Three-state irreversible disease model ($Q=\left(\begin{array}{ccc}-\lambda & \lambda & 0\\ 0& -\nu & \nu \\ 0& 0& 0\end{array}\right)$)

A sequential continuous-time Markov chain has transitions $1\to 2\to 3$ with rates $\lambda ,\nu$ and an absorbing state $3$. Its probability of ever leaving state $1$ by time $t$ is $1-e^{-\lambda t}$; its probability of occupying state $2$ is $\lambda (e^{-\lambda t}-e^{-\nu t})/(\nu -\lambda )$ for unequal rates.