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Because $A$ is a real symmetric matrix, the spectral theorem gives $\Vert A{\Vert }_2=\rho (A)$. For $\beta \rho <1$, the Neumann series satisfiesLet $m={\sum }_i{X}_i(0)$ be the initial infected count. Since its entries are indicator random variables, $\Vert X(0){\Vert }_2=\sqrt{m}$, while $\Vert 1{\Vert }_2=\sqrt{n}$. The Cauchy-Schwarz inequality and the operator norm bound yieldThus a sufficient asymptotic condition for a small outbreak isFor example, it holds when $m=o(n)$ and $\beta \rho \le 1-\eta$ for a fixed $\eta >0$. For any fixed $\delta >0$, the Markov inequality then givesThis is the spectral condition for a small discrete SIR outbreak. A uniform gap and few initial infections are needed for this conclusion; the finite-population condition $\beta \rho <1$ alone does not control an asymptotically vanishing gap or an initially macroscopic infected set.