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

Starting in state $1$, the onset waiting time has the exponential distribution of rate $\lambda$. Thus the probability of onset before the next screen is $1-e^{-\lambda t}$. This counts individuals who have already progressed to clinical disease as well as those still pre-clinical.

Conditional on onset at time $u<t$, the Markov property makes the remaining progression time exponential with rate $\nu$. Thus the probability of clinical progression before the screen, conditional on that onset time, is $1-e^{-\nu (t-u)}$.

Integrating over the onset density gives the unconditional clinical probabilityThe probability of being pre-clinical at the next screen is insteadTogether with $P_{11}(t)=e^{-\lambda t}$ and the exponential distribution for a patient initially in state $2$, the transition matrix isThis equals the matrix exponential $e^{Qt}$. The assumed unequal intensities avoid a zero denominator; at $\lambda =\nu$ the continuous limit is $P_{12}(t)=\lambda t{e}^{-\lambda t}$.