 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-207/3/c/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Whenever

I > 0

, divide the first SIR model equation by the recovery equation:

\frac{d S}{d R} = - \frac{β}{γ} S .

R = 0, S = N

, giving susceptibles as a function of recovered individuals:

S (t) = N e^{- βR (t) / γ} .

Population conservation then gives the infectious count:

I (t) = N + 1 - R (t) - N e^{- βR (t) / γ} .

These identities extend continuously to the limiting state where the infectious count vanishes.

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