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Many governors are claiming they can’t relax limits on business and recreation until their states have an extensive system of “contact tracing.” It’s a worthy aspiration. But they should listen to the scientists who warn that contact tracing won’t work against the novel coronavirus.

Gov. Cuomo says he envisions hiring an “army” of thousands of “tracers” to call people or go to their homes, notify them that they have been exposed to an infected person and explain that they must get tested and quarantine, if positive.

On Tuesday, the National Governors’ Association told Congress that states need federal money for contact tracing. By some estimates, it would cost $3.6 billion to hire 100,000 tracers nationwide, and others are suggesting we need triple that number.

“While this is a lot of people and a lot of money,” says Adriane Casalotti of the National Association of County and City Health Officials, “the other side of the coin is a country where we continue to be in lockdown.”

The New York Times, too, is pushing contact-tracing, with an editorial calling for “a legion of health workers, disease detectives,” in Gotham to find those “who may have been exposed and enact a system of isolation and quarantine.”

Undoubtedly the pols pushing it have the best of intentions, though almost any governor would see the allure of a jobs program funded by the federal government.

Trouble is, a mountain of scientific evidence indicates contact tracing won’t work against the coronavirus. And given the virus’ nature, deploying it earlier probably wouldn’t have stopped the spread.

The coronavirus is fast-moving and transmitted in multiple ways, such as touching contaminated surfaces like subway poles and door knobs. It can become aerosolized when someone sneezes two aisles over in the supermarket or coughs in an elevator. The virus is found in feces and may even spread when a toilet is flushed and viral particles become airborne.

The “disease detectives” will have to ask: “Who was that at the supermarket or in the public restroom?” In many cases, they won’t have an answer.

Lancet Global Health scientists conclude that contact tracing will work when “less than 1 percent of transmission occurred before the onset of symptoms.” That’s the opposite of the coronavirus: Victims are most contagious before or just as their symptoms begin, research indicates. By the time they are diagnosed and asked for contacts, those contacts are already infecting others. Oxford University scientists also caution that the coronavirus spreads by too many mechanisms “to be contained by manual contact tracing.”

Still, digital tracing apps with built-in memory of who was near the infected person, in the supermarket or restroom, are promising. Several Asian countries used apps that put credit-card data and other personal information in the hands of government. Most Americans would be uneasy.

Apple and Google announced on April 10 they have devised a tool to alert smartphone users if they were in contact with someone later diagnosed with the coronavirus. This tool would require widespread adoption. It might be too late for this crisis; maybe for the next pandemic.

Plus, the contact-tracing governors are calling for the old-fashioned version: slow, expensive and incapable of curbing the outbreak. So what should be the new goalpost for safely reopening, if not contact tracing?

Many say mass availability of a test for the virus. But the test is a snapshot. You can test negative before work, then catch the virus in the elevator. A reliable test for immunity would be more reassuring. But it’s still unclear whether recovered patients are immune and for how long, warns Dr. Anthony Fauci.

Still another goalpost is the development of medications to make the disease treatable. That could happen within weeks, and it would make working-age people feel safe returning to their jobs. About 85 percent of New Yorkers who have died from the virus are over age 60.

But the best goalpost of all is our collective behavior. New Yorkers are proving they will wear masks and socially distance to protect themselves and others. Kudos to our courageous, resilient city.

The public wants a safe plan, and the sooner the better, but contact tracing isn’t it.

Betsy McCaughey, a former lieutenant governor of New York, is chairwoman of the Committee to Reduce Infection Deaths.