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

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