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.

• Josh says:

It doesn’t really matter that younger people get sick with Covid, they recover well. What matters is that <10% of the population is prone to bad outcomes and easily identifiable. Vaccinate those people and the the death counts and hospitalization rates will drop below that of the flu and we won't have a health emergency any more, regardless of where we are with respect to herd immunity. That may only take a couple months.
The vaccination campaign should be conducted in such a manner that harm is minimized, which is a bit different from the fastest path to total victory. I suggest breaking the population into these two groups, then re-run the numbers with different vaccination rates between them, while also calculating deaths and hospitalizations. That will allow a strategy to be chosen that saves the most lives, and probably help in convincing your grandma to get the vaccine!

• Alex says:

Given the exponentially increasing number of new daily cases, isn’t propagating a 1.2% monthly infection rate forward an extreme underestimate?

• Sven Grenander says:

How do the unvaccinated children factor into your model ?

(Great write-up by the way)

• Glen du Plessis says:

It is really getting on my nerves that we have to be slaves to some model or the other. As far as I can gather this is really a load of crap, testing proves nothing. All it does is create fear and panic. Please view the work of Ivor Cumming, and can we get real and stop this. People will die, accept it, now or later, regardless of all the childish low IQ reasons for making money, and destroying peoples dignity and lives. Strangely people have not realised that vaccine failure rates will leave it’s mark, as an additional excessive cause of death, on top of the so-called pandemic death rate + suicides + child abandonment + a spike in divorce. I am nobodies fool. Check the death rate in any country over several years. None of this will make a difference, look at the numbers.

• James MacDonald says:

The calculations assume that the %’s of the population getting vaccinated in the low, medium, and high confidence scenarios don’t already have the antibodies.

If it’s factored in that an increasing number of those being vaccinated will already have the antibodies the dates when herd immunity is reached under each scenario get pushed out.

• Cana says:

This was very informative and everything I wanted to know. Thank you!

• J. Chapman says:

At 77, with Addison’s among my underlying issues and isolated since March 9, 2020, I’m more than ready for the vaccine, but late January looks optimistic given the slow rollout and disjointed communication at every level, in particular Warp Speed’s snail’s pace disjointed coordination with Pfizer which says it has millions of doses in its warehouses waiting for the USG TO ACT – and production is operating without a hitch – GOVERNMENT lies to the contrary.

Our government’s actions have undermined our ability to deal with COVID-19 and the hangover effect will last for months if not years.

• Joe Collins says:

Right dose right person right time…totally individually determined for actual need. That is science