Epidemiology
Epidemics turn over long before the herd-immunity threshold their early growth implies. A one-parameter rule says where they turn, and contact surveys supply the parameter.
In the spring of 2020, case counts doubled in under a week. A homogeneous epidemic model reads that growth as a reproduction number near 2.5. The herd-immunity threshold is the share of the population that must be immune before each infection causes fewer than one new one.
For $R = 2.5$ the textbook value is 60%. Yet first waves turned over with 20% or less of the population infected.
Cotton (May 2020) called this the herd-immunity paradox and traced it to variation in transmission across people, places and time. The pages below develop that explanation, put a number on it, and test it against data from the United States, Spain, Sweden, England, Geneva and Manaus.
The rule
Let $R$ be the reproduction number read from early growth. The epidemic turns over when the infected share reaches
$$\text{turnover} = 1 - R^{-1/\lambda},$$where $\lambda$ is the immunity factor. It measures how much faster the effective reproduction number falls than the number of susceptible people, because the most active people are infected first.
A homogeneous population has $\lambda = 1$, and the rule reduces to the textbook $1 - 1/R$. The turnover rule derives it and gives the table.
Where the immunity factor comes from
- Age. The contact matrices of Prem et al. give $\lambda = 1.1$ to $1.7$ with nothing fitted. Age alone moves the threshold at $R = 2.5$ from 60% only to about 51-56%.
- Activity within age groups. Contact diaries show that the number of people someone meets in a day varies widely within an age group. Combined with the age matrices, again with nothing fitted, this gives $\lambda = 2.3$ to $3.0$.
- Activity that changes over time. When busy and quiet phases alternate, the immunity factor in the growth phase exceeds its long-run value by about the growth rate times the Green–Kubo number of activity. The first wave turns early, and later waves are still possible.
- Mixtures of places. In a population made of many places, early growth is dominated by the fastest of them, and each place turns over on its own. Two convexity adjustments, one on growth and one on the turnover, explain the paradox.
What the data show
| place | contacts cut? | infected at the turnover | $\lambda$ |
|---|---|---|---|
| England, 9 regions | lockdown | 1-5% | 17-94 |
| Geneva | partial lockdown | 5% | 23 |
| Sweden, Stockholm and 2 other regions | no lockdown; large voluntary drop | 1-3% | 20-45 |
| Spain, spring 2020, 52 provinces | national lockdown | 0.3-4.6% | about 60 |
| Spain, autumn 2020 | curfew | 2.6% of those still susceptible | about 7 |
| US counties, winter 2020-21 | mixed | about 7% of those still susceptible | 2-3.5 |
| Manaus | little effective control | about 17% | 2.5 |
| contact surveys, no epidemic data | – | – | 2.3-3.0 |
Where contacts were cut, by law or voluntarily, the first wave turned with 1-5% infected. The immunity factors of 20 and more there measure the cut in contacts, not immunity.
Where control was weak, the immunity factor is 2 to 3.5. That is what contact surveys predict with nothing fitted.
Manaus went on to 66-76% infected, close to the homogeneous final size. With fixed heterogeneity it would have stopped near 35%.
So the early turnover there was temporary, as time-varying activity predicts. The details are on US county waves and Serology by country.
Cite
P. Cotton (2020). Addressing the Herd Immunity Paradox Using Symmetry, Convexity Adjustments and Bond Prices. arXiv:2006.07341.
Analysis code and a dated log of results: research/epidemic-hit. Comments welcome.