Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 3 (Statistics in Medical Practice)

Use a time-homogeneous continuous-time multi-state model with states $H$ (healthy), $I$ (ill), and $D$ (dead), where $D$ is absorbing. For risk-factor indicator $z\in \{0,1\}$, let the transition intensities beHere ${\lambda }_0$ is the infection rate without the risk factor, $e^{\beta }$ is the infection hazard ratio, $\gamma$ is the recovery rate, and $\delta$ is the disease-death rate. The assumption that the risk factor affects only acquisition makes $\gamma$ and $\delta$ common to both groups. The infinitesimal generator is

Let $P_{rs}^{(z)}(t)=\mathbb{P}(X(t)=s\mid X(0)=r,z)$ denote the transition semigroup of a continuous-time Markov chain. The first person is observed in $H$ at day zero, $I$ at day seven, and $H$ at day fourteen, so the contribution is

For the second person, death is observed exactly at day six but the infection time is latent. The density of an $I\to D$ transition at day six isThus the combined contribution $L_1{L}_2$ is a function of $({\lambda }_0,\beta ,\gamma ,\delta )$. This illustrates how panel observations contribute transition probabilities while an exactly observed transition contributes a state probability times its transition intensity.

An illness episode ends at total rate $\gamma +\delta$, so its mean duration isThe probability that its terminating transition is fatal is $\delta /(\gamma +\delta )=0.1$. Therefore, in day units,

Starting healthy, the first infection time is exponential with rate ${\lambda }_z$. Henceand

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.

One analysis can treat a reported symptom-onset date as the exact $H\to I$ transition time. That adds an exactly observed infection-time density to the likelihood, but assumes symptoms begin immediately at infection, every relevant episode is symptomatic, and dates are recalled and reported without error.

A more realistic analysis treats true infection as a latent transition and symptom onset as a noisy observation. A reporting-delay distribution, and possibly probabilities of asymptomatic infection and non-reporting, can be added to a Hidden Markov model. Weekly tests then interval-censor the state transition while the symptom date refines its distribution. This approach uses more information but requires an identifiable and correctly specified symptom-delay and reporting model.