Assessing intervention strategies for non-homogeneous populations using a closed form formula for R0.

A general stochastic model for susceptible → infective → recovered (SIR) epidemics in non-homogeneous populations is considered. The heterogeneity is a very important aspect here since it allows more realistic but also more complex models. The basic reproduction number R0, an indication of the probability of an outbreak for homogeneous populations does not indicate the probability of an outbreak for non-homogeneous models anymore, because it changes with the initially infected case. Therefore, we use "individual R0" that is the expected number of secondary cases for a unique given initially infected individual. Thus, the effectiveness of intervention strategies can be assessed by their capability to reduce individual R0 values. Also a vaccination plan based on individual R0 values for fully heterogeneous populations is proposed. It is based on the recursive calculation of individual R0 values.

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 SIkR 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.