= Irreversible three-state disease model
An irreversible three-state <continuous-time multi-state model> permits only $1\to2\to3$, with state 3 absorbing. For constant rates $\lambda=q_{12}$ and $\mu=q_{23}$,
$$
p_{11}(t)=e^{-\lambda t},\qquad
p_{12}(t)=\frac{\lambda(e^{-\lambda t}-e^{-\mu t})}{\mu-\lambda},\qquad
p_{13}(t)=1-p_{11}(t)-p_{12}(t).
$$
At equal rates use $p_{12}(t)=\lambda t e^{-\lambda t}$. A <panel-observed multi-state likelihood> uses these <transition probabilities>, allowing unobserved intermediate visits between examinations.
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