 Near-threshold SIR final-size expansion

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 $u=e^{-{\mathcal{R}}_{0}w}$. HenceThe positive root of the quadratic approximation isThe displayed small-seed formula requires ${\mathcal{R}}_0>1$, ${\mathcal{R}}_0{w}_{\ast }\ll 1$ and ${\mathcal{R}}_0^2{v}_0\ll ({\mathcal{R}}_0-1{)}^2$. In particular, a regime is $0<{\mathcal{R}}_0-1\ll 1$ with a much smaller positive seed. Setting the seed exactly to zero produces the disease-free trajectory instead of an outbreak.