= Progressive illness-death model
{title2=$Q=\begin{pmatrix}-(a+b)&a&b\\0&-c&c\\0&0&0\end{pmatrix}$}
This <continuous-time multi-state model> permits progression $1\to2$, direct death $1\to3$, and death after progression $2\to3$, with no recovery and death an <absorbing state>. For constant positive rates $a,b,c$, write $\lambda=a+b$. Integrating over the time of the progression arrow gives
$$
p_{12}(t)=\int_0^t e^{-\lambda u}a e^{-c(t-u)}\,du
=\frac{a(e^{-ct}-e^{-\lambda t})}{\lambda-c}.
$$
Use $at e^{-ct}$ when $\lambda=c$. The other <transition probabilities> are $p_{11}=e^{-\lambda t}$, $p_{13}=1-p_{11}-p_{12}$, $p_{22}=e^{-ct}$, $p_{23}=1-e^{-ct}$ and $p_{33}=1$. Unlike a purely sequential <irreversible three-state disease model>, the direct-death rate contributes to the first-state exit rate. A <mixed panel and exact-death likelihood> accounts for panel visits and exact death times with different kinds of factors.
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