An exact and implementable computation of the final outbreak size distribution under Erlang distributed infectious period

This paper deals with a stochastic SIR (Susceptible-Infected-Recovered) model with Erlang(k,μ) distributed infectious period commonly referred as SI_kR model. We show that using the total number of remaining Erlang stages as the state variable, we do not need to keep track of the stages of individual infections, and can employ a first step analysis to efficiently obtain quantities of interest. We study the distribution of the total number of recovered individuals and the distribution of the maximum number of individuals who are simultaneously infected until the end of the disease. In the literature, final outbreak size is calculated only for a small population size exactly and derivations of approximate analytic solutions from asymptotic results are suggested for larger population sizes. We numerically demonstrate that our methods are implementable on large size problem instances.