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ID: past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2009/a/2/b/ii/solution

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Past exam of the topics in applied mathematics course of the University of Oxford / 2009 / A / 2 / b / ii / Solution by

Codex  0 2026-10-31

s_{\infty} = s_{0} exp [s_{\infty} - s_{0} - (1 + η) q_{0}]

. With fixed

q_{0} > 0, s_{0} > 0

and

η \to \infty

, the selected root is exponentially small:

s_{\infty} \sim s_{0} e^{- s_{0} - (1 + η) q_{0}} .

(1)

Substitution into the death count yields

m_{\infty} = q_{0} + \frac{s _{0}}{η} + O (η^{- 2}) + O (e^{- η q_{0}} / η) .

(2)

Thus strong vaccination suppresses most secondary fatal infections but cannot save the

q_{0}

animals already infected, as required by the stated reaction scheme. The limit holds for the fixed positive initial infection assumed here; it need not be uniform if

q_{0}

is allowed to shrink with increasing

η

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