It’s time to stop looking at exponential graphs (and start testing!)

As someone who makes a hobby of tracking trends and weather, I’ve been following the Covid-19 pandemic across multiple states and countries, and it seems to me like the media and even many of the “experts” have the narrative wrong in the present moment.

We’ve all been spending a lot of time looking at exponential graphs lately.

Credit: Very Finnish Problems, Twitter

And it’s true that as a pandemic begins, it spreads exponentially. The number of people infected by each infected person is known as R, and if R is greater than 1 we will see exponential growth. For Covid-19, in the absence of any interventions, R is approximately 3 and cases double every 3-6 days.

When we talk about flattening the curve, the models that show a prolonged outbreak with a peak some time in May are using a lower R that is still greater than 1, say 1.25 or 1.5.

“Flattening the curve”, Source: New York Times

Importantly, if at any point R is reduced to less than 1, actual cases will peak in a week or so when everyone currently infected becomes symptomatic, and documented cases will peak in 2-3 weeks which is the time required for some fraction of cases to become severe and require medical attention. If R is less than 1, we are not “flattening the curve,” we are arresting the curve, and there is a big difference.

We can get an estimate of what R might be by examining the changes in our social interactions. I made up the values on the graph below, but it should be fairly representative of reality.

It seems clear to me that, under the current stay-at-home order, my social interactions have been reduced by more than a factor of three, and for many people – schoolchildren, office workers, and teachers to name a few – it is probably more than a factor of 10. Thus I would expect to see R reduced to a value lower than 1 given our current state of social distancing.

The first evidence to support the effectiveness of social distancing was a dramatic nationwide decline in fever data, thanks to the Kinsa network of smart thermometers (

Source: Updated as of April 1, 2020, click on link to see current version.

Looking at that graph, we see flu season declining through February, then around March 1 the trend starts to deviate from the expected line, which most likely corresponds to Covid-19 reaching the detection limit. The hardest-hit areas like New York, New Jersey, Florida, and Michigan show a much more significant spike in fevers through the middle of March. Then, suddenly on March 20, roughly 5-9 days after social distancing measures were enacted, the line begins to plummet. Social distancing is breaking the chain of transmission for the flu, and – presumably – also for Covid-19. As of April 1, only 1.14% of Kinsa users had fevers, vs. an expected value of 3.09% following flu season trends.

It has often been said that the US is nine days behind Italy. So let’s look at a graph of Italy’s outbreak. I should note that the media seems to prefer graphs of cumulative case counts, which can be more difficult to interpret since the numbers are always increasing. The graph below shows new cases per day.


Italy imposed a nationwide lockdown on March 9, and true exponential growth ended a few days later with new cases reaching a peak on March 21. Generalizing this curve, it looks something like the one below, with the length of lockdown between 5 and 8 weeks – provided that we have measures in place to keep R below one while relaxing social distancing, which I’ll get to in a minute.

Cases per day, generic curve following social distancing “lockdown.” Note that the drop is more gradual than the rise; this is assuming that we can’t push R lower than 0.5 or so with social distancing while maintaining essential services. Figure of my own creation.

Daily case data for all states and countries can be visualized at this site. The data are fairly messy, and case numbers are certainly undercounted and confounded by insufficient and gradually increasing testing. Nevertheless, it is possible to get a sense of where we are at on the curve.

Daily case count progression, by state

States that imposed social distancing earlier are closer to the top, or just past it, while states that imposed restrictions later are still seeing daily increases (but mostly not exponential increases at this point). A model released yesterday by the Washington state-based Institute for Disease Modeling shows Oregon at or just past the top of our curve.


What happens next?

At the moment there is a growing disparity between the headlines – based on models from several weeks ago and still discussing exponential growth – and the data which indicate that we are likely approaching peak illness. That difference will be resolved in the next week one way or the other, but I hope that my analysis provides a ray of optimism amidst an ocean of bad news.

As cases begin to decline, it will be important to maintain social distancing, since that is the only thing saving us from a return to uncontained exponential growth and the millions of deaths (and many millions more harrowing two-week illnesses) predicted in that scenario.

At the same time, as the number of infections decreases, it becomes increasingly difficult to justify the economic disruption imposed by social distancing, and increasingly important to find alternative ways to keep R below 1.

Right now, we are looking at this pie:

In most areas 1% or less of the population is currently spreading SARS-CoV-2 (higher in a few hotspots), but we are all staying home because we don’t know if we’re part of that 1% or if the people we encounter are part of that 1%, since up to half of infections may be asymptomatic.

In order to begin to relax social distancing measures, we need to be able to see this pie instead:

Once we can quarantine 5% of the population instead of 100% of the population, we can begin to reopen our restaurants and schools.

To get there, and to start getting back to normal in a reasonable amount of time, we’re going to have to ramp up testing dramatically. It makes clear economic sense. If a day’s work is worth $200, and a test costs $50, and a test provides clearance to go back to work for just a few days, then it is clearly worth it. I shared an article last month about a small Italian town that was able to completely eliminate new infections following two rounds of testing everybody and strictly quarantining those infected.

We may not actually need to test all 328 million Americans multiple times, which would be a gargantuan (though probably not impossible) undertaking. We should start by conducting random samples of, say, several thousand people in a city or region, to determine the level of infection in the population. Some rural communities likely have zero actual cases, and can be cleared to end social distancing with protection against importation of new cases. For areas that have some level of infection, it will probably be necessary to test whole towns or neighborhoods at once, in order to identify and quarantine those without symptoms. The more we can know who is infectious, the freer the rest of us can be. Coarser scale datasets like Kinsa’s fever network can be used to identify budding local outbreaks and reimpose social distancing before it spreads out of control. We don’t have to be perfect; we just have to keep R at or below 1 until an effective treatment or vaccine is developed.

In areas that have had widespread cases, like the New York area, antibody tests will be able to identify those who have been infected and are now immune, and those people can then be cleared to return to social contact and to work with vulnerable populations.

I’m normally not a fan of the American Enterprise Institute, but their plan for reopening strikes me as a well thought out strategy overall, and it is heavily dependent on testing.

We can’t know what the future will bring, but this is neither the first nor the worst pandemic that our species has lived through, and we will get through it. It is the first major pandemic in the age of globalization, and it is the first event in my lifetime that has rapidly galvanized the entire world to work together toward a common cause. I am hopeful that it can catalyze a change for the better, and that we might be able to channel some of our newfound unity toward addressing more global challenges in the years ahead.

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