Two numbers and a shape
Every uptake forecast for a new vaccine reduces to three things: the peak share of the eligible population that will eventually be vaccinated, the time it takes to get there, and the shape of the curve between launch and peak. Most arguments about a forecast are really arguments about one of those three, conducted as if they were about the whole. Separating them is the first step to a model a sponsor can question and trust.
Peak uptake is mostly set by the strength of the recommendation and by how willing the target population is. Time to peak is mostly set by access: coverage, provider stocking and where the vaccine can be given. The shape is set by how adoption spreads, whether it starts fast and levels off or starts slowly and accelerates. Different methods handle each of these differently, and the choice of method should follow from which of the three is most uncertain.
Analog curves from prior launches
The most common method is to borrow the curve from vaccines that launched before. Take the observed uptake of one or more analog vaccines, align them on time since recommendation rather than calendar date, scale each to its own peak, and apply the resulting shape to the new vaccine's population and assumed peak.
Analog curves have the great advantage of being real. They contain the effects of stocking delays, coverage lags, seasonal timing and hesitancy as they actually happened, without anyone having to model those effects separately. Their weakness is that they carry the conditions of their own launch. A vaccine that launched during a supply shortage, or whose recommendation was changed partway through, or that was introduced before pharmacies could give it, will have a shape that reflects those conditions, and those may not apply now.
Sources for analog curves include immunization information systems maintained by states, national surveys such as the National Immunization Survey, and claims data. Each has blind spots. Registries vary in completeness for adults. Surveys rely on reporting by respondents or providers. Claims miss doses that are not billed, such as doses given through public programs. Use more than one source where possible and note which one each curve came from.
The Bass model in plain terms
The Bass diffusion model, published by Frank Bass in 1969 for consumer durables, describes adoption as coming from two sources. Some people adopt on their own, independent of how many others have adopted. Others adopt because people around them already have. The model has three parameters.
Innovation, imitation and market potential
m is the eventual number of adopters, which for a vaccine is the eligible population times the peak uptake. p, the coefficient of innovation, is the rate at which people who have not adopted do so independently in each period. q, the coefficient of imitation, scales how much adoption increases as the share who have already adopted rises. In each period, the chance that a person who has not adopted does so equals p plus q times the share of eventual adopters who already have.
A high p with a low q gives a curve that starts fast and slows steadily. A low p with a high q gives the classic S-shape: slow start, rapid middle, leveling off. Adoption peaks at a time equal to ln(q/p) divided by (p + q). With illustrative quarterly values of p = 0.02 and q = 0.30, the peak arrives at about 8.5 quarters, a little over two years, by which time just under half of eventual adopters have been reached.
For vaccines, the imitation term rarely means people copying their neighbors. It captures everything that builds on prior adoption: clinicians becoming familiar with the product, practices adding it to standing orders, electronic record prompts being configured, pharmacies stocking it once they see demand, and social proof among patients. The innovation term captures the people who seek the vaccine out or are reached directly by the recommendation.
Published values from consumer products are not a substitute for vaccine-specific evidence. A widely cited meta-analysis of consumer durables found average values near p = 0.03 and q = 0.38 per year, but vaccines are recommended, covered and administered in ways no consumer product is. The value of the Bass model for vaccines is its structure: it separates the speed of independent uptake from the speed of spread, and it can be fitted to analog curves to extract p and q that apply to the vaccine context.
Logistic and other S-curves
The logistic curve is the simplest S-shape. It has a ceiling (the peak uptake), a growth rate and a midpoint at which uptake reaches half its ceiling. It is symmetric around the midpoint, so the time to go from 10 to 50 percent of the ceiling equals the time to go from 50 to 90 percent.
That symmetry is its main limitation. Vaccine uptake often rises quickly in the first season among the most motivated and the most reachable, then approaches its ceiling slowly as the remaining population is harder to reach. The Gompertz curve is a common alternative with an asymmetric shape: its fastest growth happens at about 37 percent of the ceiling, so it rises earlier and approaches the ceiling more slowly. The Bass model can produce either shape depending on the balance of p and q.
Simple S-curves are useful when there is little data and the sponsor needs a transparent, three-parameter model they can discuss. They are less useful when the shape itself is the question.
The methods compared
| Method | How it works | Strengths | Weaknesses |
|---|---|---|---|
| Analog curve | Borrow the observed shape from prior launches, scaled to a new peak | Real data; captures effects nobody had to model; easy to defend | Carries the conditions of the analog's launch; limited by how good the analog is |
| Bass diffusion | Adoption from independent uptake plus uptake that builds on prior adoption | Separates speed of initial uptake from speed of spread; fits analog data well | Parameters are hard to set without analogs; assumes a fixed market potential |
| Logistic | Symmetric S-curve with ceiling, rate and midpoint | Simple, transparent, three parameters | Symmetry rarely matches vaccine uptake, which often slows toward the ceiling |
| Gompertz | Asymmetric S-curve with fastest growth early | Matches fast early uptake and slow approach to the ceiling | Less familiar to sponsors; same data needs as logistic |
| Driver-based | Builds uptake from channel-level access, coverage and acceptance assumptions | Every lever is visible; links directly to scenarios | Many assumptions; can look precise while resting on guesses |
In practice the strongest models combine methods: fit Bass or Gompertz parameters to two or three analog curves, then use driver-based adjustments to account for differences between the analogs' launch conditions and the new vaccine's.
What shapes the curve
Six drivers account for most of the variation between vaccine launches. Each pushes on peak uptake, on time to peak, or on both.
Strength of the recommendation. A routine recommendation for an age group sets a higher ceiling than a risk-based recommendation or one based on shared clinical decision-making, because it makes vaccination the default. Insurance coverage at launch. Coverage that lags the recommendation slows uptake in the first months, because providers hesitate to give a vaccine they may not be paid for. Provider stocking. Practices must buy vaccine in advance, store it correctly and accept the financial risk of unused doses. A new, expensive vaccine with uncertain demand is often stocked late. Pharmacy access. Where pharmacists can give the vaccine, uptake among adults is typically faster, but the age groups and vaccines pharmacists may administer differ by state. Awareness. Patients and clinicians have to know the vaccine exists and that it applies to them. Hesitancy. Some share of the eligible population will decline, and that share differs by vaccine, by population and over time.
How Much Each Driver Usually Moves the Curve
Relative influence on peak and timing combined, in our judgment. Not measured data. The order can change for a vaccine where hesitancy is unusually high.
Peak uptake and time to peak are separate assumptions
A common error is to treat a slow early curve as evidence of a low ceiling, or a fast early curve as evidence of a high one. The two are driven by different things. A vaccine with a strong recommendation but delayed coverage can start slowly and still reach a high peak. A vaccine sought eagerly by a small motivated group can start fast and stall early.
Model them separately and let each have its own range. Peak uptake should be tied to the recommendation scenario and to hesitancy evidence for the target population. Time to peak should be tied to access: when coverage starts, how long stocking takes, when pharmacies can give it. This separation makes scenarios coherent and makes calibration possible, because early data tells you much more about time to peak than about the eventual ceiling.
Two special cases need their own handling. Seasonal vaccines reset each year: model the within-season curve and the season-to-season trend separately. Multi-dose series need initiation and completion modeled separately, because completion is always lower and the gap is an assumption in its own right.
Building a scenario set
A scenario set is not a grid of every combination of driver values. Most combinations are incoherent: a narrow recommendation paired with full coverage on day one and peak analog uptake describes a world nobody expects. Build three to five named scenarios, each a consistent story, and state each in words before assigning numbers.
Scenario Set Build
Name the scenarios after their story, not their size. "Routine recommendation, coverage lag of one season" tells the reader what has to be true. "High case" does not, and invites the reader to assume the high case is the optimistic one when it may be the most likely.
Calibration after launch
Once a vaccine launches, the forecast should be calibrated against real uptake data. Two cautions apply. Early data constrain time to peak far better than peak uptake. Fitting all three Bass parameters to the first few months of data produces unstable estimates, often with an implausibly low ceiling. Hold the ceiling at its scenario value, informed by analogs and recommendation, and let early data update the timing parameters first.
Second, uptake data arrive with lags and gaps. Registries receive records late, claims take months to complete, and doses given through public programs may not appear in claims at all. Compare like with like, and do not revise a forecast sharply on the basis of a month of incomplete data. A Bayesian approach that starts from analog-based priors and updates as data accumulate handles this well, and it makes the size of each revision explainable.
Watch for uptake that runs ahead of the forecast as closely as uptake that falls behind. A fast start often reflects pent-up demand from the most motivated group, which is used up within a season. Reading that early surge as a higher trajectory, and raising the ceiling to match, is the mirror image of the more familiar error of cutting the forecast after a slow first quarter. In both directions, ask which driver changed before changing the curve: if coverage, stocking and the recommendation are as assumed, the likeliest explanation is timing, not a new ceiling.
Communicating uncertainty to a sponsor
A sponsor reading an uptake forecast wants to know three things: what is most likely, how wrong it could be, and what would make it wrong. A single curve answers only the first.
Show the named scenarios as separate curves on one chart, with the base scenario marked. Add a band if the scenarios have been run probabilistically. Then state, in one sentence per decision, which scenarios clear the sponsor's threshold and which do not. Most important, say which assumption distinguishes them. If every scenario that misses the threshold does so because coverage lags by a season, the sponsor knows where to focus effort before launch.
Finally, commit in advance to how the forecast will be updated: which data, at what intervals, and what size of deviation triggers a revision. A sponsor who knows the update rules will trust the forecast more, and will be less surprised when it changes.
Frequently asked questions
A model of how a new product is adopted over time, with adoption coming from two sources: people who adopt independently and people whose adoption builds on others having adopted. It has three parameters: the eventual number of adopters, a coefficient of innovation and a coefficient of imitation. For vaccines, the imitation term captures clinician familiarity, standing orders, pharmacy stocking and social proof.
Use both. Analog curves provide real-world shapes, and fitting a Bass or Gompertz model to them extracts parameters that can be adjusted for differences in launch conditions. Analogs alone carry the conditions of their own launch. Fitted models alone have no grounding.
In our experience, the strength of the recommendation has the largest effect, followed by whether insurance coverage is in place at launch and whether providers stock the vaccine. Pharmacy access, hesitancy and awareness also matter, and hesitancy can dominate for some vaccines and populations.
Early data start informing the timing of uptake within the first season, but the ceiling is poorly constrained until uptake has clearly started to slow. Hold peak uptake at its scenario value, update timing first, and allow for data lags in registries and claims before revising.