The final outcome of SIR epidemics with infectivity profiles

Abstract A stochastic model for the spread of an SIR (susceptible $\to$ infective $\to$ removed) epidemic is considered. Infectives have independent and identically distributed infectivity profiles , which describe their infectiousness as a function of time since infection. The individual-to-individual infection rate depends also on the number of susceptibles present in the population. Exact results are derived for the distribution of statistics defined on the final outcome of the epidemic, including its final size. These are proved by using a generalisation of a Sellke construction to show that the distribution of the final outcome of the epidemic is the same as that of an associated discrete-time epidemic process, in which infectives are considered one at a time, and exploiting connection with death processes to analyse the final outcome of the latter. The results generalise easily to multipopulation epidemics.