Note: Don’t miss the related post “Avoiding Economic Carnage from the Coronavirus: There are Better Policies than Sending Everyone $1000.”

The novel coronavirus behind COVID-19 is extremely good at spreading. It is likely to take a long time to get an effective vaccine. Restrictions on interaction that apply to the vast bulk of the population quickly come to seem quite onerous. As a result, the most serious part of the pandemic is likely to end only when a substantial fraction of the population has had COVID-19 and is immune. (As of today, it is still not known for sure how strong the immunity is from having had COVID-19. I’ll assume the best on that score—that having had COVID-19, even asymptomatically, confers immunity.)

The idea that a large fraction of the population will end up getting infected may seem grim. Is it possible, though, that

• those who have only a small risk of dying if they get the disease—the young and otherwise healthy—will get exposed and become immune, ultimately bringing the pandemic under control because of their immunity, but

• those who have a higher risk of dying if they get the disease—the old or already compromised—could largely escape?

To answer this question, I need to modify the SIR model that divides people into the Susceptible, the Infectious and the Recovered. Above and at this link is a wonderful video that provides a primer on the SIR model. I need to modify the SIR model into a model that tracks the fraction of the population in each of these categories within group A and within group B. I will think of group A as the young and otherwise healthy, who are less careful about social distancing but also at lower risk of death should they get infected by the coronavirus. Group B is then the old or compromised, who I assume understand the risk they face and so are more careful at social distancing. (The old or compromised may also suffer a lower economic and lifestyle cost of social distancing.)

With six subgroups (SIR within group A and SIR within group B) the dynamic solution to the differential equation for the modified SIR model is quite complex, but the conditions for the number of infected in group A and the number of infected in group B to be shrinking can be readily analyzed. Define this notation. First, notation for population shares:

SA = share of the overall population that is both in group A and is susceptible

IA = share of the overall population that is both in group A and is infectious

RA = share of the overall population that is both in group A and is recovered (or dead)

SB = share of the overall population that is both in group B and is susceptible

IB = share of the overall population that is both in group B and is infectious

RB = share of the overall population that is both in group B and is recovered (or dead)

Note that SA + IA + RA + SB + IB + RB = 1. Second, notation for constants of transmission:

KAA = constant indicating the risk of transmission between two people in group A

KAB = constant indicating the risk of transmission from someone in group A to someone in group B

KBA = constant indicating the risk of transmission from someone in group B to someone in group A

KBB = constant indicating the risk of transmission between two people in group B

Third, the rate of resolution of an infection:

C = rate at which those who are infectious make the transition to “recovered” which includes being dead, since the dead, like the recovered, will not contribute to further transmission.

All of these are nonnegative quantities.

The two crucial inequalities in this modified SIR model are the conditions for the number of infected within a group to be shrinking. Those two inequalities are:

d IA/dt = KAA * IA * SA + KBA * IB * SA - C * IA < 0

d IB/dt = KBB * IB * SB + KAB * IA * SB - C * IB < 0

Let’s divide each inequality by the share of infected in the corresponding group to get the logarithmic growth rates:

d log(IA)/dt = (1/IA) d IA/dt = [KAA + KBA * (IB/IA)] * SA - C < 0

d log(IB)/dt = (1/IB) d IB/dt = [KBB + KAB * (IA/IB)] * SB - C < 0

Now, let’s define x as the key ratio IB/IA:

x = IB/IA

Substituting into the logarithmic form of the crucial inequalities yields:

d log(IA)/dt = [KAA + x KBA] * SA - C < 0

d log(IB)/dt = [KBB + (1/x) KAB] * SB - C < 0

Note that

d log(x)/dt = d log(IB)/dt - d log(IA)/dt = [KBB + (1/x) KAB] * SB - [KAA + x KBA] * SA.

There will be a (temporary) steady-state level of x when

d log(x)/dt = [KBB + (1/x) KAB] * SB - [KAA + x KBA] * SA = 0.

We can solve this for a temporary-steady-state level of x that might be a good approximation. Let’s plug in some plausible numbers to get some sense of what will happen.

First, let’s start with almost everyone susceptible. In particular, assume that, approximately,

SA = 2/3

SB = 1/3

Let’s use a week as the time unit. A two-week course of the disease can be represented by C = .5/week. And the idea that someone infected infects roughly 3 other people when not putting much effort into social distancing can be represented by KAA = 1.5/week. Let’s imagine that KBB = KAB = KBA = .3/week because those in group B are putting great effort into social distancing. (This is optimistic.)

This yields a quadratic equation in x. It is the positive root that is relevant: .108 with these parameters.

By contrast, suppose that no one was doing social distancing so that KAA = KBB = KAB = KBA = 1.5. Then the temporary-steady-state level of x is .616.

At the temporary-steady-state level of x, looking at either of the crucial inequalities will give us an equivalent answer for when the number of infectious people will begin declining. Let’s look at the first of the crucial inequalities, plugging in the numbers when almost everyone is still susceptible and when group B is making strenuous efforts to do social distancing but group A isn’t:

d log(IA)/dt = [(1.5/week) + .108 (.3/week)] * 2/3 - (.5/week) = .522

This clearly fails to satisfy the inequality for a shrinking number of infectious people of group A, and the number of infectious people of group is an approximately a constant ratio to this.

If half of group A had the disease (many perhaps asymptomatically) and became immune, that would reduce SA to 1/3, and the inequality would be close to being satisfied. Not quite, because x would go up from .108 because of the influence of SA getting lower faster than SB gets lower on the quadratic equation. But when a bit more than half of group A becomes immune, then the number of those who are infectious will begin to drop.

The bottom line is that if these parameters are in the right ballpark, things will only end when a large fraction of group A has had the disease and becomes immune or dies, which will also involve a substantial but considerably smaller fraction of group B has had the disease and becomes immune or dies.

Note that this ending of the pandemic doesn’t require continuing restrictions on group A. Indeed, the assumption was that group A never really took the restrictions seriously in the first place. Restrictions on group B may be necessary for quite a while.

The Key Question

The biggest unknown about COVID-19 right now is what fraction of people have already had the disease and become immune. We may know within a couple of weeks: scientists are eager to deploy tests for antibodies to the virus (which would be a good indication of immunity) on random samples of the population.

Because we know quite well how many people have died from COVID-19, it would be good news in several ways if we find that a large fraction of the population has already had the disease and has become immune. First, and most important, it would mean that the fatality rate when infected is lower than people have been thinking. Second, it would mean the symptoms are often too mild for people to have gotten diagnosed already during a period when, for the most part, only those who seemed quite sick were tested. Third, if a large fraction of people have already had the disease and have become immune, it means we are closer to the moment when it is safe to lift restrictions on the bulk of the population, which would lessen the economic impact of measures to control the pandemic.

Note that a large fraction of the population already having had the disease and having become immune would go along with the coronavirus having been fiendishly good at spreading, which is quite possibly true.

One thing that doesn’t depend on knowing how many people have already been infected is the disaster of overloaded hospitals in the next few weeks. For the next few weeks, the number of people who are in critical condition and the growth rate are enough to enable prediction. (However, one does need to guess how effective any change in social distancing measures will be.)

Discussion of Similar Perspectives in the News and the News as It Relates to Likely Values of the Relevant Parameters and Values of State Variables for the Coronavirus

Clive Cookson in the Financial Times: “Coronavirus may have infected half of UK population — Oxford study: New epidemiological model suggests the vast majority of people suffer little or no illness”

The new coronavirus may already have infected far more people in the UK than scientists had previously estimated — perhaps as much as half the population — according to modelling by researchers at the University of Oxford.

If the results are confirmed, they imply that fewer than one in a thousand of those infected with Covid-19 become ill enough to need hospital treatment, said Sunetra Gupta, professor of theoretical epidemiology, who led the study. The vast majority develop very mild symptoms or none at all.

Max Colchester and Denise Roland in the Wall Street Journal: “U.K. Trials Coronavirus-Immunity Tests for Home Use: The tests, if successful, would ease disruptions and lockdowns, but some experts have doubts”

In Britain, the government said Wednesday it is trialing personal blood-testing kits that it hopes to distribute as soon as next week. The test—if it functions—could clear the way for people who have caught and recovered from Covid-19 to return to work or volunteer to help others, potentially easing the disruption caused by the pandemic.

Across the world, laboratories and diagnostics companies are racing to fine-tune a cheap, portable, mass-testing device that can quickly show if a person has acquired immunity to the virus. The U.S. Centers for Disease Control and Prevention has said it is also working on its own antibody test for the virus.

…

If the test does work, it could not only help a portion of the population go back to work but also help answer questions about the virus itself including how widespread it is.

“In China, why is it going down? Is it because the virus is dying out or is it because many more people are getting infected than we think and are developing antibodies?” said Mr. Wraith.

Rebecca Ballhaus and Stephanie Armour in the Wall Street Journal: “Trump Says Parts of U.S. Could Go Back to Work in a Few Weeks: White House weighs proposing workplace coronavirus testing, but capacity remains limited; governors suggest they will go their own way”

Tedros Adhanom Ghebreyesus, director-general of the World Health Organization, warned in a briefing Wednesday that in the absence of necessary preparations, the virus could resurge once restrictions are eased. “The last thing any country needs is to open schools and businesses, only to be forced to close them again because of a resurgence,” he said.

Feliz Solomon, Betsy McKay and Jon Emont in the Wall Street Journal: “When Will It Be Safe to Loosen Coronavirus Lockdowns? Governments prepare for the long haul as authorities grapple with how much economic pain countries can endure”

Months of isolation, social distancing and economic distress could lead to more health problems than Covid-19 itself, said David Katz, founding director of Yale University’s Prevention Research Center. Job losses lead to lack of health insurance, food insecurity, stress, and sometimes drug and alcohol abuse and suicides, he said.

He has proposed a stratified approach in which those most vulnerable to Covid-19—people ages 75 and older, and those of any age with heart disease or another serious underlying condition—would remain isolated or maintain social distancing. Meanwhile, people under age 60 would largely go back to their daily lives after a few weeks, while following hand-washing and other precautions.

“What I’ve been concerned about is we can hurt people if we let them get this infection, and we can hurt people with an all-out war that destroys their lives in other ways,” said Dr. Katz, who said he has spoken with two U.S. governors about his proposal. “I’m saying how about we do this in phases.”

The first phase is the current one: strict social distancing for a few weeks to avoid sharp peaks of infection and prevent the health-care system from being overwhelmed, he said. Testing should be expanded in the U.S. to figure out how many people are sick, he said.

Then experts in epidemiology, virology, mathematical modeling and other fields should analyze the data and figure out a way to protect the vulnerable while allowing others to get on with their lives, he said.

Holman Jenkins in the Wall Street Journal: “The Happy Few Are the Cured: Now let’s get on with saving the economy while protecting the vulnerable.”

The giddiest among us soon will be those who tested positive and now have it behind them. The world will be their oyster. A Craigslist page will soon appear for coronavirus antibody-positive personal services. People will get paid hundreds of dollars an hour if they can document their immunity. Starbucks is opening up again in Wuhan. In the U.S., Seattle or New York will get the first antibody-positive Starbucks: Every barista will be able to prove they’ve had the coronavirus.

Eran Bendavid and Jay Bhattacharya in the Wall Street Journal: “Is the Coronavirus as Deadly as They Say? Current estimates about the Covid-19 fatality rate may be too high by orders of magnitude.”

Eran Bendavid and Jay Bhattacharya do a good job of assembling the shreds of evidence relevant to how many people are infected but don’t have serious symptoms. But they dramatically overinterpret this evidence. When comprehensive samples certain classes of people are tested for the coronavirus, the infection will be detected, on average, several days before it would have been detected if they were only tested after showing symptoms. With prevalence growing so fast, that makes the numbers from comprehensive samples comparable only after adjusting for how early in an infection something is detected.

Plots like the one just above suggest the incidence of the virus was doubling every two or three days when the data was collected, so a difference of a little over a week in how early an infection is detected could reduce the adjustment factors Eran and Jay adduce by a factor of ten. That is, the mortality rates and serious symptom rates might need to be adjusted upward by a factor of ten to account for growth over a little more than a week to make the numbers comparable. Still, these numbers give some hope that COVID-19 is better at spreading and less deadly given infection than most people have been thinking. Here are the shreds of evidence Eran and Jay assemble:

Population samples from China, Italy, Iceland and the U.S. provide relevant evidence. On or around Jan. 31, countries sent planes to evacuate citizens from Wuhan, China. When those planes landed, the passengers were tested for Covid-19 and quarantined. After 14 days, the percentage who tested positive was 0.9%. If this was the prevalence in the greater Wuhan area on Jan. 31, then, with a population of about 20 million, greater Wuhan had 178,000 infections, about 30-fold more than the number of reported cases. The fatality rate, then, would be at least 10-fold lower than estimates based on reported cases.

Next, the northeastern Italian town of Vò, near the provincial capital of Padua. On March 6, all 3,300 people of Vò were tested, and 90 were positive, a prevalence of 2.7%. Applying that prevalence to the whole province (population 955,000), which had 198 reported cases, suggests there were actually 26,000 infections at that time. That’s more than 130-fold the number of actual reported cases. Since Italy’s case fatality rate of 8% is estimated using the confirmed cases, the real fatality rate could in fact be closer to 0.06%.

In Iceland, deCode Genetics is working with the government to perform widespread testing. In a sample of nearly 2,000 entirely asymptomatic people, researchers estimated disease prevalence of just over 1%. Iceland’s first case was reported on Feb. 28, weeks behind the U.S. It’s plausible that the proportion of the U.S. population that has been infected is double, triple or even 10 times as high as the estimates from Iceland. That also implies a dramatically lower fatality rate.

The best (albeit very weak) evidence in the U.S. comes from the National Basketball Association. Between March 11 and 19, a substantial number of NBA players and teams received testing. By March 19, 10 out of 450 rostered players were positive. Since not everyone was tested, that represents a lower bound on the prevalence of 2.2%. The NBA isn’t a representative population, and contact among players might have facilitated transmission. But if we extend that lower-bound assumption to cities with NBA teams (population 45 million), we get at least 990,000 infections in the U.S. The number of cases reported on March 19 in the U.S. was 13,677, more than 72-fold lower. These numbers imply a fatality rate from Covid-19 orders of magnitude smaller than it appears.

John Cochrane in the Wall Street Journal: “Flatten the Coronavirus Curve at a Lower Cost: A total shutdown could cost the economy $1 trillion a month. We need more tailored measures.”

A blanket lockdown can’t go on. Keeping every business closed and every worker at home until a vaccine is available won’t work. Replacing the private economy with borrowed federal money for months on end won’t work. If this were the plague, with 50% of the infected dying, it might be a different story. But people won’t put up with losing many trillions of dollars to flatten the curve of this virus.

…

State and local governments need to work with businesses to figure out a satisfactory combination of personal distance, self-isolation, frequent testing, stricter rules for those who must interact with customers, cleaning protocols and so on. Each industry will likely be different. Even onerous rules, which can be eased as officials and businesses gain information and experience, are better than a blanket ban.

…

Government officials need to work with a scalpel, not a sledgehammer. Isolate old people and those with pre-existing health conditions, who are much more likely to end up needing emergency care, while letting the young and healthy get back to work, carefully. Retired people have income streams that aren’t as disrupted by the virus. They can stay home. Lock down hotspots, but not entire states. Follow the Taiwan, South Korea and Singapore models: extensive testing, contact tracing, detailed people tracking. But keep the economy open, subject to stringent safety rules.

Wall Street Journal Editorial Board: “Parks and Virus Recreation: Cutting off access to outdoor space strains the cooped-up public.”

Officials seeking to slow the spread of coronavirus have imposed sweeping restrictions on roughly one in four Americans. Several states have closed churches, restaurants and bars, gyms and other businesses. Some governors and mayors are now moving to limit access to parks and other outdoor spaces. The goal as always is to slow the virus’s spread, but with cabin fever raging for shut-ins, we have to wonder whether closing down large open spaces does more harm than good.

…

Mr. Cuomo also suggested opening some streets to pedestrians only. Increasing the amount of outdoor space would be helpful in highly populated cities like New York, and it’s feasible now that vehicle traffic has fallen amid the shutdowns. But other officials have threatened or acted to decrease the amount of outdoor space.

…

Surely some middle ground can be found between asking for public compliance with personal distancing and then making large public spaces inaccessible. Compared to grocery stores or pharmacies, outdoor spaces are lower-risk for contagion. Especially in big cities, people can’t be expected to stay cooped up in their tiny apartments indefinitely. The phrase “stir crazy” comes to mind. If officials push too far, many people will ignore both reasonable and unreasonable restrictions.