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

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- Solution i

Solution

Treat the diagnosis counts as the observed incidence series and define their total infectiousness by

s_{t} = \sum_{τ = 1}^{t} g_{τ} I_{t - τ} .

A conditionally independent Poisson observation model for the renewal process is

I_{t} ∣ R_{t}, g, I_{0}, \dots, I_{t - 1} \sim Poisson (R_{t} s_{t}), p (I_{1}, \dots, I_{T} ∣ R, g) = \prod_{t = 1}^{T} \frac{e ^{- R_{t} s_{t}} ( R _{t} s _{t} ) ^{I_{t}}}{I _{t} !} .

Initial infections before the observation window can be included in the definition of

s_{t}

.

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