= Near-threshold SIR final-size expansion
{title2=$w_\infty\simeq2(\mathcal R_0-1)/\mathcal R_0^2$}
Normalize the closed <SIR model> by its initial susceptible count, writing $u(0)=1$, $v(0)=v_0>0$, $w(0)=0$ and $\tau=\gamma t$. Then $u+v+w=1+v_0$, and the <SIR susceptible-recovered identity> becomes $u=e^{-\mathcal R_0w}$. Hence
$$
\frac{dw}{d\tau}=v_0+1-w-e^{-\mathcal R_0w}
=v_0+(\mathcal R_0-1)w-\frac12\mathcal R_0^2w^2+O((\mathcal R_0w)^3).
$$
The positive root of the quadratic approximation is
$$
w_*=\frac{\mathcal R_0-1+\sqrt{(\mathcal R_0-1)^2+2\mathcal R_0^2v_0}}{\mathcal R_0^2}.
$$
The displayed small-seed formula requires $\mathcal R_0>1$, $\mathcal R_0w_*\ll1$ and $\mathcal R_0^2v_0\ll(\mathcal R_0-1)^2$. In particular, a regime is $0<\mathcal R_0-1\ll1$ with a much smaller positive seed. Setting the seed exactly to zero produces the disease-free trajectory instead of an outbreak.
\Image[/past-exam-of-the-short-options-course-of-the-university-of-oxford/2017-epidemic-models.png]
{title=Recruitment and closed-population depletion give different epidemic trajectories}
{description=Original diagrams and generic model calculations generated by the adjacent Python script.}
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