As several states — including South Carolina, Tennessee and Florida — rush to reopen businesses, the sudden relaxation of restrictions will supply new targets for the coronavirus that has kept the United States largely closed down, according to experts, math models and the basic rules that govern infectious diseases.

“The math is unfortunately pretty simple. It’s not a matter of whether infections will increase but by how much,” said Jeffrey Shaman, a leading epidemiologist at Columbia University.

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Closing America was hard. But it came with one simple instruction: Everyone stay at home.

There are no easy answers for the phase that comes next, especially with a continued lack of testing, contact tracing and detailed guidance from federal health agencies, disease experts said. Instead, every state will conduct its own improvised experiment with thousands of lives in the balance.

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Many of the earliest reopenings will probably be confusing, chaotic, risky affairs — especially for states restarting their economies before most infectious-disease experts and some mayors and residents believe it’s safe to do so.

South Carolina’s governor issued an executive order this week reopening department stores and retailers previously regarded as not essential. Tennessee’s governor said he plans to allow most businesses to reopen once his “safer-at-home” order expires next week. Governors in Mississippi and Ohio have said the same. And Colorado’s Gov. Jared Polis (D) said some businesses could reopen on Friday.

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Some of those same states are, however, still struggling to contain outbreaks.

In Ohio, where businesses are expected to reopen by next week, a prison has become one of the most worrisome outbreaks in the country, with more than 2,000 inmates testing positive. In South Dakota, more than 700 infections have shut down a Smithfield Foods meatpacking plant. And because South Dakota remains one of the few states without a stay-at-home order, one business said it plans to go forward on Saturday with a car race drawing 700 spectators.

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Georgia, according to some models, is one of the last states that should be reopening. The state has had more than 830 covid-19 deaths. It has tested less than 1 percent of its residents — low compared with other states and the national rate. And the limited amount of testing so far shows a high rate of positives at 23 percent.

On Monday, Georgia Gov. Brian Kemp (R) explained his decision to reopen tanning salons, barber shops, massage parlors and bowling alleys, saying: “I see the terrible impact of covid-19 on public health as well as the pocketbook.” Kemp said he will urge businesses to take precautions, such as screening for fevers, spacing workstations apart and having workers wear gloves and masks “if appropriate.”

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President Trump, who has pushed to reopen parts of American life, said on Wednesday he told Georgia’s governor that he disagreed “strongly with his decision.” But, Trump added, “at the same time, he must do what he thinks is right.”

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Kemp did not directly respond but in tweets Wednesday night said he appreciated the president’s “bold leadership” and pledged that “our next measured step is driven by data and guided by state public health officials.”

In recent days, other governors have defended their decisions to reopen quickly as an economic necessity, an exercise in states’ rights and a matter of freedom.

“What I’ve seen across the country is so many people give up their liberties for just a little bit of security, and they don’t have to do that,” South Dakota Gov. Kristi L. Noem (R).

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“We can’t wait until there’s a cure to this,” said Mississippi Gov. Tate Reeves (R), who plans to reopen some businesses after a stay-at-home order expires Monday. “We can’t wait until every single person can get tested every single day to open up our economy.”

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But even states proceeding more slowly, such as Massachusetts and California, will have to walk their residents through the coming experiment with competing pressures and voices threatening to drown out public health instructions.

“As a country, we’re unprepared not just logistically but mentally for this next phase,” said Michael T. Osterholm, a University of Minnesota infectious-disease expert. He worries most Americans do not grasp the long, hard months facing them and the likelihood of repeated surges of the virus.

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“For a while, people were told all we need is to get past the peak. Then, they started hearing all we need is testing. Meanwhile, the president keeps telling everyone that things are going to reopen in a matter of weeks,” Osterholm said. “The way you prepare people for a sprint and marathon are very different. As a country, we are utterly unprepared for the marathon ahead.”

Rebounds inevitable

This is the central problem: The vast majority of Americans are still believed to be uninfected, making them like dry kindling on a forest floor. Barring a vaccine or treatment, the virus will keep burning until it runs out of fuel.

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“The trick is to keep that burn at a controlled rate,” Osterholm said. “We have focused so much on how we are dying from the virus that we have not focused enough on how to live with the virus.”

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Epidemiological models suggest the best strategy for keeping the burn rate under control is to drive the number of infections as low as possible before restoring economic activity. That would then provide time to react if cases flare.

The economic devastation that would cause is significant. But those same models suggest that opening prematurely increases the likelihood that communities will have to shut back down once infections reach a certain level, creating multiple open-shut cycles. Adding to those concerns, the director of the Centers for Disease Control and Prevention said Tuesday that a second wave of infections next winter would be even more devastating because it would coincide with flu season.

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There’s no simple, one-size-fits-all protocol for reopening the economy, said David Rubin, director of PolicyLab at Children’s Hospital of Philadelphia. Rubin is developing a model to forecast how reopening 260 large U.S. counties on May 15 would play out if residents maintained only half the social distancing measures now in place.

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The good news, Rubin said, is that modest-size, relatively spread-out cities will probably have room to make adjustments. But if restrictions ease too much, New York and similarly dense cities will rapidly see infections spike again.

“It comes back really quick, and the peaks are much higher than what you’re seeing right now,” Rubin said. “It was sobering. I was more optimistic before we did our models.”

This is why epidemiologists are cautioning state leaders to inch toward reopening with tentative, staggered steps.

The fraught science of reopening

Newly emerging science illustrates just how complicated and fraught those steps may be.

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Dine-in restaurants are one sector Trump and some governors have repeatedly mentioned. To reopen, owners may have to rethink not just how closely diners sit together and how food is served but also how ventilation systems and airflow around diners may need to be retooled.

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A recent case study — published by the CDC — examined how a single patron infected nine others at an air-conditioned restaurant in China. The infected person, a 63-year-old retired woman, did not begin running a fever and coughing until after her lunch Jan. 24 at the Guangzhou restaurant. But over the next two weeks, it became apparent the virus had spread to four diners at her table and to five people sitting at adjacent tables roughly three feet away.

Researchers studying the seating arrangements believe an air-conditioning unit propelled tiny viral droplets over distances that are normally safe between the tables.

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“To prevent the spread of the virus in restaurants, we recommend increasing the distance between tables and improving ventilation,” the researchers concluded.

Donald Milton, a professor of environmental health at the University of Maryland School of Public Health, said the restaurant appears to have had an air-conditioning unit popular in China — and increasingly used in the United States — that recirculates warmed or cooled air, with no air intake or filtration.

Milton said measures to make such a situation safe would include a ceiling fan paired with better air filtration and ultraviolet lights that kill germs. But he noted such measures would need to be designed to suit specific establishments.

Studies emerging in the past week are also changing scientists’ understanding of how the virus spreads, which will make efforts to reopen society even harder.

A growing body of evidence suggests the virus is most contagious in people before they develop a fever or even feel a tickle in their throat. That suggests silent spreaders are seeding new cases.

When severe acute respiratory syndrome (SARS) — the cousin to this new coronavirus — emerged in 2002, Asian countries were able to stop it because people became physically ill roughly at the same time they became contagious. That made it far easier to isolate and prevent the spread of disease.

A study published in the journal Nature Medicine last week estimated that people infected with the novel coronavirus are contagious almost two and a half days before symptoms appear — and that peak contagiousness occurs about 17 hours before people start feeling sick. In a sample of patients from China, the study estimated 44 percent of cases spread from person to person before symptoms appeared.

A study of Iceland’s population found that 43 percent of the people who tested positive didn’t have symptoms at the time of the test. And a recent New England Journal of Medicine study of 210 pregnant women in New York found that 14 percent tested positive but had no symptoms.

Warnings from abroad

Warning signs are also emerging from abroad.

For months, Singapore has served as an exemplar, with its pandemic response praised and emulated around the world. Despite its proximity to China and early cases, Singapore used massive testing and contact tracing to keep its disease curve flat. It even deployed police to trace people’s movement with security camera footage and credit card records.

Those painstaking efforts kept schools and businesses open and its economy afloat — until this month, when the virus found and exploited a weak point: low-wage migrant workers living in densely packed dormitories.

“It only takes a few people to let down their guard, and the virus will slip through,” said Prime Minister Lee Hsien Loong, who begged citizens to maintain discipline.

The daily number of new cases has soared from 200 last month to 1,426 on Monday. In recent days, the government shut down schools, made masks mandatory and forced hundreds of thousands of migrant workers into quarantine.

“What worries me looking at Singapore is how much capacity they have on testing and contact tracing compared to the U.S. and yet months into the pandemic, even they are having to become more restrictive,” said Tom Inglesby, director of the Johns Hopkins Center for Health Security. “Meanwhile, we in America have nowhere near that testing and tracing, yet all we’re talking about is loosening up our restrictions.”

The need for warning systems

Given the dangers involved in reopening, what states desperately need are a warning system and suppression tool to prevent infections from cresting again into the deadly peaks the United States saw in March and April.

But states are jumping into their experiments without the two tools deployed by almost every other advanced nation: massive testing and contact tracing.

Governors — Republicans and Democrats — from Virginia to Washington to Ohio continue to plead for federal authorities to fix shortages of swabs, chemical reagents and testing kits, a national supply problem they cannot solve independently.

Similarly, local health departments, decimated by decades of budget cuts, lack the money and the hundreds of thousands of workers needed to trace and quarantine everyone who comes into contact with infected people.

By pushing responsibility for the pandemic response and reopening onto the states, experts said, Trump has freed himself to play the role of criticizer-in-chief. Already, he is criticizing governors for not reopening immediately, but if cases rise uncontrollably, he can criticize state leaders for reopening too early or mishandling it.

“It might be a clever and effective political strategy, but it leaves our country without any way to pull itself out of the current mess,” said Jeremy Konyndyk, who was in charge of U.S. foreign disaster assistance during the Obama administration.

In Maryland, Republican Gov. Larry Hogan’s wife — who was born in South Korea — struck a deal to buy 500,000 tests from South Korea. Hogan said this week that he turned for help from a foreign government rather than the federal government after Trump “made it clear over and over again” that states “have to go out and do it ourselves.”

In Massachusetts, state leaders have partnered with a nonprofit that works mainly in the developing world to hire and train contact tracers.

But such efforts, born of desperation, will go only so far without federal intervention and funding, public health experts say. Even if a handful of states find some way to shore up testing and contact tracing, the virus could rage on in neighboring states, throwing off sparks that can ignite new outbreaks.

“The only tool the governors have had so far is the clampdown because it deprives the fire of oxygen. The second you let that up, the fire comes roaring back,” said Konyndyk, who oversaw the U.S. government’s Ebola response in West Africa. “But until you have water or sand, that’s all you can do. And it remains to be seen whether we as a country are going to figure out a way to get that bucket of water to start putting out the fire.”