The Exponential Power of Now

Working in Excel, she composed a graph showing the power of one case averted now, today — versus one case averted a week from now — in an effort to convey to people what she called “the positive side of exponential growth.”

Using the current case counts from the U.S. as of Friday morning, she calculated what the epidemic would look like if cases grew by 30 percent every day for a month — they have been growing by 30 to 40 percent every day for a week — then looked at what would happen if just one of those infections were prevented tomorrow instead of a week from now.

The following is an edited version of my phone and email conversations with Dr. Britta Jewell, in London, and Dr. Nick Jewell, who for the moment is staying put in Berkeley, Calif.

What is happening in this graph?

BJ The graph illustrates the results of a thought experiment. It assumes constant 30 percent growth throughout the next month in an epidemic like the one in the U.S. right now, and compares the results of stopping one infection today — by actions such as shifting to online classes, canceling of large events and imposing travel restrictions — versus taking the same action one week from today

The difference is stark. If you act today, you will have averted four times as many infections in the next month: roughly 2,400 averted infections, versus just 600 if you wait one week. That’s the power of averting just one infection, and obviously we would like to avert more than one.

The principle is that, with the exponential growth phase of an epidemic, individual and institutional actions such as social distancing taken early on can have a much greater impact than if the same actions are taken even a week later.

NJ The magnitude of the outbreak creeps up on you; it doesn’t look like things are growing very much, and then suddenly they are. Today, the U.S. is up to at least 1,714 known cases and we’re only a couple of days on from when it was 1,004. It’s going to be 4,000 by Monday, and then it’s going to be 8,000 by next Wednesday, and then it’s…. Exponential growth is staggering when it takes over.

In a nutshell, what is exponential growth?

BJ Exponential growth is a classic pattern in which numbers stay small initially, but then you end up with very large numbers very quickly. If you start with a certain number, and then multiply that number by a growth factor every day, depending on what that growth rate is, you’ll see the cumulative number doubling over a certain time period.

What really matters is how high that growth rate is. In the U.S. right now, according to Our World in Data, confirmed Covid-19 cases are increasing by about 30 to 40 percent per day and the total number is doubling about every two days.

NJ Think about your family tree and how the number of your ancestors (or descendants) grows with every generation. Or there’s the story about the rabbits breeding. Two rabbits breed four rabbits and four rabbits breed eight rabbits and eight rabbits breed 16 rabbits, and if they’re breeding every six or seven days, very soon you have a lot of rabbits.

BJ If you start with two rabbits and the number doubles every week, you’ve got about 1,000 rabbits after 10 weeks. That doesn’t seem so bad. But another 10 weeks later? You’ve got a million. It’s intuitively very hard to grasp how quickly these numbers go up beyond a certain point — people tend to anchor on the low numbers at the beginning, when the curve is relatively flat.

But the same exponential effect is equally powerful with mitigation efforts — staying home now, for example. How do the “exponential now” decisions play out later down the line?

BJ Part of the advantage of an individual action removing a single infection is not necessarily that the person I don’t infect tomorrow will never be infected, but that the infection will happen later than tomorrow, maybe in a few weeks or so.

And at some point at least some of the newly infected people will be surrounded by more immune (recovered) individuals among their contacts, so the newly infected will in turn infect fewer people, and so on.

Effectively, each of us can choose to reduce our personal R0 — the number of people each one of us would go on to infect if we developed Covid-19 — and this will help flatten the curve. If you can drive the effective R0 below 1, you can slow down and stop the spread.

According to analysis from the London School of Hygiene and Tropical Medicine’s center for mathematical modeling, it looks like Hong Kong was able to do that in February, and their epidemic trajectory looks quite different as a result. The effective reproduction number isn’t set in stone — it’s vulnerable to interventions.

What is your top research priority as you look forward?

NJ We’re trying to model and project health care demands over the immediate future, looking three weeks ahead: given our best guess of where we might be with cases in three weeks, and the likely age groups, how many hospital beds will be needed, how many I.C.U. beds, how many respirators. Health care providers are feverishly trying to plan how much they need to ramp up their capacity to meet the anticipated demand as best they can, to avoid being swamped.

BJ Not overwhelming the health care system is certainly the most pressing issue right now, and hopefully we can achieve this if we flatten the curve. But long term, I’m concerned about keeping social distancing sustainable. It’s a fine line to walk, because if you actually do slow down the epidemic substantially, it will look like everyone overreacted. It may be hard to convince people that at least some social distancing measures need to stay in place — like not shaking hands, or avoiding large crowds if you’re vulnerable — after the initial peak of the epidemic.

A lot of people seem to be asking: Is it really likely to get as bad in the United States and Canada as it is in Italy?

NJ Yes. Tell me one reason it shouldn’t. I see no reason. The United States are maybe two weeks behind Italy. I think everyone got a shock this week, and there was the sense of a shift, with the N.B.A. canceling and the president speaking from the Oval Office with the travel ban, and then Tom Hanks and Rita Wilson — bang, bang, bang.

I think next week we are going to get an even bigger shock. Because we are going to have, as I said, 5,000 to 10,000 cases. People are going to realize it is everywhere. I can’t go to the store, I can’t take public transportation, I’m surrounded by infected people. We’re not far away from that. Unless we do something — and the sooner we do it the better.

That is scary.

NJ It is very scary.

Look at the final-size equations that people are throwing around. The final size — the total number of people who get infected — is related to the rate of the exponential growth, the reproductive number. Flattening the curve is an attempt to get that reproductive number down. It makes for a longer epidemic, but less intense.

If you have an R0 of 2 — two more cases on average for every person infected — the math tells you that if it is not impeded in any way, that fire burns through the forest, and in the end it will have burned half of the forest — half of the world population or half of the U.S. population will be infected. Angela Merkel said that, worst-case or eventually, she anticipates that 70 percent of Germans will be infected.

Then it will die out, if we have lasting immunity as a herd. The virus can’t find new, susceptible hosts very easily and it dies out. This is just mathematics. The only hope we have is to lower the reproduction number. And the only way we know how to do that, absent a vaccine, is to contact trace with testing, find infected cases and isolate them. And social distancing — that’s where you reduce the contact rate between infected and susceptible people, you make it harder for an infected to bump into a susceptible. That’s our only hope at the moment. And I don’t know that that message has really gotten out. I think we’re getting there, but I’m not sure.

BJ A big question now is how long we can maintain these measures, if we choose to take them either as individuals or governments.

If we’re not expecting a vaccine for 12 to 18 months, we also need to find a way — at least after the initial first peak of the epidemic — to make sustainable changes. If social distancing measures keep a large proportion of the population susceptible now, that means those individuals may still be infected in a second, or third, wave of the epidemic if life goes back to normal.

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