Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 2 / a

Use three states of a continuous-time Markov chain: disease-free $1$, pre-clinical $2$, and clinical $3$. The transition intensities are $\lambda =q_{12}>0$ for onset and $\nu =q_{23}>0$ for progression. Clinical disease is an absorbing state. The mandatory pre-clinical phase excludes a direct $1\to 3$ jump, and this untreated progression model has no reverse transitions.

With row-vector probabilities and state order $(1,2,3)$, the generator matrix isIts rows sum to zero, so only the two off-diagonal transition intensities are unknown. This is a continuous-time multi-state model with sequential irreversible progression.