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

The first subscript in ${\beta }_{t,\tau }$ is the calendar time $t$, and the second is the infection age $\tau$, the time elapsed since the source individual became infected. Thus ${\beta }_{t,\tau }$ is the rate at which an individual of infection age $\tau$ generates infections at time $t$. The corresponding discrete infectious disease renewal equation isup to a separately modelled term for imported infection. The upper limit may instead be a fixed maximal infectious age, with unavailable terms set to zero.

The instantaneous reproduction number at time $t$ isIt is the expected number of secondary infections that one infected individual would produce if the transmission conditions at time $t$ applied throughout that individual's infectious life. The case reproduction number, also called the effective reproduction number in this question, isthe expected number actually produced by a person infected at $t$ as calendar-time conditions subsequently change. The instantaneous quantity is easier to estimate in real time because it depends on current and past incidence; the case quantity depends on future conditions.

Assume the infectivity profile is separable:Then $g_{\tau }$ is the discretized generation-interval distribution, $R_t$ is the instantaneous reproduction number, and the infectious disease renewal equation becomesHence $R_t={\Delta }_t/{\Lambda }_t$ whenever the total infectiousness ${\Lambda }_t$ is positive.

If incidence grows exponentially, ${\Delta }_t=C{e}^{\rho t}$, substitution in the infectious disease renewal equation gives the discrete Euler-Lotka equationFor $X\sim \mathrm{Geometric}(p)$ on $\{0,1,\dots \}$, $g_i=\mathbb{P}(X=i-1)=p(1-p{)}^{i-1}$. The finite geometric series therefore yieldsand consequently

Treat the diagnosis counts as the observed incidence series and define their total infectiousness byA conditionally independent Poisson observation model for the renewal process isInitial infections before the observation window can be included in the definition of $s_t$.

Use the shape-rate convention $R_k\sim \mathrm{Gamma}({\alpha }_0,{\beta }_0)$. The factors involving $R_k$ in the posterior density areBy Poisson-gamma conjugacy,so its posterior mean is $(I_k+{\alpha }_0)/(s_k+{\beta }_0)$.

Use a three-state continuous-time multi-state model with transient infected state $I$ and absorbing recovered and dead states $R$ and $D$. If recovery and death have constant transition intensities $\gamma$ and $\mu$, its Q-matrix iswith state order $(I,R,D)$. This is also a competing risks model: recovery and death are the two mutually exclusive first events.

The two event clocks have independent exponential distributions with rates $\mu$ and $\gamma$. The competing exponential clocks identity gives

The minimum of the recovery and death clocks has an exponential distribution with rate $\mu +\gamma$. Its expected value is therefore

The transition probability matrix has entriesunder the time-homogeneous Markov property. For the finite-state model in part d, the transition semigroup of a continuous-time Markov chain is

Assume a negative test identifies the recovered state. Person 1 remains infected through day 7 and recovers during $(7,14]$, so their interval-censored contribution isPerson 2 remains infected through day 7 and makes an exact $I\to D$ transition at day 10. A transition at an exact time contributes a transition probability density, givingAssuming independent individuals, their joint likelihood contribution is

The model assumes a time-homogeneous Markov property: transition intensities depend only on the current state, not on calendar time, infection age, or earlier history. A Semi-Markov multi-state model could let recovery and death hazards depend on time since infection or entry into the current state. Weekly tests reveal recovery only by interval censoring, however, so the relevant entry and transition times are latent. Fitting the less restrictive model then requires integrating over unobserved paths, and the available data may contain too little information to identify a flexible duration-dependent hazard.

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.