Adjacency-matrix bound for a discrete SIR epidemic ($\mathrm{Pr}(i\text{ ever infected})\le {\sum }_{k\ge 0}[(\beta A{)}^{k}X(0){]}_i$)

The conditional union bound gives $\mathbb{E}{X}_i(k+1)\le \beta {\sum }_j{A}_{ij}\mathbb{E}{X}_j(k)$ in a discrete SIR epidemic on a graph. Iteration bounds the infection vector at step $k$ by $(\beta A{)}^{k}X(0)$. Each vertex is infected at most once, so summing the probabilities in time gives the displayed bound. Powers of the adjacency matrix of a graph count walks, which can overcount impossible reinfections without invalidating the upper bound. Initially removed individuals must be counted separately when bounding all eventual removals.

When $\beta \rho (A)<1$, summing the adjacency-matrix bound for a discrete SIR epidemic gives the displayed Neumann series bound on the expected total infected count $Z$. For an undirected graph, the adjacency matrix of a graph is a symmetric matrix, so its Euclidean operator norm equals its spectral radius. The Cauchy-Schwarz inequality then yields $\mathbb{E}Z\le \sqrt{nm}/(1-\beta \rho )$ for $m$ initially infected vertices.

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.