INTERACTIVE COMPANION · WINTER 2025 · ORIGINAL RESEARCH

The Transmission Bench

A live model accompanying “When Most People Infect Nobody”

Two benches below. The first grows one outbreak chain at a time from a single case, so you can watch what "most people infect nobody" actually looks like as a shape rather than a percentage. The second lets you split a fixed control budget between two levers, the same two the article compares in §8, and reports the resulting extinction probability instantly from the closed-form root the article validates, not from a slow simulation.

Model 1. Grow an outbreak

Every generation, every currently infectious case independently draws a number of new cases from a negative binomial distribution with mean \(R\) and dispersion \(k\), exactly the model in the article. The chain is drawn live, one generation at a time. It stops the moment a generation produces zero new cases (EXTINCT), or once the chain has produced more than 60 cases in total, at which point it is frozen and counted as a LARGE OUTBREAK rather than drawn out to its true, much larger size.

Mean reproduction number R 2.5 2.5 is the article's own value throughout. Push it under 1 and watch every chain die out eventually, whatever the dispersion — that is the subcritical case, where extinction is certain. Dispersion k 0.1 Same seventeen values as the article's sweep. Drag it from 0.1 down to 0.01 and watch chains die out almost every time, in one or two generations; up to 10,000 and most chains grow steadily until they hit the display cutoff. Pace normal How long each generation stays on screen before the next one is drawn. Has no effect on the numbers, only on how easy the growth is to watch.

generation: 0 cases so far: 1 this chain: growing

chains run: 0 extinct: – large outbreak: – theoretical extinction probability: 86.07%

Model 2. Split a fixed control budget

Fix the baseline at the article's own realistic values, \(R_0 = 2.5\) and \(k_0 = 0.1\). Choose a total effort \(E\), the fraction of the uncontrolled mean transmission you intend to remove, then choose how much of that effort goes to thinning every case's transmission uniformly (Strategy A) versus capping only the largest events (Strategy B). Both ends of the split spend the same total \(E\); only the mechanism changes. The chart and the extinction probability below update instantly, computed from the same closed-form root as Model 1, not simulated.

Total effort E 0.45 0.45 is where the article's two strategies are furthest apart. Push E past 0.6 and uniform thinning alone drives R to 1 or below, guaranteeing eventual extinction regardless of dispersion. Split, all-A ←→ all-B 0% to B Left end spends the whole budget thinning everyone uniformly (Strategy A). Right end spends it all capping the biggest gatherings (Strategy B). Drag slowly across the middle and watch which end the extinction probability favours — it is not the end most people expect.

R after control: 1.375 gathering cap used: – extinction probability: 94.59%

pure A at this effort: 94.59% pure B at this effort: 89.00% A − B: 5.60 pts

Model 1 default (R=2.5, k=0.1): theoretical extinction 86.0685%
Model 2, split all to A, E=0.45: 94.5928%, matches the article's Table in §8 exactly
Model 2, split all to B, E=0.45: ≈89.00% with this page's integer-rounded cap, against 89.64% in the article's 20,000-replicate simulation with an exact fractional cap

What would break this

Model 1 draws real negative-binomial variables, generated in your browser as a Poisson count with a Gamma-distributed mean, the identical construction numpy uses internally and the identical mathematics behind equation \(\mathrm{Var} = R + R^2/k\) in the article. It is not, however, running anything like 40,000 replicates: it is one chain at a time, so the running "extinct" percentage above will be noisy for the first few dozen chains and only settle in after a few hundred, precisely the point Figure 2 in the article makes about Monte Carlo convergence. Model 2's "gathering cap" idealises a ban as a clean mathematical ceiling, exactly as the article's §9 admits it does; drag the split slider and you are exploring that idealisation, not a policy costing.