# Exact recursive expressions for final size

Author: Sangeeta Bhatia @sangeetabhatia03

Editor: Simon Frost @sdwfrost

Date: 2018-10-02

In a closed population, there is often interest in the total number of people infected, also known as the final size. Although simulations can be used (either deterministic or stochastic) can be used, there are more efficient methods for computing the final size.of a simple SIR model but the method can be generalised to other models.

The details are available in Black and Ross (2015). Instead of keeping tab of the population counts for each compartment ($S$, $I$, $R$), we keep a record of the number of infection and recovery events. At time t, let the number of infection events be $Z_1$ and the number of recovery events be $Z_t$. Then the state of our model is $(Z_1, Z_2)$. From one state to another, we can have at most one infection event or a single recovery event. Thus from $(Z_1, Z_2)$, the system can transition to $(Z_1 + 1, Z_2)$, $(Z_1, Z_2 + 1)$. To estimate the distribution of the final size, one needs to calculate the probability of all paths leading up to the state from which the system cannot transition $(Z, Z)$. The probabilities can be calculated easily by the states are indexed.