptibility to infection, and infectiousness after infection. I denote a vector of random variables X = {X1, X2, …} to describe factors which influence susceptibility or infectiousness. For example for the standard SEIR model of infection the random variables would be the durations of the latent period (L) and the infectious period (D), i.e. X = {L,D}. Let f (X ) denote the probability distribution of these random variables amongst new infections (taking into account differences in susceptibility), defined such that