The infectious disease renewal equation expresses current <incidence> as a <convolution> of past incidence with an <infectivity profile>, multiplied by a time-varying transmission level.

Infectious disease renewal equation

Assume the <infectivity profile> is separable:
$$
\beta_{t,\tau}=R_tg_\tau,
\qquad g_\tau\geq0,
\qquad \sum_{\tau\geq1}g_\tau=1.
$$
Then $g_\tau$ is the discretized <generation-interval distribution>, $R_t$ is the <instantaneous reproduction number>, and the <infectious disease renewal equation> becomes
$$
\Delta_t=R_t\Lambda_t,
\qquad
\Lambda_t=\sum_{\tau=1}^{t}g_\tau\Delta_{t-\tau}.
$$
Hence $R_t=\Delta_t/\Lambda_t$ whenever the total infectiousness $\Lambda_t$ is positive.

Solution

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

An infectivity profile describes how an infected individual's transmission rate varies with <infection age>.

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 Incoming links: Infectivity profile