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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-2/6a/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 6A / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

The infective equation is

Y^{'} = Y {βX - (μ + ν)} .

Since

X \leq N

, the threshold population is

N_{c} = \frac{μ + ν}{β} .

If

N < N_{c}

and

δ = μ + ν - βN > 0

, then

Y^{'} \leq - δ Y, Y (t) \leq Y (0) e^{- δ t} ⟶ 0.

Also

Z (t) = e^{- μ t} Z (0) + ν \int_{0}^{t} e^{- μ (t - s)} Y (s) d s ⟶ 0.

Any steady state with

Y > 0

would require

X = N_{c} > N

, which is impossible. Thus the infection cannot be maintained, and both infected and immune populations tend to zero from every admissible initial condition.

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