= Solution
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.
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