= Solution
The permitted transitions form the following diagram. Each arrow is labelled by its fitted <transition intensity> in units of inverse years; the absence of an arrow is a structural restriction, not an estimated small effect.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-37-transition-diagram.png]
{title=Five-state model: permitted transitions and fitted rates per year}
State 1 can move directly to 2, 3 or 5; state 2 can move to 4 or 5; state 3 can move to 4 or 5; and state 4 can move only to 5. There are no recovery transitions, and state 5 is an <absorbing state>. The rates of leaving the four living states are the sums of their outgoing arrows:
$$
\boxed{\lambda_1=0.085,\quad\lambda_2=0.181,\quad\lambda_3=0.369,\quad\lambda_4=0.513\quad\text{per year}.}
$$
The diagonal generator entries are their negatives, so every row of $Q$ sums to zero. These diagonal entries are not arrows or negative event rates. For example the direct $1\to5$ rate is $0.004$ per year, while death after reaching another state can occur at a much larger rate. The printed fit has no <covariates> on the intensities, so it estimates one pooled rate system and contains no treatment-effect comparison, despite the trial's treatment randomization. The reported $-2\ell$ describes the fitted <likelihood> level; by itself it is not a goodness-of-fit test without a reference or competing model.
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