= Spectral condition for a small discrete SIR outbreak
{title2=$\frac{\sqrt{m/n}}{1-\beta\rho}\to0$}
The displayed condition and the <resolvent bound for a discrete SIR epidemic> imply $\mathbb EZ=o(n)$. The <Markov inequality> then makes $Z/n\to0$ in probability. A sufficient special case is a sublinear initial infected count $m=o(n)$ with a fixed positive gap $1-\beta\rho\geq\eta>0$. The inequality $\beta\rho<1$ at each finite population size alone does not control a shrinking gap or an initially macroscopic outbreak.
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