What does the Delta variant mean for you? It’s time to put back on your mask!

Ellie Murray
6 min readAug 4, 2021

The CDC’s recent announcement that the Delta variant has an average reproductive number of 6–9 has a lot of people concerned & many are asking me about whether this means we need COVID boosters.

Let’s talk through the math of how we might decide!

First, we need to make sure everyone is on the same page about what the average “reproductive number” is. This term means the average number of new cases that will be infected by a single infectious individual.

When we’re talking about a population that is all susceptible, we use R0 (pronounced R-naught, like we’re fancy & British) or the basic reproductive number.

When we’re talking about a pop with mixed susceptibility & immunity, we use R or Re for effective reproduction number.

Regardless of whether we’re talking about R0 or Re, if this number is bigger than 1 any infections will lead to an outbreak that grows & grows until we stop it or it runs out of people to infect.

A R0 or Re number less than 1 means an infection either doesnt cause an outbreak or, if it does, the outbreak fades away on it’s own.

A major goal of public health is to find any viruses with an Re or R0 more than 1 and intervene until the Re is less than 1.

Now that we know what these mean, we can use them for decision-making.

The next epidemiology lesson you need is that whenever we want to make a decision we need a “base case”: we can’t decide what to do unless we know what would happen under our typical default action!

In the case of COVID there are two base cases we might want to consider.

First, how well did COVID originally spread before all the variants & before vaccines?

This base case #1 will give us intuition about the pandemic because it represents most of what happened LAST year.

The R0 for the non-variant COVID was estimated at around 2–3. Let’s go with 3 so we have a single number for our calculations.

We can see that this number (3) is larger than 1, which is why we had outbreaks.

But all the lockdowns, masking, and similar public health precautions we did over the past year helped to lower Re to close to 1 **while those things were happening** & in some cases even below 1. A huge win! Until we stopped doing them🤦🏼‍♀️

What did vaccines do to Re? Let’s calculate it!

We can get a rough calculation of Re with the following formula:

Re = R0*(1-X*Ve)

where X is the proportion of people vaccinated and Ve is the vaccine efficacy.

(Another way to write this is: Re=R0*S
where S is the proportion of people who are susceptible).

Because it’s a nice easy number, let’s assume all the vaccines have the same efficacy as Pfizer/BioNtech (ie 95%). Let’s also assume (although we know it isn’t true, that there’s no natural immunity or partial immunity — I’ll talk anout how to relax those assumptions at the end.)

For the original non-variant COVID, we find that the impact of the vaccine *if 100% of people got vaccinated* would be:

Re=3*(1–1*.95)=0.15 <<< 1!

Now, we obviously are not at 100% vaccinated. So, what about if we take the current % of fully vaccinated people in the US?

As of Aug 2, that was 49.9%, so let’s say 50% for simplicity.

Re=3*(1–0.5*0.95) = 1.575

That’s still bigger than 1 so we still need other precautions.

Plus, non-variant COVID isn’t the only COVID circulating. So, for our second base case, let’s look at the Delta variant. The CDC says it’s R0 may be 6–9. Let’s choose 8.

Recent observational data suggests Pfizer vaccine efficacy is a bit lower against Delta, let’s say 85%.

What do these numbers give us for the Re of Delta in the US as of Aug 2?

Re=8(1-.5*.85) = 4.6 >>>>>>1!

That’s definitely lower than 8 but wow that’s high! Higher even than the R0 of non-variant COVID back at the start of the pandemic!

This is why we all need masks again!!

Now let’s think about some interventions we could do.

A) What if we increase vaccination rates to 75%? That’s a big lift but not impossible.

B) What if we gave *all vaccinated people* a booster? Let’s assume the booster restores the original 95% efficacy.

For base case #1 (non-variant covid),
Option A gives us: Re=3(1-.75*.95)=0.86 <1!!
That’s less than 1 & means we COULD rely on vaccination alone! What a win!

(Option B doesn’t change our base case because the non-variant VE was always 95%.)

So we can see that *if we were dealing with non-variant COVID only*, then the best course of action would be to get the total number of people vaccinated up and then we could stop with all the masking & closers & everything!

What about Delta?

Option A (more people vaccinated) gives us:
Wow! That’s still bigger than 1, so we still need masks & other precautions, but MUCH much better than 8!!

Option B (boosters) gives us: Re=8(1-.5*.95)=4.2
<sad trombone> wow, underwhelming!
Giving EVERY vax person a booster that gets vaccine efficacy up to 95% (a fantastically high number) will only reduce Delta’s Re from 4.6 to 4.2! That’s still HIGHER than non-variant COVID!

From these calculations it seems pretty clear that vaccinating 75% of people (ie half as many again as we’ve done so far) is better than getting boosters to the original 50% of people who have been fully vaccinated.

But NO MATTER which we choose we still need other precautions!

What about if we aim a little lower? How many *new* people would need to be fully vaccinated to achieve the SAME benefit as giving all the vaccinated a booster? For this, we need to do a little algebra:
The answer: 56% of people, or 6 percentage points more than are currently vaccinated!

So how feasible is it to increase the percent fully vaccinated in the US from 50% to 56%? Well, as of Aug 2, 58% were at least *partially* vaccinated!

That means that by ensuring everyone who has started the vaccine series gets their second dose, we can get as much protection as giving EVERY vaccinated person an additional booster dose.

To me, that seems like a no brainer!

The calculations above are simplifications, because I’m ignoring the partial immunity people who are partially vaccinated have, as well as any natural immunity in people who are infected. I’m also not including the benefits of public health interventions like masking, which we know reduce susceptibility.

Part of the reason I’ve left these out is so the equation is simple, but another reason is that we really do not know that much about partial or natural immunity.

But if you had some guesses for those other immunity values, and you want to try to incorporate them, you can extend the equation above as follows:

Re = R0(1-(X*Ve + Z*Vn + A*Vp))

Here, Z is the percent of people who have been infected recently enough to still have natural immunity (although we don’t quite know how long that is), Vn is how well natural immunity prevents future infection on average, A is the percent of people partially vaccinated, and Vp is the vaccine efficacy of partial vaccination. This is still a simplification because it assumes these groups are mutually exclusive, but it’s less of a simplification than before.

No matter which equation you use, if you run the numbers, you’ll find that getting vaccines to unvaccinated people is pretty much ALWAYS a better move than giving boosters to the already vaccinated. At least until we’ve gotten almost everyone vaccinated.

So, you can decide, but for me, I want vaccines for all!

We can also see that if Delta is really this bad, we’ll need masks & other precautions for a long time to come.

That’s a real bummer & what’s worse, we could have avoided this by doing a better job controlling infections last year.

Let’s not make the same mistakes twice!

UPDATE: A few people have asked how this all compares to the flu. The R0 value for influenza is between 1 and 2. So you can see now why when we did all those precautions against respiratory viruses last year the flu just about went away completely!

Another common question is what individuals should choose for themselves. If you are at average risk, it’s better for you to convince your friends & family & neighbors to get vaccinated than to get a booster yourself. The only scenario where boosters might be more important is for people who have specific immunocompromising health conditions — if they know they didn’t respond very well to the first 2 doses, it’s possible they may benefit from a 3rd dose. But that’s not true for most of us.

This post began life as a tweetorial, or twitter tutorial, full of amusing gifs. To read the original tweetorial, click here.



Ellie Murray

Assistant Professor of Epidemiology at Boston University School of Public Health. Follow for causal inference, epidemiology, & data science. Twitter: @epiellie