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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-207/3/c/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 207 / 3 / c / ii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

The untreated rates are

λ_{U} = 0.2

and

μ_{U} = 0.25

; treatment changes them to

λ_{T} = 0.2 \times 0.8 = 0.16

and

μ_{T} = 0.25 \times 2 = 0.5

. Their stationary distributions give symptoms fractions

π_{2, U} = \frac{0.2}{0.45} = \frac{4}{9}, π_{2, T} = \frac{0.16}{0.66} = \frac{8}{33} .

Consequently

\frac{π _{2, T}}{π _{2, U}} = \frac{8/33}{4/9} = \frac{6}{11} .

Thus the estimated long-run symptoms fraction under treatment is about 54.5% of the untreated fraction. This combines less frequent onset with faster recovery; it is not either individual hazard ratio alone.

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