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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-3/11e/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 11E / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

For

β > 0

and positive initial populations, summing the coordinates of the attracting equilibrium point gives the steady total population

N_{*} (d) = ⎩ ⎨ ⎧ 1 - d, d / β, 1, 0 \leq d \leq β / (1 + β), β / (1 + β) \leq d \leq β, d \geq β .

(1)

Its minimum is

1/ (1 + β)

d = β / (1 + β)

. Small mortality leaves a fully infected population with only modest demographic loss. Raising mortality first reduces that population; after the threshold it suppresses infection sufficiently for the uninfected population to recover. Beyond the invasion threshold the disease cannot persist.

Figure 1.
Steady frog population against disease mortality, with transmission rate beta equal to two
.

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