 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-207/2/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Starting in state

1

, the onset waiting time has the exponential distribution of rate

λ

. Thus the probability of onset before the next screen is

1 - e^{- λ 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

ν

. Thus the probability of clinical progression before the screen, conditional on that onset time, is

1 - e^{- ν (t - u)}

Integrating over the onset density gives the unconditional clinical probability

P_{13} (t) = \int_{0}^{t} λ e^{- λ u} (1 - e^{- ν (t - u)}) d u = 1 - \frac{ν e ^{- λ t} - λ e ^{- ν t}}{ν - λ} .

The probability of being pre-clinical at the next screen is instead

P_{12} (t) = \int_{0}^{t} λ e^{- λ u} e^{- ν (t - u)} d u = \frac{λ}{ν - λ} (e^{- λ t} - e^{- ν t}) .

Together with

P_{11} (t) = e^{- λ t}

and the exponential distribution for a patient initially in state

2

, the transition matrix is

P (t) = e^{- λ t} 00 \frac{λ ( e ^{- λ t} - e ^{- ν t} )}{ν - λ} e^{- ν t} 0 1 - \frac{ν e ^{- λ t} - λ e ^{- ν t}}{ν - λ} 1 - e^{- ν t} 1 .

e^{Qt}

. The assumed unequal intensities avoid a zero denominator; at

λ = ν

the continuous limit is

P_{12} (t) = λ t e^{- λ t}

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