Add a hospital state $H$ and estimate the enlarged <Q-matrix> from the cohort. If infection starts in state $I$, the expected hospital occupancy generated by one infection is the <Markov reward model> integral
$$
m_H=\int_0^\infty P_{IH}(t)\,dt.
$$
Equivalently, if $Q_T$ is the subgenerator on all transient states, $m_H$ is the $(I,H)$ entry of the <fundamental matrix of an absorbing continuous-time Markov chain> $(-Q_T)^{-1}$. The number infected in the population has expectation $Np$, so <linearity of expectation> gives total expected hospital use $Np,m_H$ days. If only occupancy within a finite period counts, replace the upper integration limit by the remaining follow-up time for each infection and average over infection times.

Solution

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

{title2=$(-Q_T)^{-1}$}

For a finite absorbing continuous-time Markov chain with transient subgenerator $Q_T$, the entry $(-Q_T)^{-1}_{ij}$ is the expected time spent in transient state $j$ before absorption when starting from state $i$.


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 Incoming links: Fundamental matrix of an absorbing continuous-time Markov chain