where S, I are the proportion of individuals susceptible and infectious, respectively. The mathematical formalism behind deriving such sets of ordinary differential equations from Markov chains is given by Kurtz [113] , and a summary of the application of this methodology to infectious disease modelling is given in Diekmann and Heesterbeek [114] . However, it should be clear that (8) is familiar as the basis of all random-mixing epidemiological models.