The asymptotic behaviour of incidence I (t ) is determined by reproduction numbers [21, 25] . Two intuitively defined reproduction numbers are the case reproduction number, which I denote R c (t ), and the instantaneous reproduction number, which I denote R (t ). The case reproduction number R c (t ) is a property of individuals infected at time t, and is the average number of people someone infected at time t can expect to infect. For a person infected at time t it is the total infection hazard from time t onwards, i.e.