= Solution
Assume a negative test identifies the recovered state. Person 1 remains infected through day 7 and recovers during $(7,14]$, so their <interval-censored> contribution is
$$
P_{II}(7)P_{IR}(7).
$$
Person 2 remains infected through day 7 and makes an exact $I\to D$ transition at day 10. A transition at an exact time contributes a transition probability density, giving
$$
P_{II}(7)P_{II}(3)q_{ID}=P_{II}(10)\mu.
$$
Assuming independent individuals, their joint <likelihood contribution> is
$$
P_{II}(7)P_{IR}(7)P_{II}(10)\mu.
$$
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