Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 3 / c

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- Solution c

Solution

Using

S = N - I

reduces the system to the logistic differential equation

\dot{I} = β I (1 - I / N) .

Separation of variables and the initial values give

lo g \frac{I}{N - I} = βt - lo g θ, I (t) = \frac{N}{1 + θ e ^{- βt}} .

Differentiation yields the incidence

Λ (t) = \dot{I} (t) = \frac{θNβ e ^{- βt}}{( 1 + θ e ^{- βt} ) ^{2}} .

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