Spectral condition for a small discrete SIR outbreak ($\frac{\sqrt{m/n}}{1-\beta \rho }\to 0$)

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