On a recent Skype call with my grandmother, I broached the topic of the fast-arriving Covid-19 vaccines.

Advanced age brings wisdom, but it also brings an elevated risk of severe illness from infection with the coronavirus, so I wanted to prime her to get an FDA-approved vaccine as early as possible.

But as I was extolling the benefits of vaccination, I noticed a furrowed brow, a frown, and a look of uncertainty on her face. That took me by surprise. Surely someone who gets a flu shot every year and who raised two doctors shouldn’t feel anxious to get the Covid jab. But she clearly was.

That conversation left me worried not only about her safety but about the safety of our country. If my Grandma was feeling hesitant, millions of other Americans are probably feeling the same way. A nagging question popped into my mind: If vaccine distrust plagues our country, when will we be able to achieve herd immunity and transition to a new normal?

Herd immunity occurs when a critical mass of people become immune to a pathogen like SARS-CoV-2, the virus that causes Covid-19. With enough people immune to the virus, the chain of transmission is halted, which provides indirect protection to individuals who aren’t immune.

Inspired by Dr. Jacob E. Jones, a family medicine physician at my hospital who made some predictions on the time required to achieve herd immunity based on vaccine adoption, I set out to answer my question with a model that uses the following variables and definitions:

The basic reproduction number R0 (pronounced R-naught). This number represents how infectious a pathogen is. An R0 of 2 means one individual infected with SARS-CoV-2 is likely to infect two other people. Currently, most estimates of R0 are between 2.5 and 4. For the sake of this thought experiment, I assumed an R0 of 4.

Base prevalence. This is the percentage of people immune to the virus at a given moment in time, either from acquired infection or vaccination.

Monthly infection rate. This is the percentage of people who become infected and acquire immunity to the virus every month.

Using just the basic reproduction number, it’s possible to calculate the percentage of people needed to achieve herd immunity:

If R0 is 4, then 75% of the population needs to acquire immunity to the virus in order to halt transmission.

In late September, a Stanford study estimated that 9.3% of Americans have antibodies against SARS-CoV-2. To be sure, antibody testing may suffer from low positive predictive value when the prevalence of infection is low, but this is the best estimate we have so far. I’ll use that as the base prevalence.

If the base prevalence at the end of September — eight months from the onset of the epidemic in the United States on January 21, 2020 — was 9.3%, the coronavirus has an infection rate of approximately 1.2% of the population per month. This back-of-the-envelope calculation is in line with estimates from the medical literature, with one study estimating 52.9 million infections in the U.S. from February 27 to September 30, or an infection rate of 1.3% per month.

Using the herd immunity threshold, the base prevalence, and the monthly infection rate, it’s possible to calculate the number of months (m) to achieve herd immunity:

If the virus is left to spread at its current rate with no vaccine, it would take 55 months from October 2020 to achieve herd immunity. That means May 2025. Even if I had assumed an R0 of 3, it would still take 48 months to reach herd immunity.

This white-knuckle approach would consist of several years of misery, morbidity, and mortality, not to mention continued economic hardship.

Through a miraculous feat of science, we have a shortcut to herd immunity: two vaccines with over 90% efficacy, and possibly more on the way. Infectious disease expert Dr. Anthony Fauci has predicted that Covid-19 vaccines, one of which is being rolled out this week to frontline health care workers, should become widely available to the public by April 2021.

For the sake of simplicity, I’ll ignore the phased rollout and imagine that all Americans have the choice to get vaccinated this April. Since both vaccine candidates are a two-shot series separated by three to four weeks, it may take at least an additional month to gain full immunity. At that point, in May 2021, the base prevalence of infection will be 17.7% (1.2% per month from October 2020, when the base prevalence was 9.3%).

We’re almost there: Calling the percentage of Americans who get the vaccine Pv and considering that 90% of people who get vaccinated will develop immunity, here’s an equation for an estimate of time to herd immunity starting from May 2021, using 39% as Pv — percentage of Americans who said they would get a vaccine in an NBC News poll:

With just 39% of Americans getting a government-approved vaccine, the time to herd immunity is 19 months, meaning December 2022.

Nineteen months is no walk in the park. If 17.7% of Americans are already immune by May, and another 39% would readily get a Covid-19 vaccine, that leaves about 43% of the population who are vulnerable to infection but skeptical of vaccination.

Convincing roughly half of these skeptical Americans to take the shot, boosting Pv to 60.7%, shortens the time to herd immunity to two months, meaning July 2021.

Convincing just a few more, hitting a critical mass of 63.7%, would let us achieve herd immunity as soon as the second wave of vaccinations has been completed.

 Scenario Herd immunity achieved No vaccine May 2025 Low confidence in a vaccine (39% get vaccinated by May 2021) December 2022 Medium confidence in a vaccine (61% get vaccinated by May 2021) July 2021 High confidence in a vaccine (64% get vaccinated by May 2021) May 2021

This model, like every model, makes some assumptions. One is that the coronavirus does not mutate. If it behaves like influenza and other coronaviruses and starts accumulating minor mutations (a phenomenon called antigenic drift), then the virus could require yearly immunizations to cover new strains, similar to the flu. This scenario would likely lead to an endemic (baseline) level of infection in the population.

Another assumption is that coronavirus antibodies are long lasting. If antibodies from natural infection or immunization last only six or 12 months, booster shots will be required to maintain immunity.

Lastly, these scenarios assume an ideal circumstance where everyone vulnerable to disease is identified and offered the vaccine at the same time.

The goal of this thought experiment is to highlight the immense power of vaccines and the impetus to immunize. In the era of misinformation, it may not be sufficient to debunk myths and simply provide people with facts. Traditional methods of vaccine promotion may have paradoxical effects.

A large study examined the effects of medical professionals trying to promote the measles, mumps, and rubella (MMR) vaccine for children among vaccine-skeptical parents. Educational interventions, from debunking myths (such as the link between MMR and autism), to teaching about the dangers of measles, mumps, and rubella all made anti-vaccine parents less likely to vaccinate their children.

Meanwhile, after enacting stricter vaccine laws following a 2014 measles outbreak, California’s MMR vaccine rates rebounded to the critical herd immunity threshold.

There is an arsenal of robust vaccine policies that can fast-track us to herd immunity: financial incentives, limits on personal or philosophical exemptions, and compulsory requirements for businesses and schools (once a vaccine for children is approved).

My Grandma was less moved by my equations and statistics than by my sustained concern for her health. After weeks of nudging her in the right direction, she is now more open to the shot, but still hasn’t fully decided if she will get it when her turn comes.

Not all vaccine skeptics have a physician in the family to guide them. Most don’t personally know someone who works inside a hospital who can describe the horror and devastation this virus has wrought.

To defeat this virus quickly, we need to embrace policy rather than persuasion.

Zach Nayer is a transitional year resident physician at Riverside Regional Medical Center in Newport News, Va., and an incoming ophthalmology resident at Harkness Eye Institute at Columbia University in New York City.

• Hal says:

All models are flawed, some models are useful. So I thank the author for providing a useful if (by definition) flawed model.

Nearly everyone I talk to is (a) looking forward to being vaccinated and (b) in no hurry to do so. Doctors I’ve talked to report that is what they are hearing from patients as well. What we all want to see is a few months of experience by millions of others so side effects that weren’t uncovered by trials (or whose probability of occurrence was misestimated) are found and understood. At that point the public attitude will change and vaccine uptake will grow significantly. So the main problem with the numbers here are that while we may eventually get high confidence in the vaccine, that won’t kick in until late Spring shifting herd immunity to the right. September 2021 seems like a more reasonable prediction than either May or July.

Vaccine uptake may be further delayed by expert/government statements that despite the vaccine rollout we will be subject to the same life restrictions as if there was no vaccine. If obtaining immunity brings no benefit to oneself in the short to medium term, that alters the risk/reward calculation. If you are in the low risk of severe illness group, i.e. most of the population, then why rush for a vaccine that still has you unable to dine in, go to concerts, go to movies, attend family gatherings, travel, etc.? If we want to goose vaccine uptake we need to provide incentives. And those aren’t “this will all end sometime later in 2021, or maybe 2022, if you get vaccinated”. Tell someone they can go visit their mother in an assisted living facility in a currently travel restricted state once they’ve been vaccinated and they’ll be standing outside a clinic tomorrow demanding the vaccine. Tell them that isn’t going to occur for many months and they’ll wait. And if enough wait, we will be looking at 2022 instead of 2021 for life to get back to normal.

• William E King says:

There are numerous flaws with this analysis. First, HIT must be discounted for the susceptibility heterogeneity, which appears to be high for COVID. Most HIT estimates from epidemiologists are now in the 50-60% range. Second the rate of natural infections is not a straight line. At the end of Sept, when the 52MM estimate was made, the confirmed infection rate was 2.1%. As of yesterday (75 days later) it is 5.1% and we are adding 1MM per week. We will probably be at 100MM+ infections by the end of the year (30% of population). Even if that slows down after the holidays, with even a modest # of vaccinations (say 80MM) – we get to the 50-60% range by late spring.

• Erin M Blum says:

I think the author should have used an exponential growth function to model the disease spread, rather than a steady monthly growth rate. Exponential growth better matches the data of the past 9 months. Starting from 10% of the population already infected and using a 1.2% daily growth rate (based on data for my state, nj), 75% of the population will have had it in about 180 days, so March-April timeframe. The exponential growth model is very sensitive to the assumed growth rate. You can find calculators online that make it easy to test different assumptions. I think this is a major flaw in this article.

• Greg says:

My recollection is that the Stanford study found infection / recoveries in New York’s and in several Sunbelt States we’re already about 25% of the state populations or higher. On Face the Nation last week, Dr. Gottlieb conceded that several States were in the 30% to 50% range and that they may be seeing early signs of herd immunity.

Just looking at the CDC weekly mortality reports for California, every week, the CFRs are at about 0.5% and have been consistently declining every month, just as they have been in almost every state and Nation.

Dr. Gottlieb recently said that only 1 in 8 cases gets confirmed. So, 250,000 cases per day may correspond to 2 million infections per day or 60 million per month. 4 years to achieve herd immunity seems spectacularly long.

By the time the vaccine is available for the general population, covid deaths may have declined to zero in some states (thanks to wise use of the vaccine and to natural immunity). Vaccinating healthy people (hospital workers) seems like it will cost lives.

States that imposed lockdowns while their hospitals were empty have much lower levels of natural immunity. I think the Stanford study concluded that California was at 3 or 4% infected/recovered. They are obviously a lot higher now. States that went cuckoo for lockdowns may face a fourth surge in about April (the surges seem to come every 4 months).

• A. Eaves says:

Considering the side effects of these vaccines it’s almost out ways the desire for getting them. You are left with six in one had and half a dozen in the other.

• Alex says:

Useful but you missed a few ponts:

1. Covid19 is not fatal to more than 1% of the population. About 5% need to be hospitalized. In the rest it behaves just like a bad flu. Hence there is no way to push through with a mandatory policy for taking the vaccine.

2. You are missing a very basic fact about human behavior: nobody wants to go first. Early adopters are always hard to find and rightfully so. Evolution taught us to be careful. But as the number of vaccinated people grows and the number of “dangers” fails to materialize more and more people will take the shot. Voluntarily.

3. As the vaccine becomes easily accessible the governments and the insurance companies will most likely NOT cover any costs unless you can prove to be vaccinated. And faced with the risk of a 10K+ bill most people will take the shot. Only the most foolhardy will still hold out. And they deserve to pay for their foolishness.

• Mike says:

About your point 1: your figures imply up to 3.3 million dead and up to 16.5 million people hospitalized, presumably within the next 12-18 months. I wouldn’t want to live with that kind of carnage on my conscience.

• Tony says:

Exactly, Jon and Mike. Perhaps Zach’s grandmother is wise not to take her medical advice from her kinfolk.

• Arjun says:

I found this article to be useful and informative and I agree with the conclusion, but I am using the simple model with different assumptions. By my calculations using Worldometers it looks like 2% of us are currently testing positive per month, and if the tests are only capturing half of all new cases then 4% of us are getting infected per month. If we are at 20% now then it will take us 10 months to get to 60%. If 8% of us get vaccinated per month then the time to herd immunity will be reduced by 6 months and we could save half a million lives.

• Michelle says:

Your numbers are specifically referencing a percentage of the population that may be testing positive for covid, not the percentage of deaths from covid. Maybe you should try looking at it from that aspect. Also, make sure even those numbers are accurate because there are many deaths blamed on covid but they died from a different cause entirely. Hospitals can’t increase the funds they receive without high numbers of covid deaths.

• David O'Connor says:

This article contains a gigantic blunder in its estimate of the number of Covid-recovered persons in the US, and therefore gives a silly estimate for how long it should take to achieve herd immunity. The Stanford study the author cites for his estimate of 9.3% was based on residue from blood samples of dialysis patients at the end of June. Because dialysis patients have very low social mobility (they have an unemployment rate of around 80%, for example), even at that time, 9.3% was too low for the general population. I have no idea why he plugged this number into his simple model. He cites a recent study that estimates 53 million Covid-recovered persons as of the end of September, so a reasonable estimate now would be closer to 100 million, consistence with estimates of the Infection Fatality Rate (IFR) of about 0.3-0.4% and total deaths of 300,000. The vast majority of Covid-recoverd persons are already immune, so it is a waste of resources to force them to be vaccinated right now.

• Jon goldstein says:

Isn’t Dr Bayer confusing the vaccine’s excellent rate of protection from getting severe covid 19 symptoms with its still unknown capacity to prevent virus transmission?

• Jon goldstein says:

Sorry for typo, I mean Dr Nayer.

• Mike says:

Yes! The calculations are based on a huge unverified assumption, i.e., that vaccination will prevent asymptomatic transmission. Right now we have only one partial data point from the Moderna vaccine, which showed that at the time of the second shot there were 60ish% fewer asymptomatic cases (I don’t have the figures handy) in the vaccinated group than in the placebo group.

The risk is that by vaccination we are creating hundreds of thousands, and later millions, of potential asymptomatic spreaders, who would also be (or feel) invulnerable because of the vaccine, so with far less incentive to wear masks and socially distance.

• David says:

Great point, Jon Goldstein!