In 1964, David Daley and David Kendall proposed a model for how rumors spread through a population, borrowing the machinery of infectious-disease epidemics. Every person is in one of three states: ignorant (hasn't heard it), spreader (knows it and is actively telling others), or stifler (knows it but has stopped telling — either bored of it or told someone who already knew).
α.β — telling someone who already knows the news kills the urge to keep telling it.k = β/α largely determines the "final size" — the fraction of the population that ever hears the rumor before it dies out. Higher α (people love repeating it) spreads it further; higher β (people quickly get bored) chokes it off sooner.The same ignorant–spreader–stifler mathematics has been reused to model the spread of memes, chain letters, computer worms, and even how quickly a piece of misinformation exhausts itself on social media — long before "going viral" was a phrase anyone used.
A 3D crowd of agents shows a rumor spreading through a population: ignorants who haven't heard it, spreaders actively telling others, and stiflers who know it but have stopped repeating it.
Spreaders convert ignorants into new spreaders at rate α, but turn into stiflers at rate β whenever they contact someone who already knows — so a rumor can burn out from the inside even while ignorants remain.
Adjust population size, contact rate α and stifling rate β, then watch the crowd's colors shift from blue (ignorant) through orange (spreader) to grey (stifler) as the rumor rises and fades.
Daley and Kendall's 1964 model shows a rumor almost never reaches everyone — typically a fixed "final size" fraction stays forever ignorant, no matter how long you wait.