Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 2

Use a two-state continuous-time Markov chain with uninfected state $S$ and infected state $I$:The generator is $Q=\left(\begin{array}{cc}-a& a\\ b& -b\end{array}\right)$. Exponential holding times give expected time from recovery to the next infection $1/a$ and expected infection duration $1/b$.

The hypotheses imply $a=1/5=0.2$ and $b=1/1.25=0.8$, so $a+b=1$. Conditional on the negative result at month 1, the likelihood is the probability of remaining in $S$ at month 2 and moving to $I$ by month 3. The supplied matrix exponential givesThe Markov property therefore gives

Acquired immunity means that the reinfection rate is no longer the common constant $a$: it should depend on infection history, usually decreasing immediately after recovery and perhaps rising as immunity wanes. The two observed states are then not Markov unless the state is enlarged to record time since infection, number of prior infections, or latent immune class. Monthly binary tests identify infection status only at visits, so infection and recovery times are interval-censored and past short episodes may be missed. The sparse histories contain little information for estimating several history-dependent transition intensities.

The SIR model isThe expected infectious duration is $d=1/\gamma$.

The SEIR model inserts a latent exposed state:The expected time from infection through the end of infectiousness is $d=1/\sigma +1/\gamma$; the duration during which transmission occurs is $1/\gamma$.

The SIRS model has waning immunity:An infection lasts $d=1/\gamma$, while immunity lasts $1/\omega$ on average.

Under frequency-dependent mass-action kinetics, if $I(t)$ of the $N(t)$ individuals are infectious, the force of infection isOne newly infectious person in an otherwise susceptible population transmits at total rate approximately $\beta$ for mean infectious duration $1/\gamma$. Hence the basic reproduction number iswhere $d_I$ is the infectious, rather than latent-plus-infectious, duration.

Hospital admissions are a delayed thinning of infections. If $f(s)$ is the delay density and $\lambda (t)$ denotes the population infection incidence rate, then the convolutionis the admission incidence rate, with an additional term for infections before time zero if the observation window starts during an epidemic.

For $f(s)=\sigma e^{-\sigma s}$,Differentiating the convolution, equivalently integrating by parts, givesThus admissions follow a first-order lag: they move toward $p$ times current infection incidence at adjustment rate $\sigma$.