Written by Dr Flavio Toxvaerd

Herd immunity – crucial yet irrelevant

In their working paper, Dr Flavio Toxvaerd and Prof Robert Rowthorn consider a susceptible-infected-recovered type model of infectious diseases, such as COVID-19 or swine flu, in which costly treatment or vaccination confers immunity on recovered individuals.

Herd immunity diagram

Once the exclusive domain of scientists and public health officials, awareness of the control and effects of infectious diseases has recently broken into the public consciousness. As never before, non-experts pore over graphs of disease prevalence and discuss when and how the lockdown should be eased.

This is indeed an interesting though challenging time to be an infectious disease epidemiologist. As a result of this public interest in the field, two technical terms from epidemiology have recently entered everyday language, namely herd immunity and rate of reproduction  or R-rate. Judging by the frequency of these terms in discussions about the epidemic and government policy, surely these are crucial concepts that should guide policy makers in combating the disease?

On March 13, 2020, the UK government's chief scientific adviser, Patrick Vallance, stated that as a consequence of the strategy to combat the COVID-19 epidemic, 'Communities will become immune to it and that's going to be an important part of controlling this longer term'. Continuing, he said that, ‘About 60 per cent is the sort of figure you need to get herd immunity.’[1]

It is fair to say that this reference to herd immunity as part of the government's strategy set off a storm of indignation, being widely interpreted as a wholesale capitulation by a government prepared to do nothing to protect its citizens against the oncoming wave of the epidemic. Subsequent government statements sought to downplay the role of herd immunity. On March 14, Health Secretary Matt Hancock stated, ‘We have a plan, based on the expertise of world-leading scientists. Herd immunity is not a part of it. That is a scientific concept, not a goal or a strategy.’[2]

Yet, was Patrick Vallance wrong? What exactly is herd immunity and which role does­–and  should–it play in a sensible defence of the public health? The term herd immunity, also referred to as population immunity, is widely misunderstood and it does not help that professional epidemiologists themselves use the term in very different ways (see Fine, 1993 and Fine et al. 2011 for reviews).

A good definition of the concept of herd immunity is given by Fine (1993), who states that population immunity refers to ‘[...] the indirect protection afforded to nonimmune individuals by the presence and proximity of others who are immune.’ This definition recognises that some amount of herd immunity is always present when people can become immune but does not tie down the achievement of herd immunity to any particular threshold number of immune people in the population.

This brings us to the notion of a disease’s rate of reproduction, conventionally denoted by the letter R. In a series of infographics, the UK Government has indicated that restrictions would be tightened or eased depending on whether the rate of reproduction was higher or lower than 1 and even showed a map indicating R numbers for different geographical areas. Presumably local restrictions will also be guided by the level of R. So why is R compared with the magic number 1? How do the dynamics of the disease change when R creeps just above 1?

To answer this, we need to understand how epidemiologists have traditionally conceptualised herd immunity. Brauer and Castillo-Chavez (2012, p. 420), an authoritative text on mathematical biology, states that ‘A population is said to have herd immunity if a large enough fraction has been immunized to ensure that the disease cannot become endemic.’ In other words, according to this view, herd immunity is obtained only when a critical number of people are immune. The mathematical representation of this threshold number of immune individuals is simply that the effective R is less than 1.

As it turns out, the number R is important to know, but almost irrelevant for policy. How can this be? First, let us consider what information the rate of reproduction contains. This number is generally thought of as the number of secondary infections that a single infected individual causes. It depends on how many susceptible people there are in the population, on how infectious the disease is, on contact patterns (i.e. how frequently people meet and interact) and on how long infected people remain infectious to others before recovering.

Thus, knowing R is useful because it tells us something about the path of the disease. But that does not mean that we can simply tie easing or tightening restrictions to whether R is higher or lower than 1. Why so? Consider a disease like the common cold, which for most people has very mild symptoms and no significant health effects but is very highly infectious. It has a rate of reproduction of 2-3, significantly higher than 1. Should we impose a complete lockdown of the economy to combat the common cold? Of course not. Since symptoms are mild, it is not reasonable to incur large economic costs to combat this disease. Similarly, how much lockdown do you think athlete’s foot merits?

Now consider a disease like MERS. This disease has a rate of reproduction of 0.3-0.8, i.e. well below 1, but has serious health effects for those infected.[3] The case fatality ratio of MERS is estimated at 43%.[4] Should we take decisive measures to combat this disease? Almost certainly. Although MERS will die out over time even without intervention, a reproduction rate just under 1 may still mean that too many people may end up infected and so taking measures to suppress the infection may be a high priority goal.

So, we have concluded that for one of these two diseases, no particular intervention would be warranted, while for the other it definitely would, even though the former had a much higher rate of reproduction. This conclusion adopts a holistic approach to disease control, including both economic and health considerations, rather than simply focusing on their rates of reproduction.

The optimal control of an infectious disease will imply that at different stages of the epidemic, the rate of reproduction will be different. Typically, it will be higher at the early stages of the epidemic and lower at later stages. This is true even if the health authorities are making all the right decisions at all times. Recall that the optimal policy balances costs and benefits of control throughout the epidemic and this may involve periods in which the rate of reproduction is higher than 1. This is not a failure of policy, in the same way that a reproduction rate under 1 cannot necessarily be counted as a success. Therefore, a one-size-fits-all policy of tightening restrictions when R is above 1 and easing when is below 1 is unlikely to be optimal.

A clear illustration of these principles can be found in my recent work titled “On the Management of Population Immunity,” written with my colleague Bob Rowthorn. In the paper we consider a number of issues, including the use of antiviral medications, here I focus on the optimal policy on mass vaccination.

If the aim of the health authority is simply to bring the rate of reproduction below 1, then this can in principle be achieved by vaccinating a sufficiently large fraction of the population. By doing so, vaccination plays the same role as having many people recover with immunity, immediately shifting the epidemic to the post-peak phase with declining infections. In other words, by vaccinating a critical threshold, vaccinations can swiftly induce enough population immunity to ensure that the disease cannot increase further. When Patrick Vallance referred to 60% of the population that needs to be immune to achieve herd immunity, this is the critical threshold that he alluded to.

But as discussed above, whether this policy is desirable should depend on costs and benefits of reaching herd immunity. Just because something is feasible, does not mean that it is desirable. In our work, we show that the socially optimal vaccination policy is not simply a matter of achieving the famous herd immunity threshold. Instead, vaccinations should be deployed when they are most valuable, which happens to be around the peak of the epidemic. We show that following this vaccination policy gives better outcomes overall than a policy that seeks to immediately reach the “herd immunity threshold” of vaccinated individuals.

In summary, if feasible, herd immunity will be indispensable for overcoming the epidemic. But when formulating optimal health interventions, society should adopt a holistic approach and carefully trade off all the different aspects that affect social wellbeing, including both health and economic outcomes. Any policy we formulate and implement will determine the way herd immunity builds up in the population. But we need to shift the focus away from arbitrary thresholds of herd immunity and tailoring our policies to achieving these, and instead focus on formulating effective policies to increase socially desirable outcomes. With this shift in focus, the issue of herd immunity becomes secondary to that of good policies. Herd immunity becomes the consequence of policy and not the driver of it.

Read paper




Brauer, F. and C. Castillo-Chavez (2012): Mathematical Models in Population Biology and Epidemiology, 2nd edition, Springer.

Fine, P. E. M. (1993): Herd Immunity: History, Theory, Practice, Epidemiologic Reviews, 15(2), 265-302.

Fine, P., K. Eames and D. L. Heymann (2011): "Herd Immunity": A Rough Primer, Clinical Infectious Diseases, 52(7), 911-916.

Toxvaerd, F. and R. Rowthorn (2020): On the Management of Population Immunity, Bennett Institute Working Paper.

  • About the author

    Dr Flavio Toxvaerd, Visiting Fellow 2018-19

    Dr Flavio Toxvaerd is a University Lecturer at the Faculty of Economics, University of Cambridge and a fellow of Clare College. He holds degrees from the University of Copenhagen (BSc Economics, MSc Economics), the London School of Economics (MSc Econometrics and Mathematical Economics) and the ...   Learn more

    Flavio Toxvaerd