When an outbreak ends on the tipping point. An Evidence Press briefing. Epidemics, rumours, opinions and new habits all spread from person to person. Compartment models describe this by splitting a population into a few groups, such as unaware, heard it once, spreading and lost interest, and tracking how people move between them. In the classic epidemic model the last spreaders fade away exponentially: their number falls by a fixed fraction every day. In 2025, Chan, Crommelin and Mandjes conjectured that this always happens in a broad class of such models, provided two diagrams describing who can move where, and who depends on whom, contain no loops. That conjecture had already been refuted, by an earlier Evidence Press release published the day before this one, with a four-group model that fades like one over time squared. This release is its successor. It explains why the conjecture fails. Fading is exponential when the outbreak overshoots, so that each spreader ends up replacing less than one of itself. If instead the endgame lands exactly on the tipping point, where each spreader replaces exactly one, the fade-out becomes a slow power law. The release gives a four-group staged rumour model in which both diagrams have no loops, writes down its whole trajectory exactly, and shows that its spreaders fade like time to the power minus three halves. A version of a well-known two-step contagion model from the physics literature does the same. With four groups nothing fades more slowly than that. The slow tails are fragile. They need finely tuned starting points, a small nudge restores exponential decay at a predictable rate, and in the families studied they never follow an outbreak peak. Simple conditions on the model rule them out altogether. For epidemiology, slow fade-out is the signature of an endgame poised at the tipping point. But a simulation suggests real, finite populations die out much earlier than the smooth model predicts. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or independently reproduced. Published on 21 October 2026, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.