= Solution
Use an irreversible <continuous-time multi-state model> $1\to2\to3$, with state 3 absorbing. Let $a=q_{12}(50)$ and $b=q_{23}(50)$, and define log-linear age effects by
$$
q_{12}(A)=a e^{\beta_{12}(A-50)},\qquad
q_{23}(A)=b e^{\beta_{23}(A-50)}.
$$
For a closed-form <panel-observed multi-state likelihood>, freeze the age covariate at the start of each observation interval. The <transition intensities> are then $(a,b)$ over the first interval and $(\lambda,\mu)=(ae^{5\beta_{12}},be^{5\beta_{23}})$ over the second. Conditional on the initial state, the <likelihood contribution> is
$$
\boxed{L(a,b,\beta_{12},\beta_{23})
=e^{-5a}\left[1-e^{-ae^{5\beta_{12}}}
-p_{12}\!\left(1\mid ae^{5\beta_{12}},be^{5\beta_{23}}\right)\right].}
$$
Indeed, $p_{11}(5)=e^{-5a}$ and $p_{13}(1)=1-p_{11}(1)-p_{12}(1)$. In this irreversible <continuous-time Markov chain>, the intermediate state has probability
$$
p_{12}(u\mid\lambda,\mu)=
\begin{cases}
\displaystyle\frac{\lambda(e^{-\lambda u}-e^{-\mu u})}{\mu-\lambda},&\lambda\ne\mu,\\
\lambda u e^{-\lambda u},&\lambda=\mu.
\end{cases}
$$
The second expression is the continuous limit of the first and handles equal rates. The observed two-state jump across an interval is compatible with adjacent instantaneous transitions: an unobserved visit to state 2 occurs between the examinations.
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