Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 3 / c

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- Solution c

Solution

An illness episode ends at total rate

γ + δ

, so its mean duration is

\frac{1}{γ + δ} = 10 days .

The probability that its terminating transition is fatal is

δ / (γ + δ) = 0.1

. Therefore, in day units,

δ = 0.01, γ = 0.09.

Starting healthy, the first infection time is exponential with rate

λ_{z}

. Hence

1 - e^{- 30 λ_{1}} = 0.06, 1 - e^{- 30 λ_{0}} = 0.01,

and

λ_{1} = - \frac{l o g 0.94}{30}, λ_{0} = - \frac{l o g 0.99}{30}, β = lo g \frac{λ _{1}}{λ _{0}} .

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