The Daley-Kendall Model: The Mathematics of Gossip

Three states, not two

In 1965, mathematicians Daryl Daley and David Kendall proposed a model for how a rumor moves through a population, built by analogy to epidemic models but with a crucial twist. Every person is in one of three states: Ignorant (has not heard the rumor), Spreader (knows it and is actively telling others), or Stifler (knows it, but has stopped spreading it — it is old news to them). This is different from a two-state disease model, and the difference is exactly what makes rumors behave the way they do.

live demo · a rumor spreading through a contact network● LIVE

The transition rules

Every transition in the model happens through a pairwise contact between two people, at some rate governed by the population's contact rate:

Spreader meets Ignorant  → Ignorant becomes a new Spreader
Spreader meets Spreader  → BOTH may become Stiflers (with prob. β)
Spreader meets Stifler   → the Spreader may become a Stifler (with prob. β)

β = "stifling probability" — how readily a spreader loses interest
    on discovering the rumor is already known

Notice what triggers stifling: not the passage of time, but the specific experience of telling the rumor to someone who already knows it. That single social mechanic — losing enthusiasm because the story feels stale — is the entire engine behind the model's most surprising result.

Not the same as an epidemic

In a classic SIR disease model, an infected person recovers on their own after a characteristic time, regardless of who they meet. Rumor spreading works the other way around: a spreader keeps spreading indefinitely until a specific kind of encounter — bumping into someone who already knows — turns them into a stifler. Because that encounter becomes more likely as the rumor becomes more widely known, the process contains its own natural brake: the more successful a rumor becomes, the faster its spreaders are converted into stiflers.

A rumor never reaches everyone

That self-braking mechanism produces a striking, testable prediction: in the classic fully-mixed version of the model, a rumor does not eventually reach the entire population, even given infinite time. As the number of people grows large, the fraction who remain permanently ignorant converges to a fixed constant near 20%, determined by solving the model's differential equations for the final size of the ignorant population. Unlike an epidemic with R0 above 1, which can in principle sweep through nearly everyone, a rumor typically fizzles out with a meaningful chunk of the population never having heard it — because spreaders run out of enthusiasm faster than they run out of people to tell.

Two variants, one family

A related model published by Anatol Maki and James Thompson makes one small change: when a spreader meets another spreader or a stifler, only the initiating spreader risks becoming a stifler, not both parties. The qualitative story — a self-limiting spread that always leaves some fraction of the population untouched — holds for both the Daley-Kendall and Maki-Thompson formulations; only the precise final proportions differ.

Frequently asked questions

How is rumor spread different from a disease epidemic model?

In a standard SIR epidemic model, an infected person recovers spontaneously after some time, independent of who they meet. In the Daley-Kendall rumor model, a spreader only stops spreading because of a specific kind of contact — meeting someone who already knows the rumor, which makes it feel like old news. Stifling is socially triggered, not a timer.

Does a rumor always reach the whole population eventually?

No. Because spreaders lose interest whenever they run into someone who already knows, the rumor typically stops spreading while a substantial fraction of the population is still ignorant of it — in the classic fully-mixed model, roughly 20% of people never hear it at all, regardless of how long you let the process run.

What is the difference between the Daley-Kendall and Maki-Thompson variants?

In the original Daley-Kendall model, both parties in a spreader-spreader or spreader-stifler encounter can become stiflers. In the Maki-Thompson variant, only the spreader who initiated the contact becomes a stifler; the other party's status is unaffected. The two produce slightly different final proportions of people who never hear the rumor, but the qualitative dynamics are the same.

Try it live

Everything above runs in your browser — open Rumor Spread and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Rumor Spread simulation