Herd immunity

Herd immunity is a population-level reduction in transmission. It occurs when enough people are protected that an infectious person, on average, produces fewer than one new infection.

Core idea. Herd immunity does not require everyone to be immune. It requires the population to contain few enough susceptible people that infection can no longer sustain continued growth.

Start with the quantities

Symbol	Meaning
\(R_0\)	Average number of secondary infections produced by one infectious individual when the population is fully susceptible.
\(p\)	Proportion of the population that is protected.
\(1-p\)	Proportion that remains susceptible.
\(R_{\mathrm{eff}}\)	Average number of secondary infections produced under the current level of susceptibility.

Why protection reduces transmission

Suppose initially \(R_0=4\). This means that, in a fully susceptible population, one infectious person produces about four new infections on average.

If 25% of the population is protected, then 75% remains susceptible. The infectious person no longer has the same opportunity to generate four new infections, because some contacts are now with protected people.

In the simple homogeneous model, this is represented by

\[\boxed{R_{\mathrm{eff}}=R_0(1-p)}.\]

Example. If \(R_0=4\) and \(p=0.25\), then \[R_{\mathrm{eff}}=4(1-0.25)=3.\] So one infectious person now produces about three new infections on average rather than four.

When does herd immunity occur?

The key threshold is \(R_{\mathrm{eff}}=1\). If \(R_{\mathrm{eff}}>1\), infection can grow; if \(R_{\mathrm{eff}}<1\), transmission tends to decline.

\(R_{\mathrm{eff}}>1\): growth | \(R_{\mathrm{eff}}=1\): threshold | \(R_{\mathrm{eff}}<1\): decline

Deriving the herd-immunity threshold

Using \(R_{\mathrm{eff}}=R_0(1-p)\), the threshold occurs when

\[R_0(1-p)=1.\]

Therefore

\[\boxed{p_c=1-\frac{1}{R_0}}.\]

Here \(p_c\) is the critical protected proportion.

Example with \(R_0=4\). \[p_c=1-\frac14=0.75.\] So the threshold is 75% protected.

How protection changes \(R_{\mathrm{eff}}\)

The figure below simply plots \(R_{\mathrm{eff}}=4(1-p)\). The horizontal axis is the protected proportion \(p\); the vertical axis is \(R_{\mathrm{eff}}\).

At 0% protected, \(R_{\mathrm{eff}}=4\). At 50%, \(R_{\mathrm{eff}}=2\). At 75%, \(R_{\mathrm{eff}}=1\). Above 75%, infection tends to decline.

Why unprotected people can benefit

An unprotected person can still become infected. The indirect benefit arises because transmission chains are more often interrupted by protected individuals, so infection has fewer routes through the population.

Why the simple threshold can be misleading

The previous calculation treated every protected person as completely immune. Real vaccines may instead reduce the chance of infection without reducing it to zero.

Let \(e\) denote vaccine efficacy against infection. It compares the infection risk in vaccinated and unvaccinated groups:

\[\boxed{e=1-\frac{R_V}{R_U}=\frac{R_U-R_V}{R_U}},\]

where \(R_U\) is the infection risk in the unvaccinated group and \(R_V\) is the infection risk in the vaccinated group. Here, risk means the proportion of each group that becomes infected during the period being compared.

Calculating vaccine efficacy. Suppose 10% of the unvaccinated group becomes infected, so \(R_U=0.10\), while 2% of the vaccinated group becomes infected, so \(R_V=0.02\). Then \[e=1-\frac{0.02}{0.10}=1-0.2=0.8.\] Therefore \(e=0.8\), or 80% efficacy against infection. This means the infection risk in the vaccinated group is 80% lower than in the unvaccinated group in this comparison; it does not mean that exactly 80% of vaccinated individuals are completely immune.

If a proportion \(p\) of the population is vaccinated, then in this simplified model the effective protected proportion is approximately \(ep\). Therefore

\[\boxed{R_{\mathrm{eff}}=R_0(1-ep)}.\]

Example. Keep \(R_0=4\) and use \(e=0.8\). If 75% of the population is vaccinated, \[R_{\mathrm{eff}}=4[1-(0.8)(0.75)]=4(0.4)=1.6.\] Therefore 75% vaccination is no longer enough to reach the threshold.

Setting \(R_{\mathrm{eff}}=1\) gives the required vaccination coverage:

\[\boxed{p_c=\frac{1-1/R_0}{e}}.\]

With \(R_0=4\) and \(e=0.8\),

\[p_c=\frac{0.75}{0.8}=0.9375.\]

So approximately 93.75% must be vaccinated in this simplified model.

Complete protection versus an imperfect vaccine

The figure compares the same disease, \(R_0=4\), under two assumptions. The horizontal axis means the proportion vaccinated, not the proportion completely protected.

The green line is complete protection (\(e=1\)); it reaches \(R_{\mathrm{eff}}=1\) at 75% vaccination coverage. The brown-orange line is an 80%-effective vaccine (\(e=0.8\)); it reaches the same threshold only at about 93.75% coverage. This shows why vaccination coverage and effective population protection are not necessarily the same quantity.

Other factors can also change the simple threshold: immunity may wane, contact rates may differ between groups, and transmissibility may change. More detailed models represent these mechanisms explicitly.

The threshold does not mean zero cases

Reaching the threshold means sustained growth is no longer expected under the model assumptions. It does not mean transmission stops immediately or that no further infections can occur.

Key idea. Herd immunity is easiest to understand through \(R_{\mathrm{eff}}\). Complete protection gives the simple threshold \(1-1/R_0\), but imperfect protection means a larger proportion of the population may need vaccination to achieve the same reduction in transmission.