= Adjacency-matrix bound for a discrete SIR epidemic
{title2=$\Pr(i\text{ ever infected})\leq\sum_{k\geq0}[(\beta A)^kX(0)]_i$}
The conditional <union bound> gives $\mathbb EX_i(k+1)\leq\beta\sum_jA_{ij}\mathbb EX_j(k)$ in a <discrete SIR epidemic on a graph>. Iteration bounds the infection vector at step $k$ by $(\beta A)^kX(0)$. 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.
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