= Solution
\b[There are three free <transition intensities>: progression, death from the initial state, and death from the advanced state.] In the three-state <illness-death model>, state 3 is an <absorbing state>, and there is no recovery transition.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-32-multistate.png]
{title=Irreversible three-state progression model with mild, severe and absorbing death states}
{height=320}
Writing $a=q_{12}$, $b=q_{13}$ and $c=q_{23}$, with all three nonnegative, the <transition intensity matrix> is
$$
\boxed{Q=\begin{pmatrix}-(a+b)&a&b\\0&-c&c\\0&0&0\end{pmatrix}.}
$$
Each diagonal entry is minus the sum of the row's outgoing <transition intensities>, rather than an additional unknown parameter. A <continuous-time multi-state model> with these rates describes the severity labels in the observations; an explicit cured state would require a richer state space.
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