Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 35 / 3

Let $u$ be the infection time and $s=t-u\ge 0$ its incubation delay, with independent delays having probability density function $f(s)$. Each infection contributes to the symptom-onset intensity according to its delay density. Back-calculation of infection incidence uses the convolutionObserved onset intensity is infection intensity convolved with the incubation distribution. If infections begin at a known $t_0$, take $h(u)=0$ for $u<t_0$, reducing the first integral's lower limit to $t_0$. Otherwise past infections must be included; the observation window's start need not be the infection process's start.

In discrete back-calculation with endpoint cohorts, for equal-width intervals $I_k=[t_{k-1},t_k)$, write $\Delta =t_k-t_{k-1}$, $h_i$ for the expected infection count assigned to time $t_i$ under the end-of-interval approximation, and ${\mu }_k$ for the expected onset count in $I_k$. DefineAn infection assigned to $t_i$ can first contribute to $I_{i+1}$, so the discrete convolution with this timing convention isThe sum includes any infection cohorts before the observation window. With no infections before $t_0$ and cohorts beginning at $i=1$, it is ${\sum }_{i=1}^{k-1}$ and the first mean is zero in this endpoint approximation.

For unequal intervals use $q_{ik}=\mathrm{Pr}(t_{k-1}-t_i\le T<t_k-t_i)$, truncating negative endpoints at zero, and ${\mu }_k\approx {\sum }_i{h}_i{q}_{ik}$. An alternative approximation assigning cohort $i$ to $t_{i-1}$ gives ${\sum }_{i\le k}{h}_i{q}_{k-i+1}$. Both are legitimate discretisations; the subsequent solutions use the end-of-interval convention explicitly specified in part (d), so the index shift is essential.

For the stated scale-parameterisation of the Weibull distribution, its incubation survivor function is $S_T(s)=\exp [-(s/\lambda {)}^{\kappa }]$ for $s\ge 0$, with $\lambda ,\kappa >0$. The continuous back-calculation of infection incidence equation becomesSet $h(t-s)=0$ before the infection process's start if one is specified. For the equal-width endpoint approximation, integrate the incubation density over each delay bin:Thus the discrete Weibull equation isFor unequal intervals replace $q_{k-i}$ by $S_T(\max \{0,t_{k-1}-t_i\})-S_T(\max \{0,t_k-t_i\})$. These are probabilities, not point evaluations of a density; no additional factor of $\Delta$ is needed after bin integration.

Assume individual incubation delays are independent of the Inhomogeneous Poisson process of infections and of one another. This marking assumption is needed in addition to specifying a marginal incubation density.

In the discrete cohort model let $N_i\sim \mathrm{Poisson}(h_i)$ be the independent infection counts. Mark each infection by its eventual onset interval. For cohort $i$, the categories have probabilities $q_{ik}$ for observed intervals, with the remaining probability assigned to onsets outside the window. Poisson thinning makes the counts $N_{ik}$ in these categories mutually independent Poisson random variables with means $h_i{q}_{ik}$. One direct proof is their probability generating function:Different cohorts are independent. Summing their counts therefore gives independent Poisson onset counts in disjoint intervals:The same argument applies exactly in continuous time by the Independent marking theorem for Poisson point processes and mapping each marked infection to its onset time. The endpoint model approximates its means; its independent-Poisson conclusion is exact within that discrete model. A fixed cohort size would instead induce negatively correlated onset-bin counts, so the Poisson process infection assumption matters.

With the onset means ${\mu }_k(\theta ,\lambda ,\kappa )$ defined by the discrete back-calculation of infection incidence, the independent Poisson observation model yields the product likelihoodFor positive means its log-likelihood is ${\sum }_k\{y_k\log {\mu }_k-{\mu }_k-\log (y_k!)\}$. A zero mean assigns probability one to a zero count and zero to a positive count. Any unknown pre-observation infection history must be parameterised or supplied as well; it cannot silently be excluded merely because the observation series starts at $t_0$.

Use one transition per time step in the printed discrete-time multi-state diagnosis model. Put $a={\delta }_E$, $b={\delta }_N$, and $r=1-b$. Let $I_k=S_{1k}\sim \mathrm{Poisson}(h_k(\theta ))$ be the new infection count. Let $U_k$ be the undiagnosed non-early count present in state 3 at step $k$, including those retained by its self-loop. Start with $I_0=U_0=0$.

For the realised random variables, define conditionallyusing independent individual transition choices, and setThe early split's two complementary counts are not independent conditional on $I_{k-1}$, nor are the late split's two counts conditional on $U_{k-1}$. Random state incidences must not be equated to their expectations.

Explicitly, the first four realised vectors areHere $E_k$ and $D_k$ have the conditional laws just given; $D_1=D_2=0$. The corresponding expressions solely in terms of the parameters are the mean flow vectors. Writing $h_j=h_j(\theta )$ and interpreting state 3 with its self-loop as $S_{3k}=U_k$, they areThese follow from $\mathbb{E}{U}_k=(1-a)h_{k-1}+r\mathbb{E}{U}_{k-1}$. They make the minimum one-step early delay and two-step late delay explicit. If “entering state 3” is reserved for first entry only, its incidence is $I_{k-1}-E_k$ with mean $(1-a)h_{k-1}$; $U_k$ is then a separate occupancy variable. The late-diagnosis expressions are unchanged. This distinction resolves the wording's use of incidence alongside a state-3 self-loop.

The expected values of early and late new diagnosis counts at step $k$ areEach summand in $n_k$ corresponds to an infection that avoids early diagnosis, enters the non-early state one step later, remains there for the intervening steps and is then diagnosed. Define $h_j=0$ for $j\le 0$, and take an empty sum as zero. The early share of diagnosis intensity isIt is defined only when $e_k+n_k>0$; in particular there are no diagnoses at step 1 under the initial condition. With infection counts following a Poisson distribution and independent marking, early and late diagnosis counts in a fixed interval are independent Poisson random variables. Conditional on a positive total, the early count has a binomial distribution given the total with probability $p_{Ek}$, so this ratio also equals the expectation of the observed early fraction conditional on a nonzero total. Without that conditioning a sample fraction at zero diagnoses is undefined.