Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 3 / f / Solution

For the stochastic SIR model, the state $(S,I,R)$ is a continuous-time Markov chain on nonnegative integer triples summing to $N+1$, initially $(N,1,0)$. Its only transitions areRates vanish when a proposed transition would leave the state space. Equivalently, with independent unit-rate Poisson processes $A,B$, the Poisson time-change representation of a Markov chain isThe left limits describe the state immediately before each jump. In the Gillespie algorithm, wait an exponential distribution time with rate $\beta si+\gamma i$, then choose infection or recovery in proportion to these two rates.