Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 3 / d / Solution

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.