A key unanswered question about Covid-19 is how long a person can expect to be immune after recovering from illness. Answering this question is crucial for understanding how extensive our vaccination programs must be to drive down transmission, planning for next year, and identifying long-term strategies for managing Covid-19. A new study suggests that the answer may be years, not months.

To be clear, the study actually establishes that protective immunity from prior infection by SARS-CoV-2 lasts *at least* seven months. (Note: this is similar to what we currently know about immunity conferred by vaccination with the Pfizer and Moderna vaccines.) It does not, however, establish exactly how long that immunity lasts. At the end of this post, I explain how we can extrapolate from their findings to find a (speculative) duration of 4.7 to 8.4 years.

## First, what exactly did the study find?

The study followed 25,661 health care workers in the UK from June 18, 2020 to January 11, 2021. Of these workers 8,278 — about one third of the total — were known to have been infected with SARS-CoV-2 either because they had antibodies to the virus or because they had received a positive PCR test. The remaining 17,383 participants were presumed not to have been previously infected. Altogether, participants were followed for 5,018,549 person days, including PCR tests every two weeks and antibody tests every month.

The protective effect of prior infection was measured by looking at the different rates of infection (measured by a positive PCR test) in the group with prior infection compared to the group without.

Specifically, for the group that had previously been infected, a participant was considered to be a “possible reinfection” if a person received two positive PCR tests more than 90 days apart or had no antibodies when they enrolled in the study, but was found to be PCR positive four or more weeks later. A participant was considered to be a “probable reinfection” only if there was very strong, quantitative serological data or genetic sequence data, which is the gold standard for documenting reinfection. In fact, only two probable reinfections were documented along with 153 possible reinfections for a total of 155 reinfections altogether. Importantly, some of the possible reinfections may actually have been chronic infections. Thus, the analysis, which is based on all 155 reinfections, represents an upper bound to the number of people who may have been reinfected.

These 155 reinfections compare with 1,704 infections in the group not previously infected. Thus, the chance a previously infected person being reinfected was 155 out of 8278 (or 1 in 53) while 1,704 out of 17,383 (1 in 10) previously uninfected persons became infected during the study period. These crude numbers don’t tell the whole story, however, mainly because when a person was originally infected or became reinfected during the study, they were no longer “at risk,” and because the groups differ in other, relatively minor ways. Thus, the authors used a statistical model to estimate the incidence rate ratio, which is a better measure of the relative risk of infection faced by a person who had previously been infected with SARS-CoV-2 compared with one who had not.

I think the key finding of the study was that when all possible and probable reinfections were considered, the incidence of reinfection in the group that previously had been infected was reduced by an estimated 84.1% compared with the incidence of primary infection in the group that had not previously been infected.

Recall that the combination of possible and probable reinfections is an upper bound, which means that this 84.1% is a lower bound. (Well, it’s a lower bound of sorts. I have not considered the confidence interval on this lower bound, a statistical measure of reliability, which in this case establishes that the lower bound is most likely between 81% and 87%.)

So, to conclude, the study establishes that protective immunity from prior infection by SARS-CoV-2 is pretty robust, at least for seven months. It has not established how long that immunity lasts. However, we can make some rough calculations if we’re willing to go out on a limb and extrapolate.

## Now, how do we extrapolate?

First, we know that not all people who are infected with SARS-CoV-2 mount an effective antibody response. The fraction that fails to mount such a response is probably around 5% or 10%. I assume that this fraction of the previously infected cohort never actually developed any protective immunity in the first place. This is an idealization. Such people may have some partial immunity conferred by other immune mechanisms, but it’s reasonable as an approximation.

Second, we assume that those people who did develop a protective immunity lost it at a constant rate. (This compares with, for instance, a situation where everyone who develops antibodies has complete immunity for a specific period of time and then loses it all at once.) There’s presently very little information about this part of the process, but again the assumption is reasonable for a first approximation.

Given these assumptions, the *rate of waning* can be calculated from the formula:

rate of waning = ln(protective effect/(1-fraction that fail))/-study duration

From this study, the protective effect is 0.841, the fraction that fails is around 0.05 or 0.10, and the study duration is 207 days, which gives a waning rate of 0.000328 to 0.000589. Taking the inverse gives a theoretical average time until immunity is lost of 1699 to 3053 days or 4.7 to 8.4 years. Interestingly, this appears to be about double that of other coronaviruses that circulate in people.

So all in all, this is good news. A long duration of natural immunity means that vaccination programs do not need to be sustained as intensely to drive down transmission, and lends some added flexibility to future planning and long-term management strategies for Covid-19.