= Solution
Divide the susceptible equation by the removed equation, or differentiate the resulting invariant directly, to get $du/dw=-\mathcal R_0u$. The initial data give the <SIR susceptible-recovered identity>
$$
u=e^{-\mathcal R_0w}.
$$
Population conservation then gives $v=1+v_0-w-e^{-\mathcal R_0w}$. Since $w'=v$, the <Taylor expansion> for small $\mathcal R_0w$ yields
$$
\boxed{\frac{dw}{d\tau}\approx v_0+(\mathcal R_0-1)w-\frac12\mathcal R_0^2w^2.}
$$
At the end of the outbreak, the infectious count approaches zero. The positive root of the quadratic approximation is
$$
w_* =\frac{\mathcal R_0-1+\sqrt{(\mathcal R_0-1)^2+2\mathcal R_0^2v_0}}{\mathcal R_0^2}.
$$
For $\mathcal R_0>1$ and a seed satisfying $\mathcal R_0^2v_0\ll(\mathcal R_0-1)^2$, this reduces to
$$
\boxed{w_1\approx\frac{2(\mathcal R_0-1)}{\mathcal R_0^2}.}
$$
The truncation also requires $\mathcal R_0w_1\ll1$, hence a near-threshold regime $0<\mathcal R_0-1\ll1$. The seed must be small but positive: setting it identically to zero gives no epidemic. This is the <near-threshold SIR final-size expansion>, and the physical removed count is $S_0w_1$.
\Image[/past-exam-of-the-short-options-course-of-the-university-of-oxford/2017-epidemic-models.png]
{title=Replenishment permits repeated waves; the closed SIR model has a final-size relation}
{description=Original generic model calculations. The left panel shows a positive trajectory of the discrete recruitment model. The middle panel shows a closed SIR outbreak. The right compares the exact closed-model final-size root with its small-seed quadratic approximation near threshold.}
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