The term “herd immunity” first found its way into the UK’s public consciousness very early on in the pandemic. The UK’s chief scientific officer, Patrick Vallance, talked about allowing “enough of us who are going to get mild illness to become immune” and building up “some degree of herd immunity while protecting the most vulnerable”. What Vallance was referring to was herd immunity by natural infection. Without a vaccine this is the only way you can achieve herd immunity—with the attendant illness and death that accompany these natural infections. The idea drew concern from broad swathes of the scientific community and talk of herd immunity as a strategy was subsequently quashed by the government.

Despite the toxic connotations the phrase took on for many, herd immunity is an incredibly important concept. It suggests diseases can be eliminated without everyone having immunity. If you want to eliminate an established disease, then herd immunity is the way to do it. To achieve herd immunity, people can gain immunity in two different ways: through being infected and recovering with immunity, or through being vaccinated. For covid, the best way for people to get immunity is through vaccination, even for younger age groups who are at a lower risk from severe complications due to covid.

When enough people have immunity to the currently circulating variant, the disease will start to decline. But how many people need to have immunity before the disease starts to die out? To calculate the so-called “herd immunity threshold” (HIT) we need to do some mathematical epidemiology.

At the beginning of an epidemic when almost everyone is susceptible to the disease and there are no interventions in place to control it, simplistically speaking, the majority of the expected disease dynamics can be boiled down to a single number, *R** _{0}—*the basic reproduction number. This tells us how many people we expect each infected person to pass the disease onto during the course of their infectious period. This is a special case of the

*R*-number we have become so familiar with from the news—the zero-subscript indicating this is the value of

*R*at the

*beginning*of an epidemic with

*zero*interventions in place.

If we can take measures to reduce *R* so that it is below 1, then the disease will begin to die out. One way to do that is for people to gain immunity. The more people that have immunity, the fewer people each individual will be able to pass the disease onto. If *p* is the proportion of the population that is immune (and *(1-p)*, therefore, the proportion not immune) then we can calculate the current reproduction number from the basic reproduction number as *R=R*_{0}**(1-p)*. If a proportion *p=1-1/R** _{0}* of people are immune then this takes us to

*R=1*, the tipping point at which infection should start to decline. If, for example, the original covid variants circulating in the early stages of the pandemic had an

*R*

*of 3, then at least two-thirds of the population would need to have perfect immunity to bring*

_{0}*R*below 1. With more transmissible variants such as delta, which may have a basic reproduction number as high as 6, this calculation suggests that the HIT might be as high as 83%.

This is a very simplistic picture, but it can be useful for back of the envelope calculations. In reality many factors will influence the HIT. One consideration is the degree of immunity conferred. Vaccines, for example, are not 100% effective at stopping people from transmitting the virus. Immunity from natural infection is thought to be even worse. This means the HIT will be higher than suggested by the naïve calculation above. Even if vaccination reduces the degree of onwards transmission by as much as 85%, this would increase the HIT to 98%. The potential for immunity to wane also means we will need to up the numbers of people vaccinated and think about delivering booster vaccinations.

Another important factor is heterogeneity of the population—understanding that not everyone mixes with everyone else equally. This is particularly important when we are vaccinating by age brackets. Immunity will not be spread evenly. Even if we reach a theoretical HIT through vaccination, if there are large demographics that are not immune then the disease can still spread freely in these groups. By not offering the vaccine to children, for example, we are providing a large reservoir of unprotected people in whom the virus can freely circulate.

Conceptualising herd immunity as “all or nothing” is not particularly helpful. Generally as immunity builds up in the population it slows the spread. Even if we can’t reach a level of immunity that will keep *R* below one once all restrictions are relaxed (and certainly without vaccinating children it is unlikely we can reach the levels required), the more immunity we have the slower the spread will be, and the easier it will become to control covid though tested public health measures.

The government’s plan to relax almost all restrictions from 19 July 2021 makes it clear that the UK’s path towards herd immunity will, at least in part, be through natural infection with the consequences that entails. Recent spread has already seen healthcare services struggle to cope under excess pressure and will lead to many avoidable deaths and long term illnesses. Currently, around 47% of the UK population are not yet fully vaccinated. Denying everyone in the country the best chance of being protected through vaccination and relying instead on “caution, vigilance, and personal responsibility” to tackle an airborne and highly contagious infectious disease is an abdication of responsibility, which will involve exposing millions to the acute and long-term impacts of mass infection.

**Kit Yates**, senior lecturer, Department for Mathematical Sciences, University of Bath. **Twitter**: @Kit_yates_maths

**Competing interests**: KY is a member of Independent SAGE.