The real meaning of R in pandemics

Why tiny changes in behavior can have a huge effect

If you’ve been following the news about the coronavirus pandemic, you’ve probably heard about the reproduction rate R.

For a particular region and time, R is defined as the average number of people infected by each infected person. If this definition sounds a little circular, that’s because an epidemic is something circular — a vicious loop.

Put simply, if R is above 1.0 then the number of new cases will grow over time, since each infected person causes more than one other person to be infected. But if R is below 1.0, then the epidemic in that region will shrink and then end.

R does not capture all of an epidemic’s dynamics, such as clustering and random luck. But it does reflect its essence, and can tell us a lot.

The problem with R is that it’s too abstract. It’s not immediately clear how a particular value of R will affect us. How bad is R=1.2, R=1.5 or even R=2.5? What do these numbers mean for our economy and society?

In this article, I want to explain how the value of R determines three other quantities which are more concrete:

1. The growth rate in new detected cases.
2. The amount of time we need to spend under lockdown.
3. The number of infections or vaccinations that will end the epidemic.

My goal is to show how small differences in R have a surprisingly large effect on these numbers. For as long as the epidemic is ongoing, tiny changes in behavior can make a huge difference to our lives.

Growth Rate

The effect of R>1 is exactly like money with compound interest. Given a good interest rate, a small investment can yield impressive returns over time. This is because, each year, the interest is added to last year’s balance, and next year’s interest is paid on the new combined total.

For example, if you deposit $100 in the bank at the age of 20 and can (somehow) earn 20% interest per year, it will be worth $2,264,480 by the time you’re 75. You’ve turned one skipped concert into a decade of luxury cruises.

A 20% interest rate is equivalent to R=1.2. But unlike money in the bank, COVID-19 doesn’t compound on a yearly cycle. Instead, there’s an average of 4 days (called the “serial interval”) from the moment one person is infected, to when they pass it on. This usually happens before symptoms appear.

To calculate how quickly the number of cases will grow in a week, start by working out how many serial intervals occur during that time. Since a week contains 7 days, that’s 1.75 four-day intervals. There’s no problem with decimal points here, because we’re averaging over a large population, and every carrier will be at a different point in the cycle.

Now we have to compound R accordingly. If for example R=1.2, we calculate 1.2 to the power of 1.75, which comes out to around 1.38. This means that, every week, the number of cases will be multiplied 1.38x compared to the week before, i.e. it will grow by 38%. So if we’re at 100 cases/day at the start of the week, we’ll be at 138 cases/day by the end of it.

Now let’s look at some longer time frames. The average month contains 30.4 days which is equivalent to 7.6 four-day intervals. If R=1.2, we raise 1.2 to the power of 7.6, which gives almost exactly 4. So the number of cases will increase 4x per month. At the end of one month, our 100 cases/day become 400/day, then 1,600/day after two months and 6,400/day after three. And after just four months of this compound growth, we’ve reached 25,600 cases/day and a small country’s health service collapses.

But here’s the weird thing. Even though R is compounded exponentially, the value of R itself depends on our behavior in a completely linear way. Let’s say we all acted that tiny bit more carefully to prevent one out of every twenty infections (perhaps by keeping our masks over our noses). This would reduce the reproduction rate by 5%, from R=1.2 to R=1.14, which doesn’t seem like much. But actually the difference is huge. If we do the same calculation as before, the number of cases only multiplies 2.7x per month instead of 4x. And after four months we’d reach 5,300 cases/day instead of 25,600. That’s almost five times fewer, for a minuscule change in behavior.

But what if we all act a little less safely, and increased the infectiousness of the disease by 5%. This would increase R from 1.2 to 1.26, which again seems like it shouldn’t matter much. But if we do the math of compounding, we see it leads to an increase of 5.8x per month, or over 110,000 cases/day after four months. In other words, a total catastrophe.

Time under lockdown

We have seen that tiny differences in R have a huge effect on the speed at which an epidemic grows. But no matter what its exact value, if R stays above 1.0, we will need to be in lockdown some of the time. Otherwise, the number of cases will continue to grow relentlessly, until the capacity of the health system is reached.

During a lockdown, R drops below 1.0. In fact, we’ve learned from many countries’ experience that the number of cases during lockdown approximately halves from one week to the next, the equivalent of R=0.67. If we consider this rate as fixed, then the R outside of lockdowns will determine how much of the time we need to be inside lockdowns to keep the epidemic under control.

Let’s take a simple example. If the number of cases doubled each week outside of lockdown (R=1.49), then we’d need one week in lockdown to compensate for each week outside. This would keep the overall number of cases stable, since one week of halving exactly reverses one week of doubling. And it wouldn’t actually matter if we locked down for one week out of two, or one month out of two — the epidemic would still be in overall equilibrium.

Given other values of R, calculating the proportion of time required under lockdown is a little more complicated and explained in the notes at the end. But the bottom line is that we’d need to spend (ln R) / (0.4 + ln R) of the time in lockdown, where ln means “natural logarithm”. We can multiply this result by 100 to turn it into a percentage.

To follow our original examples, if R=1.2 outside of lockdowns, then this calculation comes out to 0.31, meaning 9.5 days of lockdown per month. But if R=1.14 instead, this is reduced to 7.5 days per month. That’s 21% less social and economic damage from just a 5% reduction in infectious behavior. And if R=1.26, then we’d need a little over 11 days per month, or 17% more time in lockdown because of a 5% increase in contagion.

One last comment about this. If we do need to be in lockdown some of the time, why not schedule it instead of waiting for things to go wrong? For example, a country with R=1.2 could lock down regularly for one out of every three weeks. Knowing this in advance would allow families, schools and businesses to plan properly, and reduce the psychological impact.

Herd immunity

In our discussion of R so far, we’ve ignored the possibility of immunity, the fact that some people can become protected from infection. During an epidemic, this immunity will naturally grow, as most of those people who recover from the disease cannot be infected again, at least for a while. (It’s too early to tell how long immunity lasts for COVID-19, and there have been rare but documented cases of second infections.)

Aside from this natural process, immunity from a disease can also be achieved through vaccination. While no vaccine has yet been approved for mass deployment, dozens are in various stages of testing. So it’s reasonable to hope that, during the next year, one or more will successfully cross the line.

Whether immunity is obtained through infection or vaccination, it has the effect of reducing contagion. In epidemiology, R0 (that’s R-zero) is defined as the “basic reproduction number” — the average number of people infected by each carrier in a population with zero immunity. The value of R0 is set by our behavior and the characteristics of the virus itself. But if some people do have immunity then the effective reproduction number (R) will be lower.

How do we calculate R from R0? Let’s imagine that a random one in ten people in a population is immune, due to a previous infection or vaccination. Out of the R0 average infections that each carrier would have caused, a random 10% will not now happen. This means that R is 10% lower than R0, or R0 multiplied by 0.9. For example, if R0=1.3 then R=1.17.

Now, “herd immunity” is defined as the proportion of people that need to be immune in order to bring R below 1 and so end the epidemic. Assuming these people are randomly distributed in a population, this proportion is calculated as 1-(1/R0), as explained in the notes at the end. This number can be multiplied by 100 to convert it into a percentage.

By applying this calculation, if R0=1.3, then herd immunity is reached with 23.1% of the population, or 231 out of every thousand people. But as before, this result is very sensitive to the exact value of R0. If we behave that little bit better, and prevent one in twenty infections, then R0 would drop to 1.235 and herd immunity would require 190 instead of 231 out of every thousand — that’s 18% fewer infections or vaccinations from a 5% reduction in infectious behavior. And if we pushed R0 up by 5% to 1.365, then 267 out of every thousand would need to be immune, a 16% jump in the wrong direction.

Summary

I didn’t choose R=1.2 and R0=1.3 by chance. If we look at those countries now experiencing a second wave of the coronavirus, these are the approximate values they’re seeing. Even with all of the hand washing, face covering and social distancing, each person with COVID-19 is still infecting around 1.2 additional people. And with studies showing that just 5–10% of the population in these countries have antibodies, we can infer that their R0 is approximately 1.3. All this is pretty bad news.

But there is a silver lining: Tiny changes in our behavior can yield a huge reduction in the epidemic’s social and economic damage. To summarize what we’ve already learned, if we start with R=1.2 and R0=1.3, and then act to prevent just one in twenty infections, we’ll have:

• 79% fewer cases per day after four months without lockdown.
• 21% less time under lockdown to keep the epidemic under control.
• 18% fewer infections or vaccinations required for herd immunity.

And if we really make an effort and prevent one in ten infections:

• 96% fewer cases per day after four months.
• 48% less time under lockdown.
• 37% fewer infections or vaccinations for herd immunity.

To me, at least, that looks like a great return on investment.

Notes

• To keep the math simple, I assumed that R remains constant when calculating the case growth and lockdown time. In reality, even with a fixed set of behaviors, R will change due to the weather, public holidays and gradually increasing immunity.
• I worked with a serial interval of 4 days, but estimates from studies vary. The true serial interval will affect the case growth rate and lockdown time, but not the threshold for herd immunity. For any realistic serial interval, differences in R still have a huge effect.
• At the very start of the pandemic, before social distancing, R was around 2.5 in many countries. This means that approximately 60% of their population will need to be immune in order to return to normal life.
• Deriving the time under lockdown: Let p be the proportion of time under lockdown, so (1-p) is the proportion of time outside. This means that, for every week of lockdown, we have (1-p)/p weeks outside. Given a serial interval of 4 days, the number of cases will increase by R^(7/4*(1-p)/p) during these weeks. To balance the halving during the lockdown week, this must be equal to 2. So:

R ^ (7/4 * (1-p) / p) = 2
7/4 * (1-p) / p * (ln R) = ln 2          [take natural logarithm]
(1-p) / p * (ln R) = 4/7 * (ln 2)        [multiply by 4/7]
(1-p) / p * (ln R) = 0.396               [calculate right side]
(1-p) * (ln R) = 0.396p                  [multiply by p]
(ln R) - p*(ln R) = 0.396p               [expand left side]
ln R = 0.396p + p*(ln R)                 [add p*(ln R)]
ln R = p * (0.396 + ln R)                [simplify right side]
ln(R) / (0.396 + ln R) = p               [divide by (0.396 + ln R)]
p = (ln R) / (0.4 + ln R)                [switch sides and round]

• Deriving the herd immunity rate: If the proportion of immune people in the population is q and randomly distributed, we have already shown that R=R0*(1-q). To end the epidemic we need R≤1, so:

R0 * (1-q) ≤ 1
1-q ≤ 1 / R0                             [divide by R0]
-q ≤ (1 / R0) - 1                        [subtract 1]
q ≥ 1 - (1 / R0)                         [negate both sides]

• In reality, the immunity in a population will not be randomly distributed – there will be more infections and immunity in higher risk groups.