Researchers studied when a common shortcut in epidemic modeling is valid. Epidemiologists often replace a contagious process on a network with a static random-graph model called Bernoulli bond percolation. In that percolation model, each contact (edge) is kept independently with the same probability, which makes the final outbreak sizes easier to compute. The paper asks: under what conditions does that replacement actually give the same answers as a general non-recurrent epidemic such as the susceptible-infected-recovered (SIR) model?

To answer this, the authors start from a very general description of non-recurrent infections. They describe transmission in terms of two random time intervals: the time from when a person becomes infected to the next transmissible contact with a neighbor, and the time from infection to recovery (the end of infectiousness). A contact transmits only if it happens before recovery. From these ingredients they define a transmissible network: the original contact network but keeping only those edges where transmission would occur. They also require time homogeneity, meaning these timing rules do not depend on the outbreak history.

The paper lays out three distinct kinds of equivalence between an epidemic model and Bernoulli bond percolation. Type-1 is exact isomorphism on a single network: the two models produce the same set of infected nodes in every possible outcome. Type-2 is a weaker match: every node has the same marginal probability of being infected under both models. Type-3 is an even weaker match about averages: the expected size of a large major outbreak is the same in large random networks built by the configuration model. To study type-2, the authors compare message-passing equations for epidemic percolation and for Bernoulli bond percolation. They also relate their work to prior results that showed exact mapping only when infection durations are degenerate (that is, not random).