Below one it dies out. Above one it runs away. There is nothing in between.
At R₀ = 1.5 the SIR tool infects 58.5% of the population. Push it to 0.9 and the outbreak is gone. There is no setting that infects a steady trickle.
Take the SIR model and sweep its reproduction number.
At R₀ = 1.5 — the flu preset — it reports a final size of 5,845 people, 58.5% of the population. At R₀ = 3 it reports 9,412, or 94.1%. At R₀ = 15 it reports 10,000 out of 10,000: everyone.
Now go the other way, below 1, and the tool has a preset called subcritical for the purpose. The outbreak does not shrink smoothly. It fails to happen. A few people are infected, each infects fewer than one other on average, and the chain stops.
There is no setting that produces a modest, sustained trickle. Every value either dies out or takes off, and the boundary is exactly 1.
Why exactly one, and why so sharp
Strip the biology away and what is left is a counting question. Each unit — an infected person, a neutron, a copy of a DNA fragment, a man with a surname — produces some number of successors before it is removed. Call the average R.
After n generations the expected population is Rn. That is the whole argument. If R < 1, Rn → 0, and not slowly: at R = 0.9 a chain is down to 3% of its starting size in 33 generations. If R > 1 it grows without bound until something else stops it. And at exactly R = 1 the expected size is constant forever, a knife edge that no real system balances on, because R is never known to that precision and never stays still.
The sharpness comes from the exponent. Every other quantity in these models — how fast, how many, how long — shifts the curve. R decides which of two utterly different things happens. It is the rare case where a continuous parameter produces a genuinely binary outcome, and it is why the exponential tool's sign convention matters more than any of its magnitudes.
The same threshold, four costumes
Epidemics. R₀ is the average number infected by one case in a fully susceptible population. Everything public health does — vaccination, distancing, isolation — is an attempt to push R below 1, and it does not matter which lever achieves it.
Chain reactions. The neutron multiplication factor k counts how many of the neutrons released by one fission go on to cause another. k < 1 is subcritical and the reaction dies. k > 1 is supercritical. A reactor is a machine for holding k at 1.000 by continuously adjusting it, which is the engineering version of admitting nobody can sit on a knife edge without steering.
PCR. Each cycle copies each fragment, so R here is the efficiency: how close to 2 the doubling actually gets. The PCR tool is unusual in this family because R > 1 by construction and the interesting question is how far above; but let efficiency fall below 1 through a bad reagent and the sample vanishes into noise instead of amplifying.
Surnames. Which is where the mathematics started. In 1875 Galton asked why so many aristocratic family names had died out, and Watson worked out the answer: model each man as producing a random number of sons, and extinction is certain whenever the mean is at most 1. Less obviously, it has a substantial probability even when the mean exceeds 1, because chance alone can end a line early.
Their paper got the final step wrong: they concluded that extinction was certain in all cases, missing the surviving branch. It took decades before the correction was widely known. The threshold itself, though, was right, and it is the same threshold the epidemiologists rediscovered fifty years later.
The part the average hides
One caution, because R is a mean and means conceal their spread.
Two diseases with identical R₀ = 2 can behave very differently if one infects two people reliably and the other infects nobody 90% of the time and twenty people occasionally. The averages match; the outbreaks do not. The second is driven by superspreading, dies out far more often by luck, and when it does not, explodes faster.
Watson and Galton's framework already contains this: extinction probability depends on the whole distribution of offspring, not just its mean. It is why "R is below 1" is a statement about the average behaviour of a system that may be doing something much wilder in each individual case, and why the same R can feel like two different diseases.
References (1)
- the threshold, derived first for surnames and wrong in its conclusion Watson & Galton (1875). On the Probability of the Extinction of Families. The Journal of the Anthropological Institute 4.