Homogeneous transmission models can be interpreted as twolevel hierarchical models, where the processes which guide the natural history of infection within the host are considered separate from those which drive transmission between hosts. The link between the two can be thought of as the function b (t, t) which translates the impact of changing processes within the host into changing infectiousness as a function of time since infection. The approach taken here to modelling household transmission is to study a three-level hierarchical model of transmission. The three levels are within-host, within-household, and between households. The natural history of infection is described by the individual infectiousness function b (t, t). I assume in this section that individuals are homogenous in infectiousness and susceptibility. I then use this to predict the course of epidemics within households, and derive a function b * (t, t * ) which describes the average infectiousness of a household towards other households as a function of the time since the household was infected, t * (from here-on, I use the starred symbols to denote properties of households, and un-starred symbols to denote properties of individuals). The basic idea behind this analysis is illustrated in Fig 1. To simplify the notation, and because the main aim of this section is to study the case of an epidemic growing exponentially, I consider the situation where infectiousness is independent of calendar time t. This could be relaxed, though only if variation in time is somewhat slower than the typical duration of infection within a household.