Herd : Letting the run?

Total proportion of the 1.0 population who were infected

0.8 Overshoot

0.6 Threshold for herd immunity Infected 0.4 + Recovered Recovered

0.2 Susceptible rprino population of Proportion

Infected 0.0 0 5 10 15 20 Time to reach herd immunity Time

The graph shows the proportion of susceptible, infected, and recovered individuals over time. When the number of individuals who are either recovered or currently infected (‘infected + recovered’) hits the herd immunity threshold, the does not come to a sudden halt. It has a momentum. Until the dies out, many more people get infected than required for herd immunity. Herd immunity: Letting the pandemic run? Page 2

First of all, what is herd immunity? To make which is a characteristic property of the virus). this clear, we need to look at the so-called Mathematicians have derived a formula to ob- reproductive number R of the virus. The re- tain an estimate for the required fraction. For productive number is the number of people SARS-CoV-2, for which every patient infects to whom a single infected individual trans- around 2.5 other patients during the early mits the on average. If this number phase of the epidemic, this fraction is 60%. is larger than one, the virus can spread, and Hence, if we had a and gave it to 60% more and more people become infected. If it of the population1, the virus could not spread is smaller than one, the number of infected in the population (there could still be small people declines. We can reduce this number, outbreaks though). for example, through physical distancing or The herd immunity threshold contact tracing and isolation of contacts. How- ever, this requires ongoing action. As soon as Let R0 be the basic reproductive number we stop, the reproductive number goes up of the virus. I.e. every infected person again. infects R0 other people on average in a A permanent solution is immunisation of fully susceptible population. If we take a large part of the population. If many people out a fraction x of the susceptible pop- are immune, it is likely that an infected per- ulation (e.g. through ), the son does not meet a susceptible person who reproductive number drops to R0(1 − x). could catch the disease, and the reproduction How large does x need to be to achieve number of the virus is below one. In that case, herd immunity? Well, the reproductive we speak of herd immunity. The population number needs to be smaller than one. as a whole is immune even though some in- Hence: dividuals may not be immune. We can reach R0(1 − x) < 1, this through vaccination if we have an effec- tive vaccine. However, immunity also builds which solves to up in the population over time as more and 1 x > 1 − . more people become infected, develop an im- R0 mune response, and recover. This requires, of course, that individuals remain immune for However, what does it look like if we let an extended period of time after recovery (see the pandemic spread to reach the required below). fraction of 60% immune people? By the time The buildup of immunity is why some peo- that 60% of the population is infected or recov- ple argue we should just let the pandemic ered, the reproductive number drops below run, taking no control measures. However, one and the number of infected individuals achieving herd immunity through of starts declining. This is hence the peak of large parts of the population is not the same the epidemic. However, at the peak of the as achieving herd immunity through vaccina- epidemic, a lot of people are infected. Even tion, and it doesn’t work the way many people though each of them now only infects less think. Let’s have a look at that! than one other person, this is still a very large For the reproduction number to be smaller number of new . And these newly than one, a large enough fraction of the pop- infected patients can again infect new indi- ulation needs to be immune. This fraction viduals. In that way, although the threshold depends on the reproduction number of the for herd immunity has been reached, a lot virus in a population where no one is immune of infections still occur until the virus finally and where no control measures are in place goes extinct. Again, mathematicians have de- (this is called the basic reproductive number, veloped a model to grasp the size of this ‘over- 1In reality, we need to give it to some more, since even the best vaccine is not perfect and some people may not become immune. Herd immunity: Letting the pandemic run? Page 3 shoot’. The following figure shows how many since vaccinated people cannot infect others people will get infected until the epidemic such that we are fully in control. We could dies out and the herd immunity threshold, de- try to reach the herd immunity threshold of pending on the basic reproductive number of 60% more slowly, with considerable control the virus. measures in place. This would reduce the

1.0 overshoot (the extra 30%) but it would take

Total proportion a very long time. 0.8 of the population who were infected Importantly, unlike with , which Threshold for are very safe, from these 90% that would be- 0.6 herd immunity come infected in an uncontrolled epidemic,

0.4 many will develop severe symptoms, many will require hospitalisation, leading to over-

rprino population of Proportion 0.2 burdening of the health care system, and part SARS - CoV 2 of them will die. 0.0 0 1 2 3 4 5 There is yet another problem with this ap- Basic Reproductive NumberR 0 proach to herd immunity. We do not know for We can read from this graph that for SARS- how long immunity to SARS-CoV-2 lasts. If CoV-2 (R0 ≈ 2.5), almost 90% of the popula- it wanes too quickly, herd immunity cannot tion will become infected, as can also be seen build up, and the virus will keep circulating in the figure on the first page. This is much in the population, reinfecting people over and more than the 60% required for herd immu- over again. So far, we don’t know for how long nity! This does, of course, not happen if we people remain immune following an infection achieve herd immunity through vaccination, with SARS-CoV-2.

Controlling close to the peak?

1.0

0.8 Reduction in Overshoot We can reduce the overshoot if we 0.6 implement strict control measures Threshold for herd immunity Infected once the herd immunity threshold 0.4 + Recovered Recovered has been reached (or even a bit be-

0.2 Susceptible fore). However, in reality, it is diffi- rprino population of Proportion cult to make a full stop at the right Infected 0.0 time, and it is a risky strategy. 0 5 10 15 20 Implementing control Time measures when herd immunity is reached

Dr. Hildegard Uecker, Prof. Dr. Arne Traulsen, Dr. Chaitanya S. Gokhale

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