= 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
$$
L_1=P_{HI}^{(0)}(7)P_{IH}^{(0)}(7).
$$
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 is
$$
L_2=P_{HI}^{(1)}(6)\,\delta.
$$
Thus the combined contribution $L_1L_2$ 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>.
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