R0 — the one number that decides whether a spark becomes a wildfire. It is the expected count of secondary infections from a single case dropped into a fully susceptible population; for the SIR model it is exactly β/γ. Above 1 a fire, below 1 a fizzle. Down the center, data flows: rates go in, the engine integrates the epidemic, the verdict comes out. The blue team builds and defends it; the red team tries to break it. A model with named assumptions — not medical advice.
source Kermack & McKendrick, "A contribution to the mathematical theory of epidemics," Proc. R. Soc. Lond. A 115, 700–721 (1927), doi:10.1098/rspa.1927.0118; the R0 operator definition, Diekmann, Heesterbeek & Metz, J. Math. Biol. 28, 365–382 (1990), doi:10.1007/BF00178324. Paywalled — AMBER. Rendered, not quoted.
A well-mixed population of size N splits into Susceptible, Infectious, Recovered. Two rates drive it:
β the transmission rate, γ the recovery rate. dS/dt = −βSI/N, dI/dt = βSI/N − γI, dR/dt = γI.
The number falls straight out: R0 = β/γ = infections per case × expected time infectious. The effective number Rt = R0·S/N declines as susceptibles deplete.
For the current β, γ — live quantities:
| quantity | value |
|---|
The number that decides an outbreak is R0: above 1 a fire, below 1 a fizzle. And 1 − 1/R0 is the herd-immunity line — the immune fraction at which Rt falls to 1 and growth stops.
That same threshold is the pivot of the-sir-model: R0 is the eigenvalue of its disease-free equilibrium, and every curve there is decided by which side of 1 it lands on. Each sphere is the next one's premise.
The blue team's live check: re-integrate the epidemic and confirm that R0>1 grows and R0<1 dies — that the label matches the dynamics. If red tampers with the formula, this badge is where it shows.
Two dials feed the engine: β, how fast an infectious contact transmits, and γ, how fast a case recovers (γ = 1/infectious-days). Their ratio is the whole story.
| symbol | meaning | units |
|---|---|---|
| β | transmission rate | /day |
| γ | recovery rate | /day |
| R0 | β / γ | dimensionless |
| 1−1/R0 | herd immunity | fraction |
Assumptions (AMBER): one well-mixed population, constant β and γ, no births/deaths/reinfection. Real epidemics violate all of these — this is a model, not medical advice.
Rt = R0·S/N crosses 1 exactly at the epidemic peak — the moment growth turns to decline. The curve below is integrated live (RK4, dt=0.05).
Move either dial — R0, the peak, and the whole trajectory are computed on the spot, never looked up.
What the machine produces, proven: an epidemic occurs if and only if R0>1 (the threshold theorem); it peaks exactly where S/N = 1/R0; and it is stopped by an immune fraction of 1 − 1/R0. R0=1 is the critical point between the two regimes.
The blue team's witness (left) confirms these live; the red team (right) tries to make the number lie.
R0 also says nothing about how fast, only whether. Two epidemics with the same R0 can have wildly different growth rates and peaks. Treating R0 as destiny — or as a single measurable constant of a pathogen — is a category error the panel is careful not to make.
"R0 = β·γ." Cut. R0 is the ratio β/γ — rate in over rate out. The product has the wrong units and the wrong threshold; the tamper below plants exactly this bug.
"R0 > 1 means everyone gets infected." Cut. R0>1 means the outbreak grows; the final size is set by the Kermack–McKendrick relation and is always < 100%.
"R0 is a fixed constant of the pathogen." Kept, corrected. It is β/γ — behaviour and contact structure move β, so R0 is model- and population-specific, not intrinsic.
The red team's move: swap the formula to R0 = β·γ (a product, not the ratio). A sub-critical set that should fizzle is now mislabelled an epidemic. The blue team's witness (window 7) is watching.
Flip R0 to a product and the threshold theorem breaks: β=1.2, γ=2.0 truly fizzles (β/γ=0.6) but the product reads 2.4 and cries "epidemic." The witness re-integrates, sees the dynamics disagree with the label, and turns red. Nothing is faked; the attack is real and it is caught.