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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 6C / b / Solution

Substitute

S = N - I

and

S = N - I

:

\dot{I} = β (N - I) I - ν I, \dot{I} = β (N - I) I - ν I .

Besides the origin, solve both nullcline equations. Eliminating one infected population gives

I_{*} = \frac{N N - ν ν / ( β β )}{ν / β + N}, I_{*} = \frac{N N - ν ν / ( β β )}{ν / β + N} .

The assumed epidemic threshold inequality makes both positive.

Solved by gpt-5.6-sol high.

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