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For each $z$, integrate the probability of occupying the ill state:Equivalently, this is the $(H,I)$ entry of the fundamental matrix of an absorbing continuous-time Markov chain $(-Q_{z,T}{)}^{-1}$.

There is also a direct calculation. Each episode is fatal with probability $\delta /(\gamma +\delta )$, so the expected number of episodes before death is $(\gamma +\delta )/\delta$. Each lasts on average $1/(\gamma +\delta )$, givingThe acquisition rates change the waiting time between episodes but, under this model, not the total time eventually spent ill. Thus both risk groups have the same estimate.

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 integralEquivalently, 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.