 Solution

Use a time-homogeneous continuous-time multi-state model with states $H$ (healthy), $I$ (ill), and $D$ (dead), where $D$ is absorbing. For risk-factor indicator $z\in \{0,1\}$, let the transition intensities beHere ${\lambda }_0$ is the infection rate without the risk factor, $e^{\beta }$ is the infection hazard ratio, $\gamma$ is the recovery rate, and $\delta$ is the disease-death rate. The assumption that the risk factor affects only acquisition makes $\gamma$ and $\delta$ common to both groups. The infinitesimal generator is