Branching field note · mean held still
The Average That Hides the Outbreak
Two infections can each average two new cases and still face very different futures. Hold R0 at 2, move only the negative-binomial dispersion k, and run thousands of branching outbreaks. The chance of eventual stochastic extinction (after one introduction in the branching model) changes sharply, the surviving tail grows wilder, and an idealised control policy aimed toward that tail can outperform population-wide control at the same nominal control parameter c. Every model result is recomputed live, while the fitted SARS estimate and the limits of the approximation stay visibly separate.
R0 is not wrong. It is the mean number of secondary infections in a susceptible population, and it still marks whether sustained growth is possible. It does not, by itself, tell you how that mean is distributed among people or introductions.
01 · Keep the mean, move the worldOne introduction. Thousands of possible futures.
The model starts with one infected person. Each case draws a number of secondary cases from a negative-binomial offspring distribution. Set the mean, then move k. Large k is close to homogeneous Poisson transmission. Small k puts more mass at zero and more weight in a long right tail.
Live branching process
The exact fixed point and 2,500 seeded outbreak runs are recomputed after every change.
Running 2,500 outbreaks.
At the default settings the mean remains exactly 2.0. Yet the branching tree often stops immediately, because the same distribution that permits a rare large draw makes zero a common draw. The few introductions that do escape can grow explosively. The average is maintained across all possible people and futures, not inside each realised outbreak.
02 · The fixed pointWhere extinction hides in one equation
Let Z be one case’s number of secondary infections. In Lloyd-Smith and colleagues’ model, latent infectiousness is gamma-distributed and transmission conditional on it is Poisson. Mixing the two gives a negative binomial. Its probability generating function is:
If each descendant starts an independent copy of the same process, the ultimate extinction probability is the smallest solution of q = G(q). The page iterates that equation from zero. This is not a fitted trend line and it is not read from the simulation.
What R0 still decides
If R0 ≤ 1, extinction is certain in this branching model. If R0 > 1, sustained growth has positive probability. The derivative G′(1) = R0 keeps that threshold intact.
What k adds
For the negative binomial, Var(Z) = R0 + R0²/k. Lower k raises both the zero class and the far tail. It changes the distribution of outcomes without moving the mean.
The paper reported k = 0.16 for one SARS contact-tracing dataset from Singapore, with a bias-corrected non-parametric bootstrap 90% interval of 0.11 to 0.64. That is an estimate from a particular outbreak and method. It is not a permanent property of SARS, and it is not evidence that another place, time, variant, or surveillance system has the same k.
“Fine, k changes the noise. The mean still decides whether growth is possible, so the rest is decoration.”
03 · The tail changes the leverSame nominal c, different control
A variance parameter becomes operational when control can be allocated unequally. The comparison below reconstructs policies in the paper’s supplement. They share the same nominal control parameter c, not the same reach or delivery. “Population-wide” multiplies every infected case’s latent infectiousness by (1 − c). “Targeted” fully controls a fraction c of cases on average, with a person in the top 20% of latent infectiousness four times as likely to be controlled as someone in the bottom 80%.
Control bench · same nominal control parameter c
The index case is untreated. Control begins with the second generation, as in the paper’s simulated comparison. These curves show eventual extinction, not its 100-case containment threshold.
calculating
every latent infectiousness × (1 − c)
eventual extinction · effective R = calculating
calculating
fraction c fully controlled, with fourfold top-20 selection
eventual extinction · effective R = calculating
At 20% effort, top-tail cases are controlled with 50.0% probability and bottom-group cases with 12.5% probability.
The advantage is not magic. The same nominal c represents different delivery in the two policies. At the default k, the model’s top fifth carries most expected transmission. Within the targeted policy, fully controlling equal numbers from that fifth and the lower four fifths lowers the effective reproductive number much more sharply than scaling everyone’s infectiousness.
It is also not a ready-made public-health instruction. The policy assumes latent lifetime infectiousness can be identified before it is expressed, then reduced to zero. Real control sees imperfect proxies, arrives late, changes behaviour, and operates through contact networks. The calculation establishes a conditional result: if useful predictors identify the high tail, R0 alone cannot price the value of that information.
The check · equations, constants, choices
The browser has just recomputed the fixed points, the gamma tail integral, both control curves, and a fresh seeded simulation. Three fixed anchors are recalculated below every time the page loads:
Checking q as k falls through 50, 10, 1, 0.5, 0.16, 0.1, 0.05.
Checking the current control probabilities and effective reproductive numbers.
| Class | What appears here | What it means |
|---|---|---|
| Definition | R0, negative-binomial k, q = G(q) | Mathematical objects inside the stated branching model. |
| Computed | Extinction, zero class, variance ratio, tail share, simulations, and control curves | Live consequences of the equations and the current controls. |
| Fitted | SARS Singapore k = 0.16, 90% bootstrap interval 0.11 to 0.64 | A reported 2005 estimate from 57 cases. The full-data fit is cited, not secretly recomputed from unavailable line-level data. |
| Chosen | 2,500 runs, 4,096-case viewing cap, 80-generation cap, top-20 split, fourfold targeting | Free apparatus and policy choices. They are not biological constants. |
| Open | Transfer across settings and useful prediction of the high tail | k can vary with context, time, variant, behaviour, data quality, and model choice. |
The separate no-dependency verifier repeats the calculations, runs 30,000 simulations at three anchor settings, and exits non-zero on drift: node research/the-average-that-hides-the-outbreak/verify-the-average-that-hides-the-outbreak.mjs.
The exact scope of “extinction”
The analytic q is ultimate extinction after a single introduction in an early-phase Galton-Watson branching process. It assumes a large susceptible population, independent cases drawn from the same offspring law, and no depletion, immunity, spatial structure, changing behaviour, or repeated importations. One introduction dying out does not mean a pathogen cannot return or that an established epidemic will vanish by the same mechanism.
The displayed simulation calls a run extinct only when its active generation reaches zero. It stops following a surviving run after 4,096 viewed cases or 80 generations. That cap controls browser work. The uncapped fixed point is the primary number, and the simulation is a corroboration.
Why “homogeneous” still has random offspring
In this model, k → ∞ removes variation in the latent individual reproductive number. Transmission events remain Poisson, so two otherwise identical people can still infect different numbers by chance. The page’s “Poisson-like” preset uses k = 50, which is close to, but not exactly, the infinite-k limit. The check panel computes that limit separately.
How the targeted curve is integrated
Latent infectiousness ν follows a gamma distribution with shape k and mean R0. The browser numerically finds its 80th percentile. With total effort c ≤ 0.4, the lower group is fully controlled with probability c / 1.6 and the upper group with probability 4c / 1.6. Those probabilities average back to c, with equal numbers controlled in the upper fifth and lower four fifths. Incomplete-gamma integrals then give the controlled probability generating function and its mean. No control point is hard-coded.
Primary record
J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp, and W. M. Getz, “Superspreading and the effect of individual variation on disease emergence,” Nature 438 (2005), 355–359. The supplementary information contains the generating functions, fixed-point method, targeting definition, simulations, and confidence-interval table used here.
The source paper itself warns that outbreak datasets can select for successful invasions and conspicuous superspreading events, that surveillance can miss isolated cases, and that larger datasets are needed. This page preserves those limits.