Across the United States, the number of coronavirus cases has skyrocketed, leading to calls from public health officials to mask up regardless of vaccination status. Many vaccinated people are asking whether it’s time to get a booster dose. But the math behind the spread of infectious diseases like covid-19 can help us see that it’s not.

The change in coronavirus guidance from the Centers for Disease Control and Prevention and from some local authorities came about because of the delta variant, which epidemiologists estimate has a basic reproductive number between 6 and 9. The basic reproductive number, or R0 (pronounced arr-naught), is a measure of the average number of people directly infected by a single infectious person in a scenario where no one in the population has any immunity to the disease. The original virus that caused covid-19 had an R0 of about 3, meaning that someone infected in April 2020 would, on average, infect three others. The delta variant is two to three times more infectious; on average, one infected person may infect between six and nine people if none of them are vaccinated or had a prior infection.

But the basic reproductive number alone doesn’t tell the whole story. To really understand what we can expect from the delta variant, we need to consider vaccinations, natural immunity and any precautions people are taking to lower their chances of being exposed, such as wearing masks and social distancing. To do that, epidemiologists calculate another measure, called the effective reproduction number, or Re (pronounced arr-eee). The Re helps us estimate how much a disease may spread when a population has at least some immunity. Technically speaking, the Re measures how many new people a single person infects, accounting for whatever precautions people are taking and overall immunity levels. Whether we’re talking about the Re or the R0, any value larger than 1 means trouble, because if each person can infect more than one other person, any disease outbreak will continue to spread exponentially unless we take action.

The Re for a place and time can be calculated from the R0 using a fairly simple equation: Multiply the R0 by the proportion of the population that is susceptible to the disease. This proportion can be expressed as 1 minus the proportion of the population that is immune to the disease, and it can incorporate different types of immunity. In mathematical terms, the equation would be Re=Ro*(1-x*v), where x*v is the proportion of the population that is immune, x is the percentage of people who are fully immune (i.e., vaccinated) and v is the vaccine’s effectiveness.

Understanding this formula will yield important intuition on why boosters aren’t yet helpful, so let’s walk through a simplified example supposing that everyone in the population was vaccinated.

Calculating how effective vaccines are against new variants is challenging, but epidemiologists have estimated that the two-dose vaccines are about 85 percent effective against the delta variant. That’ll be our value for v, with one important caveat — for simplicity, we are assuming that all vaccines have the same effectiveness, because we don’t have great data on how the vaccines differ for the delta variant. For demonstration purposes, we’ll use an R0 of 8 (although the range is between 6 and 9, so try out other values and see how things change). Based on these numbers, our Re estimate would be around 1.2 if we were able to vaccinate 100 percent of the population. That’s still higher than 1, but just barely, and it’s definitely within a range where other control measures such as frequent testing and contact tracing would probably be enough to stop any outbreak.

But we don’t expect everyone to be vaccinated. As of Tuesday, about 50 percent of the U.S. population was fully vaccinated. Taking into account the 85 percent efficacy of the vaccines, this means about 43 percent of the population is protected against the delta variant. The Re for this scenario is just over 4. That’s much higher than 1, and it’s even higher than the R0 of the original, non-variant form of the coronavirus. So with half the population in the United States vaccinated, the delta variant is still more infectious than the original virus was last year.

With cases on the rise, some officials and pundits are calling for booster shots for those who are fully vaccinated. The Food and Drug Administration is considering a strategy on boosters, and the Biden administration is pushing for them for some high-risk groups. Israel has already begun delivering booster doses to those older than 60. Epidemiologists, on the other hand, are not convinced this is the right approach. Let’s see what happens to the Re if we give a booster shot to everybody who is fully vaccinated, assuming that the booster raises the effectiveness of the vaccine to 95 percent, the original effectiveness of the mRNA vaccines against the novel form of the virus. With the same 50 percent of the population fully vaccinated, this gives us an Re of around 4 — still higher than the R0 for the virus last year. So giving a booster shot to half the U.S. population would only slightly reduce the Re of the delta variant.

But what if we increased the number of fully vaccinated people in the United States? Suppose we got 75 percent of the population fully vaccinated, which would probably require approving vaccines for children younger than 12 and a significant amount of outreach to increase vaccine uptake. Using the estimate of 85 percent vaccine effectiveness against delta, we calculate an Re of just below 3. That means that by putting in resources to vaccinate another 25 percent of the U.S. population, we can reduce the spread of the delta variant more than if we gave a third shot to the 50 percent of the population that’s already vaccinated. If you play around with these numbers, you will find that getting more people vaccinated is almost always the better strategy for lowering the Re — and therefore, slowing the spread — than giving booster shots.

So far, these estimates have not taken into account people who have natural immunity from previous coronavirus infection or those who have received only one dose of an mRNA vaccine (people who have received one dose of the Johnson & Johnson vaccine are fully vaccinated). Let’s see what happens if we take these into consideration (warning: This includes slightly more math).

About 9 percent of the population has received one, but not two, doses of an mRNA vaccine. According to a study from Britain, one dose of the Pfizer vaccine proved to be 60 to 70 percent effective against the original virus, so let’s suppose for the purpose of this example that one dose is 60 percent effective against the delta variant. With 50 percent of the population fully vaccinated and 9 percent partially vaccinated, we have about 48 percent of the population immune to the delta variant (0.5*0.85 + 0.09*0.6 = 0.48). The R0 value does not change, so our Re estimate taking into account those who are fully and partially vaccinated is about 4.

There is not as much information on how much natural immunity is provided by previous coronavirus infections, or how long it might last, and it is difficult to estimate what proportion of those who have recovered from infection have remained unvaccinated. (It’s also difficult to estimate accurately how many people had asymptomatic infections and never knew it.) The CDC estimates that about 35 million people have been infected; let’s assume that about half of these people are now also fully vaccinated. (The agency also estimated that only 1 in 4 cases have been reported, which would mean more than 120 million were infected, but the only hard data available is its other count, so we’ll use that for the purposes of this math.) This gives us an (imperfect) estimate of about 5 percent of the U.S. population with natural immunity but no vaccination. Let’s suppose that the effectiveness of natural immunity is 30 percent — again, there is very limited information on natural immunity, so this number is purely theoretical. Taking into account natural immunity and those who have at least one dose of the vaccine, this still leaves about 50 percent of the population susceptible to the delta variant. The Re for this scenario is 4. That means that accounting for people who have partial or natural immunity improves our situation a little, but not as much as getting more people fully vaccinated would.

Given this math, why have some countries decided on boosters? Mostly because policy decisions don’t only consider science. In Israel, all coronavirus restrictions were lifted earlier this summer when half the population was vaccinated, but indoor masking was reinstated just 10 days later as cases increased. Since then, most infections have been among adults over age 50, and the booster campaign may be an attempt to control the surge before the Jewish High Holidays in September.

The math of outbreaks helps us to understand that we cannot control the delta variant by maximizing the immunity of only a segment of the population. If you’ve already been vaccinated, your best next step for lowering your risk of a breakthrough infection is to help your friends, family and neighbors decide to get vaccinated themselves. Seeking out a booster won’t help as much, unless you are immunocompromised — in which case, do talk to your doctor.

Vaccinating everyone eligible is a lofty goal, but it’s worth aiming for. Vaccines have been a key tool of public health since the first smallpox vaccine was developed in 1796. Widespread vaccination campaigns have stopped outbreaks of polio, measles, chickenpox and many other infectious diseases. We can make sure covid-19 is next by working to vaccinate everybody, not just in the United States but around the world. As of Aug. 10, just under 16 percent of the world’s population was fully vaccinated. Our global Re is still very high, and it will take global cooperation to bring it below 1, but we have the tools and knowledge to do so. All that’s left is to act.

*This article has been updated since it published to indicate that the CDC has multiple estimates of the number of covid infections in the United States.*

Twitter: @epiellie

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