The turnover rule
An epidemic turns over when the infected share reaches $1 - R^{-1/\lambda}$, where $\lambda$ is the immunity factor. The textbook threshold is the case $\lambda = 1$.
The textbook threshold
In a homogeneous population each infected person infects $R$ others at the start. When a fraction $S$ of the population is still susceptible, only that fraction of their contacts can be infected, so the effective reproduction number is
$$R_e = R\,S .$$The epidemic stops growing when $R_e = 1$, that is when $S = 1/R$. The infected share at that point is the herd-immunity threshold,
$$1 - \frac1R .$$Unequal activity
People differ in how many others they meet. The busiest are infected first, and they are also the ones who would have passed the infection on most. Each infection therefore removes more than its share of future transmission, and $R_e$ falls faster than $S$.
The immunity factor $\lambda$ measures how much faster:
$$R_e = R\,S^{\lambda} .$$Early on, when few are infected, this is
$$R_e \approx R\,\big(1 - \lambda\,(1 - S)\big),$$so each percentage point of infection lowers $R_e$ by $\lambda$ percentage points of $R$. For activity that follows a gamma distribution with squared coefficient of variation $\mathrm{CV}^2$, and that raises both the chance of being infected and the number of people infected, the power law is exact with
$$\lambda = 1 + 2\,\mathrm{CV}^2 .$$Setting $R_e = 1$ gives the turnover:
Equivalently, the textbook formula applies with an adjusted reproduction number $R^{1/\lambda}$ in place of $R$.
The table
Infected share at the turnover:
| $R$ from early growth | textbook, $\lambda = 1$ | $\lambda = 2$ | $\lambda = 2.7$ | $\lambda = 3.5$ | $\lambda = 5$ |
|---|---|---|---|---|---|
| 1.3 | 23% | 12% | 9% | 7% | 5% |
| 1.5 | 33% | 18% | 14% | 11% | 8% |
| 2.0 | 50% | 29% | 23% | 18% | 13% |
| 2.5 | 60% | 37% | 29% | 23% | 17% |
| 3.0 | 67% | 42% | 33% | 27% | 20% |
Doubling in under a week ($R \approx 2.5$) with a turnover near 20-23% corresponds to $\lambda \approx 3.5$, and the adjusted reproduction number is then $2.5^{1/3.5} = 1.3$.
The column $\lambda = 2.7$ is the value that one-day contact diaries give for the United States, Sweden and Spain with nothing fitted (age and activity). A two-year contact panel gives 3.4-3.7 (how long busy phases last).
Against the US county waves
In winter 2020-21, 326 US county waves, grouped by their growth rate, turned over having infected these shares of the people still susceptible when the wave began (fatality rate 0.7%; details):
| $R$ at the start of the wave | textbook | $\lambda = 2$ | $\lambda = 3.5$ | observed |
|---|---|---|---|---|
| 1.08 | 7.8% | 4.0% | 2.3% | 6.5% |
| 1.16 | 13.8% | 7.2% | 4.2% | 7.0% |
| 1.23 | 18.9% | 10.0% | 5.8% | 6.9% |
| 1.34 | 25.5% | 13.7% | 8.1% | 8.8% |
The level sits between $\lambda = 2$ and $\lambda = 3.5$, well below the textbook. The observed share varies less with $R$ than the rule does, but the county growth rates are too noisy to measure that slope; see US county waves.
Where the rule stops
The immunity factor is read from deaths divided by an infection fatality rate, so it scales with the rate assumed. Across 0.5% to 1%, the county estimate runs from about 2.4 to 5.3.
If activity changes over time, the rule gives where the first wave turns, not lasting immunity. As busy and quiet people trade places the immunity factor relaxes toward 1, and the epidemic can resume (activity that changes over time). Manaus reached 66-76% infected after turning early.
Where contacts were cut by lockdowns or by voluntary distancing, the turnover mostly reflects the cut.