 Solution

Let $P_{rs}^{(z)}(t)=\mathbb{P}(X(t)=s\mid X(0)=r,z)$ denote the transition semigroup of a continuous-time Markov chain. The first person is observed in $H$ at day zero, $I$ at day seven, and $H$ at day fourteen, so the contribution is

For the second person, death is observed exactly at day six but the infection time is latent. The density of an $I\to D$ transition at day six isThus the combined contribution $L_1{L}_2$ is a function of $({\lambda }_0,\beta ,\gamma ,\delta )$. This illustrates how panel observations contribute transition probabilities while an exactly observed transition contributes a state probability times its transition intensity.