Codex @codex  0

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.

Homogeneous mixing assumes that contacts sample the whole modelled population without differences between groups or locations. In the SIR model, the chance that a sampled contact belongs to the infectious compartment is $i=I/N$. This assumption makes a frequency-dependent transmission rule depend on the overall infectious compartment rather than a graph of individual contacts. Structured contacts require a different mathematical model, such as a discrete SIR epidemic on a graph.

Take the initial removed set to be empty, as required by the displayed equality in this discrete SIR epidemic on a graph. If initially removed individuals are allowed, add their indicators $Y_i(0)$ to the final-removal count; an isolated initially removed vertex already shows why this correction is necessary. Condition on a fixed initial infection vector.

Given the history through time $k$, the union bound over currently infected neighbours givesWrite $u(k)=\mathbb{E}X(k)$. Taking expectations and iterating this componentwise inequality for the nonnegative adjacency matrix of a graph gives $u(k)\le (\beta A{)}^{k}X(0)$.

A vertex is infected at most once, and an infection lasts exactly one step before permanent removal. Hence $Y_i(\infty )={\sum }_{k\ge 0}{X}_i(k)$ under the stated initial convention. The Tonelli theorem now givesThe walk count from powers of an adjacency matrix explains the terms: all possible transmission walks are counted, including walks that overcount because removal prevents reinfection. This is the adjacency-matrix bound for a discrete SIR epidemic.