A new app, a piece of gossip and a flu strain look nothing alike, but they spread in strikingly similar ways. Three models cover most of what you will see, and the shape of the adoption curve tells you which one is at work.
Broadcast: one source, many listeners
In the broadcast model, people hear about something from a single central source, such as an advert, a news alert or a product launch email. Each period, a fixed share of the people who haven't heard yet will hear.
Because the pool of people who haven't heard shrinks, growth is fastest on day one and slows steadily after that. The curve is concave, shaped like the letter r. If your sign-ups spike at launch and then taper, broadcast is doing most of the work, and growth stops when the broadcast does.
Contagion: person to person
In a contagion model, people learn from each other. The SIR model tracks three groups: Susceptible, Infected (actively spreading) and Recovered (no longer spreading). The key number is R₀, the average number of new cases one case causes in a fully susceptible population: the transmission rate times the contact rate, divided by the recovery rate.
- If R₀ is above 1, each case causes more than one more, and the spread grows.
- If R₀ is below 1, the spread dies out.
- Spread stops growing once the share of people already immune reaches 1 − 1/R₀. With R₀ = 3, that threshold is two-thirds of the population.
Contagion produces an S-curve: slow at first, because few people are spreading it; then explosive; then slow again, as it runs out of new people to reach. Ideas behave the same way. A meme's R₀ is how many people each sharer convinces to share again.
The Bass model: both at once
Most real products spread through both channels. The Bass model combines them. The share of the remaining market that adopts each period is:
p + q × (share who have already adopted)
The parameter p captures innovation, meaning adoption driven by outside sources such as advertising. The parameter q captures imitation, meaning adoption driven by people who have already adopted. Studies across many consumer products often cite averages of around p ≈ 0.03 and q ≈ 0.38, so word of mouth is usually the bigger force by an order of magnitude.
When q is much larger than p, you get a strong S-curve with a clear peak. Adoption per period peaks at time ln(q/p) / (p + q). With the typical values above, that is about six periods in. The useful insight is that the ratio of q to p sets how long the slow start lasts.
Thresholds: why some things never take off
Contagion models assume everyone is equally persuadable. In reality, people have thresholds: I'll join once enough others have. Granovetter's threshold model shows how fragile this makes collective behaviour.
Picture 100 people whose thresholds are 0, 1, 2 and so on up to 99. The person with threshold 0 starts, which tips the person with threshold 1, then 2, and the whole crowd joins in. Now change a single person's threshold from 1 to 2. The first person starts, nobody else's threshold is met, and nothing happens. Two crowds with nearly identical people produce opposite outcomes. When adoption stalls, the cause is often a gap in the threshold distribution rather than a weak product.
Networks: the paths matter
Spread can only travel along connections. Percolation shows that there is a sharp critical point: below a certain transmission probability, outbreaks stay small and local; just above it, they can reach most of the network. Tightly knit clusters are great at reinforcing an idea and poor at exporting it. Bridges between clusters are where a local hit becomes a global one.
Reading your own curve
| Curve shape | Likely mechanism | What it means |
|---|---|---|
| Fast start, steady slowdown (r-shape) | Broadcast | Growth is paid for; it stops when spending stops |
| Slow start, sharp rise, plateau (S-shape) | Contagion or Bass with high q | Word of mouth is working |
| Early rise, then a stall well below the market size | Thresholds or a fragmented network | Something is blocking the next group from joining |
One warning: fitting p and q from early data is notoriously unstable. Before the inflection point, many different parameter values fit equally well. Treat early forecasts as ranges, not numbers.
Levers you can pull
- Raise p with advertising, press and partnerships: useful to start, expensive to sustain.
- Raise q by making use visible and sharing easy: referral rewards, shareable outputs, team invites.
- Lower thresholds with social proof, free trials and reversible commitments.
- Seed the bridges. Find the people who connect otherwise separate groups and reach them first.
Try the interactive labs on the Bass and SIR pages to see how small parameter changes reshape the whole curve.