Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 2

The crude incidence-rate estimator isper person-month. This treats all 3000 person-months as susceptible time and assumes infection times are observed exactly, follow-up is complete, and each first infection is detected. In practice tests occur intermittently, so infection is interval-censored; latent, brief, or asymptomatic infections may also be missed.

Let $P(t)=e^{Qt}$ be the transition probability matrix of the four-state continuous-time Markov chain, and put $N=\{S,R\}$ for a negative test and $P=\{E,I\}$ for a positive test. Starting susceptible at time zero, the likelihood contribution isEquivalently, with indicator diagonal matrices $D_N,D_P$ and the susceptible basis vector $e_S$ it isThis sums over every hidden state sequence compatible with the three test results.

For transition-intensity vector $q$ and generator $Q(q)$, the expected infectious occupancy over three months from $S$ is the Markov reward model quantityInsert the fitted intensities to obtain $m(\hat{q})$, evaluating the matrix exponential and integral numerically if necessary.

If $\eta =(\log q_{SE},\log q_{EI},\log q_{IR})$ has estimated covariance $V$, calculate the numerical gradient $g={\nabla }_{\eta }m(e^{\eta })$ at $\hat{\eta }$. The delta method gives estimated variance $g^{T}Vg$ and the approximate confidence intervalSimulation from $N(\hat{\eta },V)$ followed by transformation through $m$ gives a useful asymmetric alternative.

The linear SEIR model with a constant susceptible-to-exposed intensity isAdding the four equations confirms conservation of population: $S+E+I+R=N$.

The separable equation $\dot{S}=-q_{SE}S$ with $S(0)=S_0$ has solution

Substitute part (ii) into $\dot{E}+q_{EI}E=q_{SE}{S}_0{e}^{-q_{SE}t}$. Multiplication by the integrating factor $e^{q_{EI}t}$ givesSince $E(0)=0$, integration and rearrangement yield

At an interior maximum, $\dot{E}(t)=0$, equivalentlyThusThis is positive for distinct positive rates. Its continuous limit when $q_{EI}=q_{SE}=q$ is $1/q$.

The $S\to E$ transition must be redefined. A constant $q_{SE}$ gives each susceptible the same external infection pressure even when no infectious person remains. For influenza, the force of infection should depend on the current infectious prevalence. The progression $E\to I$ and recovery $I\to R$ can retain rates $q_{EI}$ and $q_{IR}$.

With frequency-dependent transmission parameter $\beta$, the nonlinear SEIR equations are

While $I>0$, divide the susceptible equation by the recovered equation:Taking $R(0)=0$ and integrating givesAt the end of the epidemic $E_{\infty }=I_{\infty }=0$, so conservation gives $R_{\infty }=N-S_{\infty }$. Definingwe obtain the final size relation for an epidemic