# How Many Coronavirus Infections? Rewards for Testing May Provide an Answer

In March, Andrew Atkeson, an economics professor at the University of California, Los Angeles, tried to assess the mortality rate from the new coronavirus based on what is known about its spread.

If two-thirds of the population becomes infected, as would be needed to achieve what is known as “herd immunity,” the difference between those mortality rates would amount to two million deaths nationwide.

But the answer was elusive. “I wanted to see if we can estimate where we are,” Mr. Atkeson said. “I realized no, we can’t.”

He was missing one critical number: the infection rate in the general population. Three economists — Magne Mogstad and Alexander Torgovitsky of the University of Chicago and Andres Santos of U.C.L.A. — have devised a technique to provide it, by figuring out how to coax even people who believe they are healthy into taking a test for the coronavirus.

Short of testing everybody — an onerous exercise, even if enough test kits were available — figuring out the infection rate across the population requires a representative sample. The sample will include sick people clamoring to be tested but also people who have little interest in going through the ordeal.

That raises a problem. In Iceland, which performed tests on a big share of its population, researchers said the results might have been skewed because those concerned about potential infection were more likely to take part.

The method is crafty. Say you split the sample into 11 equal random slices and offer people in each a reward. For simplicity, say the reward for taking the test scales up from nothing in the first group to \$200. This method would yield the share of people who would take the test for nothing, the share who would do it for \$20 and so forth.

To figure out who would take the test for \$100 but not for \$80, you would subtract the share of people who responded to the \$80 reward from those who took it for \$100. The infection rates for the group could be computed in a similar fashion, to reveal the pattern about how testing changes and the infection rate changes as rewards increase.

There are likely to be people who decline to take the test even at the highest reward. But the researchers can approximate their infection rate by extrapolating from the information they have. If they know how the participation rate and the odds of testing positive change as the reward rises from \$160 to \$180 to \$200, they can deduce what the results would be at \$220 and \$300.

The underlying assumption is that people’s behavior changes smoothly as incentives change. If that holds, the researchers could construct a decent estimate of the infection rate in all or most of the population.

As the virus becomes more and more prevalent across countries, such testing will become critical. “What is the probability that if I go to the store I will become infected?” asked Marc Suchard, a professor of biomathematics at U.C.L.A. “You can’t nail that down without knowing the underlying rate of infection.”

Notably, as Mr. Atkeson found out, it’s hard to tell what share of infected people die from the virus if you don’t know how many have had it. You could get the same number of deaths with a high mortality rate and a low infection rate or the other way around, if the virus propagated very fast but killed fewer people.

Different findings would call for different strategies. A slow spread with a high death rate might suggest the containment policies deployed around the world are working, and would support maintaining a strict lockdown. The alternative could suggest that more people have some resistance to the virus and support relaxing quarantines across U.S. cities and states.

“Distinguishing between these two scenarios is key,” Mr. Mogstad said.

And yet nobody has measured this statistic. Countries testing for the virus have mostly focused on testing people suspected of being sick — people with symptoms and those who have come into contact with them.

Take the National Institutes of Health. In a two-year study about to begin, it will perform an antibody test on 10,000 volunteers from around the country. The number is large. But the N.I.H. decided to apply it on an “opt in” basis, to anybody who asked for it, so it will not amount to a representative sample of the population.

It will represent only those who want it most, capturing mostly Americans who fear they are sick. It will miss many asymptomatic carriers of the virus, who see no point in getting tested. The exercise will then fail to provide a faithful picture of how the new coronavirus has spread across the country.

There is other valuable testing to be done. Some health experts argue that, early in an epidemic, it is more useful to test specifically those suspected of carrying the virus, to trace their contacts and introduce more nuanced separation and quarantining policies.

“I don’t care about the prevalence of the infection in the population,” said Stefano Bertozzi, a professor of health policy and management at the University of California, Berkeley, who was the last director of the World Health Organization Global Program on AIDS. “I care about when the infection rate will overwhelm the capacity of the health system.”

There are more urgent questions at this stage than the overall infection rate in the population, he noted. For instance, what is the share of infected people who get sick, and how fast is the infection rate growing across thousands of cities? There is hope that epidemiological modeling based on testing of infected people can approximate this answer effectively, he said.

And yet, Mr. Mogstad argued, tracking the infection rate in the population is essential to track the evolution of the virus, including asymptomatic cases, and how it affects people of different races and income levels. That is critical to deciding when to relax restrictions, evaluating the effectiveness of measures taken and calibrating epidemiological models.

Figuring out this rate of infection is not that costly. “It’s not about how many you test — it is about who you test,” Mr. Mogstad said, adding that 5,000 tests could do the trick.

The economists’ technique could find other applications. Norway’s statistical agency is planning to use it to improve the response rates to surveys about people’s reaction to the virus and their engagement in social distancing, especially among the young and old.

The University of Chicago is raising funds to roll out three sets of antibody tests using the incentive technique — one on a nationwide sample and two on samples in Chicago, one of people living below the poverty line, and another of people living above the poverty line. The idea is to provide a proof of concept so the technique can be put into broader use. It could apply equally to antibody tests, which determine which people have been infected, and RNA tests to determine which people are currently infected.

There are other challenges to understand the spread of the coronavirus. Notably, tests are far from perfect. In particular, there are not yet good antibody tests that can faithfully pinpoint people who have been infected with a specific strain of coronavirus. Even when one is developed, tests will have to be deployed across a range of populations and over time, to understand the evolution of the virus and its distribution.

And yet the urgency to figure out the reach of the virus is increasing as it spreads. As Mr. Suchard pointed out, individual tracing of the virus can be useful in closed communities or places where it is less common, but in some cities where the virus has become prevalent, “we need population-based sampling” to save lives.