Herd immunity

When enough people in a population cannot catch an infection and pass it on, each case fails to replace itself, and outbreaks shrink even among people who have no protection of their own. That is herd immunity. How many is “enough” is not a fixed fact about a disease. It comes out of a formula, and the formula's inputs change from place to place.

What is being claimed

Fine, Eames and Heymann point out that the term, coined almost a century ago, is used in several ways: for the share of a population that is immune, for a threshold that share should cross, and for a pattern of immunity that keeps a new infection from invading. The sharp version is a threshold theorem they credit to Smith in 1970 and Dietz in 1975. If immunity is handed out at random, and people mix at random, the number of new infections falls once the proportion immune exceeds 1 − 1/R₀.

R₀, the basic reproduction number, is in Randolph and Barreiro's words “the average number of secondary infections caused by a single infectious individual introduced into a completely susceptible population”.

Why it matters

The protection reaches people who were never vaccinated. Randolph and Barreiro name people who cannot be vaccinated, including the very young and the immunocompromised. Fine and colleagues give a measured case: after conjugate vaccines against pneumococcal and Haemophilus infections arrived, falls in disease among people too old to have been vaccinated accounted for one to two thirds of the total reduction in some populations.

Where 1 − 1/R₀ comes from

Fine's first figure uses R₀ = 4. One case leads to 4, then to 16. Now make three quarters of the population immune. Of each case's four contacts, three are immune, so each case passes the infection to only one person, and the number of cases holds steady. Make the immune share any larger and the count declines.

In general, if a share p is immune, each case causes R₀ × (1 − p) new ones. Randolph and Barreiro call that the effective reproduction number, R_e, and the aim of vaccination is to push it below 1. It falls below 1 exactly when p exceeds 1 − 1/R₀. For R₀ of 2, 3, 4 and 5 that is 50%, 67%, 75% and 80% (our arithmetic for 2).

Crossing the line does not stop cases on the spot. Incidence starts to decline. In Randolph and Barreiro's model of an epidemic left to run, the number infected peaks at the threshold, and they note it takes a long time after that before new cases stop.

The vaccine has to stop spread, not just illness

Real vaccines are imperfect. Call E the vaccine's effectiveness against transmission. Fine and colleagues give the coverage needed as (1 − 1/R₀) / E. With R₀ = 4 and E = 0.9, that is 0.75 / 0.9, or 83.3% (our arithmetic). They add that if E is below 1 − 1/R₀, vaccinating the whole population cannot eliminate the infection. At R₀ = 15 the threshold is 93.3%, so a vaccine with E = 0.9 would need 103.7% coverage (our arithmetic): impossible.

The extreme case is a vaccine that prevents disease but not infection. Then, they write, “there would be no indirect effect, and no herd immunity.” Randolph and Barreiro add that immunity which wanes, as it does for pertussis and rotavirus, leaves room for periodic outbreaks.

R₀ is not one number

Delamater and colleagues write that R₀ “is not a biological constant for a pathogen”. It depends on how often people meet, so it belongs to a pathogen in a particular population. Vaccination lowers R_e, never R₀. The often-quoted measles R₀ of 12–18 rests on United States data from 1912–1928 and English and Welsh data from 1944–1979. That range gives thresholds of 91.7% to 94.4%. Pertussis at 12–17 gives 91.7% to 94.1%. The measles literature holds more than 20 values between 5.4 and 18, thresholds of 81.5% to 94.4%, and a 2017 review found feasible values of 3.7 to 203.3, thresholds of 73.0% to 99.5% (all thresholds our arithmetic). Randolph and Barreiro quote estimates of 2 to 6 for SARS-CoV-2, which spans 50% to 83.3%.

Where the immunity sits

The formula assumes immunity is spread evenly and people mix at random. Fine and colleagues warn that clusters of unvaccinated people are vulnerable to outbreaks, and Randolph and Barreiro add that such pockets are at risk even when the population as a whole is over the threshold. The interactive shows this with a toy model of our own. Take R₀ = 4 and 80% coverage with a perfect vaccine, and R_e is 0.8. Split the population into two halves at 100% and 60%, with nine in ten contacts inside one's own half, and R_e becomes 1.44 (our arithmetic).

Unevenness can also help. In Fine's two-group example, high-risk cases each infect five high-risk people and low-risk cases one low-risk person. R₀ is 5, so the simple rule asks for 80%, yet vaccinating 80% of the high-risk group alone could in theory prevent outbreaks.

Two more cautions from Fine and colleagues. Herd immunity is not the same as personal immunity: people protected only by it “remain fully susceptible”. And because every threshold rests on simplified assumptions, “the sensible public health practice is to aim for 100% coverage”.

In short

Each case must, on average, infect fewer than one other person. With random mixing that happens when more than 1 − 1/R₀ of people are immune, which with a vaccine of effectiveness E means vaccinating more than (1 − 1/R₀)/E of them. R₀ varies between places and eras, and immunity can cluster, so herd immunity is a threshold you calculate for a population, not a single number a disease carries.