 Solution

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.

Writing $a=q_{12}$, $b=q_{13}$ and $c=q_{23}$, with all three nonnegative, the transition intensity matrix isEach 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.