Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 1 / f

The transition probability matrix has entriesunder the time-homogeneous Markov property. For the finite-state model in part d, the transition semigroup of a continuous-time Markov chain is

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 isPerson 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, givingAssuming independent individuals, their joint likelihood contribution is