Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 36 / 1 / a / Solution

In the SI model, one infection reduces the susceptible count by one, and the total event rate is $\lambda X(t)Y(t)/n=nb(X_n(t))$, where $b(u)=\lambda u(1-u)$. The Poisson time-change representation of a Markov chain therefore giveswhere $N_n$ is a unit-rate Poisson process. Subtracting its clock from its count givesThus $X_n(t)=a-{\int }_0^{t}b(X_n(s))ds-{\epsilon }_n(t)$. The clock stops when the susceptible count reaches zero, so the representation also includes the absorbing state.