= Solution
Starting in state $1$, the onset waiting time has the <exponential distribution> of rate $\lambda$. Thus \b[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 \b[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 probability
$$
P_{13}(t)=\int_0^t\lambda e^{-\lambda u}\bigl(1-e^{-\nu(t-u)}\bigr)\,du=1-\frac{\nu e^{-\lambda t}-\lambda e^{-\nu t}}{\nu-\lambda}.
$$
The probability of being pre-clinical at the next screen is instead
$$
P_{12}(t)=\int_0^t\lambda e^{-\lambda u}e^{-\nu(t-u)}\,du=\frac{\lambda}{\nu-\lambda}(e^{-\lambda t}-e^{-\nu t}).
$$
Together with $P_{11}(t)=e^{-\lambda t}$ and the <exponential distribution> for a patient initially in state $2$, \b[the transition matrix] is
$$
\boxed{P(t)=\begin{pmatrix}e^{-\lambda t}&\frac{\lambda(e^{-\lambda t}-e^{-\nu t})}{\nu-\lambda}&1-\frac{\nu e^{-\lambda t}-\lambda e^{-\nu t}}{\nu-\lambda}\\0&e^{-\nu t}&1-e^{-\nu t}\\0&0&1\end{pmatrix}.}
$$
This equals the <matrix exponential> $e^{Qt}$. The assumed unequal intensities avoid a zero denominator; at $\lambda=\nu$ the continuous limit is $P_{12}(t)=\lambda t e^{-\lambda t}$.
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