By Marc Lipsitch
How many people in a population will need to get a COVID-19 vaccine before herd immunity is achieved? This question has been all over the news lately, with speculations of 70%, 80%, or other numbers. The math needed to answer this question is simple, but the question is, in my opinion, unanswerable at this point because the inputs to the math — the numbers to plug into the relevant equation — are uncertain and may vary according to context.
The simple math of herd immunity
When a new infection enters a fully susceptible population, as SARS-CoV-2 infection did in early 2020, each infectious case on average infects R0 other cases. As the disease spreads, it leaves in its wake immunity to infection in some or all of those who have recovered. Moreover, once society recognizes the threat, measures can be put in place (masks, distancing, handwashing, movement restrictions, etc.) that may further hamper spread. At some time t when the disease has been spreading for a while, the average case in a population will infect R(t) others in that population, where usually R(t) < R0, thanks to the combined effect of immunity to infection that has accumulated, and control measures. As has become common knowledge thanks to the teachable moments of this pandemic, when each individual infects on average more than one other, the number of cases will rise, and when each individual infects on average less than one other, the number of cases will fall.
Vaccines can help reduce R(t). We often refer to a critical vaccine coverage f*, which is the proportion of randomly chosen individuals in the population that must be vaccinated to achieve R(t) < 1. If we have a vaccine that reduces transmission by a factor x, then we want to create the situation where Rvac(t) = (1 – xf*) Runvac(t) = 1 — that is, where the average case in a vaccinated population infects less than one additional person (we use an equals sign because we are trying to get the value of f* that just barely achieves this threshold; any value of f > f* will create Rvac(t) < 1. If we rearrange that expression, we get : the proportion that needs to be vaccinated is greater for large values of Runvac(t) and for smaller values of x. Textbooks often write that , which makes sense because as we noted in the previous paragraph, R(t) < R0 in most cases, because of immunity and control measures. Therefore if we can control spread without any other immunity or control measures, we can control it even better with the help of those things. When we think about “getting back to normal” we are thinking of controlling the disease with vaccination alone, and not the control measures we have imposed.
A simple calculation
Early in the SARS-CoV-2 pandemic, many estimates of R0 were published for this new infection, mainly based on data from early spread in Wuhan and other early sites of transmission. These estimates tended to cluster in the range of 2-3 [1–3], though some were higher.
Two trials of mRNA vaccines against SARS-CoV-2 have come out with efficacy against COVID-19 near 95%, with some uncertainty . In one published study, on the Pfizer vaccine, the 95% CI for vaccine efficacy was 90.3% to 97.6%.
If we take the upper end of the 2-3 range for R0 and assume a vaccine reduces infection by 90%, then our calculation of gives a value of f* = 74%, squarely within the range that many people have been discussing.
Why that calculation may be too simple
That calculation may be too simple because its assumptions about the inputs — x and R0 — may be wrong, and in particular, overly optimistic. It may also be too simple because while it assumes x and R0 are fixed numbers, they may actually vary. In particular, the protection against infection may x decline over time, and R0 likely varies from location to location. There are also some factors that could help, reducing the proportion we have to vaccinate to achieve Rvac(t) < 1.
Uncertainty and variation in R0
As noted above, there were many early estimates of R0 in the early days of COVID-19. In all new epidemics, these estimates are difficult to make. Most estimates use the rate of exponential growth in cases and the serial interval, or time between an infection and a transmission from that infection . Each of these quantities is measured with error, in ways that have been long understood. For example, early case numbers are subject to varying sensitivity of detection, as surveillance systems scramble to meet a new challenge. Over time, surveillance may become more sensitive, as more testing happens, or less sensitive, as systems are overwhelmed ; these time trends can masquerade as faster or slower growth of the epidemic, respectively. Delays in reporting can make the recent part of an epidemic curve look less steep, leading to underestimates of growth rates . Serial intervals can change (usually shorten) as control measures are put in place [8,9], biasing estimates of R0 in complex ways. Apart from these biases, R0 is a social, as well as a biological variable, reflecting demography and behavioral patterns in the populations, so it is expected to vary between populations even for the same infectious disease . The upshot is that some of the higher estimates, well above the range of 2-3 for R0 of SARS-CoV-2, cannot be ruled out, at least for some populations with high rates of contact .
Uncertainty in x, part 1
The primary endpoint in all randomized clinical trials of SARS-CoV-2 vaccines to date has been reduction in COVID-19 — symptomatic disease caused by SARS-CoV-2. This is the endpoint on which the estimates of ~95% have been obtained for the two mRNA vaccines. Vaccines can have three different kinds of beneficial effects . They can reduce:
- Susceptibility to infection. This means that a vaccinated person exposed to the virus will remain uninfected with greater probability than if they had been unvaccinated. The efficacy component that reflects this activity is called vaccine efficacy for susceptibility, or VES.
- Progression to symptoms. This means that a vaccinated person who does become infected is less likely to experience symptoms than if they had been unvaccinated and become infected. This component is known as vaccine efficacy against progression, VEP.
- Infectiousness to others. This means that a vaccinated person who becomes infected will infect fewer other people than if they had been unvaccinated and become infected. This component is known as vaccine efficacy against infectiousness, VEI.
The primary endpoint measured in trials is a combination of VES and VEP. An individual can avoid having the endpoint (symptomatic infection) by avoiding infection or by getting infected but avoiding symptoms. From the primary outcome alone, we can infer something about the possible values of VES and VEP, but not separate them. For example the 95% result could be because individuals were purely protected against susceptibility to infection with no effect on progression (VES = 95% and VEP = 0) or just the opposite — no protection against susceptibility to infection but protection against progression (VEP = 95% and VES = 0).
For herd immunity we care about VES and VEI. If a person doesn’t get infected, they can’t transmit, so VES contributes to x. If they get infected but are less infectious to others, that will contribute to herd immunity, so VEI contributes to x. In fact, the relationship is this: x = 1 – (1 – VES)(1 – VEI). Larger values of VES and VEI translate into larger values of x, and if either is 100% then x is 100%.
The primary endpoint doesn’t tell us anything about VEI and while it gives us some information about likely values of VES, it doesn’t rule out any of them .
Two of the completed vaccine trials have some information about VES. The Moderna vaccine trial showed a 63% (95%CI 32-80%) reduction in PCR+ swabs in vaccine vs. placebo participants when they were swabbed at the time of their second vaccine injection. In one of the two vaccine doses given in the Astra-Zeneca chimp adenovirus-vectored vaccine trial , there was a 58.9% (1.0-82.9%) reduction in asymptomatic carriage on swabs taken from vaccinated vs. placebo recipients, with no statistical difference in the other dose. Together, these results probably reflect positive values of VES and/or VEI and thus positive values of x, but the uncertainty is wide. Some but not all animal studies of the vaccines discussed so far in this note suggested effects on susceptibility and/or infectiousness, so seeing some evidence of an effect in humans is not surprising.
It is worth noting that many other viral vaccines do dramatically reduce transmission; for example measles  and influenza . However, several vaccines for respiratory pathogens, such as pneumococcal vaccines  and acellular pertussis vaccines  appear to offer substantially more protection against disease than against infection and transmission.
All things considered, it is very likely that the existing vaccines with very high efficacy against symptomatic infection also make some contribution to reducing transmission. However it seems very possible that they will provide only partial protection against infection and transmission, and the amount of protection matters to the calculation of x and thus to the estimate of the critical vaccination fraction f*.
Uncertainty in x, part 2
The efficacy of vaccines against SARS-CoV-2 has been measured very rapidly, with the consequence that most of the data come from a period very shortly (~2 months) post vaccination. It is completely normal for vaccination to produce antibody concentrations that decline with time, rapidly over months and then more slowly over years , sometimes with a concomitant decline in protection. In other vaccines, the antibody concentration needed to protect in the nasopharynx (where SARS-CoV-2 infection happens) is higher than that needed to prevent disease . So it is possible that all of the vaccine efficacy measures, including those against transmission, will decline with time since vaccination.
Other unknowns and a tentative bottom line
While these are the most important sources of uncertainty about herd immunity and the existing data on the existing vaccines for SARS-CoV-2, other factors could affect herd immunity:
- It is possible that the virus could evolve to be more transmissible (selection pressure makes it unlikely that it would evolve to be less), increasing R0 and thus f*.
- It is possible that the virus could evolve to partially or fully escape immunity from the vaccine, reducing x and thus increasing f*.
- It is possible that lasting change in our behavior could reduce our degree of contact, making a permanent change in transmission conditions such that R(t) < R0 even without vaccine, which would reduce f*.
- It is possible that our patterns of vaccination will be nonrandom, as has been proposed by most health authorities. Vaccinating those who are not particularly important in transmission could increase the proportion who need to be vaccinated, while vaccinating key transmitters could reduce it. Most prioritization schemes include individuals in both groups (e.g., community-dwelling very elderly, probably less central to transmission, and essential workers of various kinds, probably more central).
- Immunity from natural infection, to the extent it is protective against infection and transmission , will supplement that from vaccination, though to a declining degree in a well-vaccinated population . This will at least temporarily reduce the requirement for vaccination.
Bottom line: It is almost certain that most vaccines against SARS-CoV-2 will reduce transmission and thus contribute to population (or “herd”) immunity. The degree of that contribution is unknown, but evidence from animal and human studies to date and evidence from other infections leads this observer to think that the vaccine may reduce transmission 50-70%. This is not a confident prediction and could well be wrong in either direction. If it were in this range, and if R0 were upwards of 3 in many parts of the world (also uncertain), it is quite possible that achievable levels of coverage might not be enough, on their own, to prevent sustained transmission, though they might be close.
Importantly, sustained herd immunity is not the only value of a vaccine or the only way it could help us return to a more normal life. If high coverage can be achieved in those most at risk of severe outcomes, we could achieve a state where virus continues to circulate (at a level reduced by partial herd immunity) but the toll on the health system and the mortality toll is dramatically reduced because fewer highly vulnerable people are infected, and even fewer of those experience symptoms, thanks to direct protection by the vaccine. In my personal opinion, this is the most likely path to a more normal life in many countries.
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