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The coronavirus has infected far more people in the United States than testing has shown so far, and stringent measures to limit social contact in parts of the country not yet seeing many cases are needed to significantly stem the tide of illness and death in the coming months.

Those are the conclusions of Columbia University researchers who used a New York Times database of known cases and Census Bureau transportation data to model how the outbreak could evolve based on what is known about the virus. The estimates are inherently uncertain, and they could change as America adopts unprecedented measures to control the outbreak.

But they offer a stark warning: Even if the country cut its rate of transmission in half — a tall order — some 650,000 people might become infected in the next two months.

The growth is driven by Americans with mild symptoms who are carrying and spreading the virus without being aware that they have it, the researchers say. The number of undetected cases — 11 times more than has been officially reported, they estimate — reflects how far behind the United States has fallen in testing for the virus.

New York City, Seattle, Boston and parts of California already have such large outbreaks that they will probably see significant growth even after taking extraordinary measures over the past week, the researchers say. New York City’s outbreak, the nation’s largest, grew to more than 4,000 known cases on Friday and is likely to increase many times over even in a favorable scenario.

On the other hand, parts of the country without large clusters of cases could still avoid the worst of the outbreak — if they impose measures like closing schools, banning mass gatherings and testing and quarantining sick people and their contacts. The epidemic would then spread inland at a much slower pace and strike with less severity, the estimates say.

But controls would need to be put in place immediately, and everywhere.

“We’re looking at something that’s catastrophic on a level that we have not seen for an infectious disease since 1918,” said Jeffrey Shaman, a professor of environmental health sciences at Columbia and the leader of the research team, referring to the Spanish flu. “And it’s requiring sacrifices we haven’t seen since World War II. There are going to be enormous disruptions. There’s no easy way out.”

Controls in many places have already changed the fabric of American life. Restaurants and schools have closed, people are working from home and travelers have put off trips — all changes that should reduce the rate of infection. In California, New York and Illinois, officials have announced more stringent directives aimed at stopping the spread. That means that at least 1 in 5 Americans will be under orders to stay home in the coming days, and more states are expected to follow suit.

But other places have resisted. Bars and restaurants remain open in Wichita, Kan., in a county that reported its first case on Thursday night, and in Knoxville, Tenn., where reports of the virus have been scarce. Some malls remain open in Chicago, recreational marijuana shops in Oregon, and some retail chains nationwide. Many businesses are still requiring workers to come into the office.

“I’m not really supportive yet of restricting things further or closing down,” said Pete Meitzner, head of the Board of County Commissioners in Sedgwick County, which includes Wichita, a city of about 400,000. Mr. Meitzner said he was growing increasingly anxious at the possibility that people who do not know they are ill might spread the virus, but that he also needed to consider the fate of workers as the community decided whether to close buildings.

“Some of our public offices and buildings are, but I do feel horrible for our restaurant workers and our hospitality workers at our hotels,” he said. “They are really suffering.”

Of the area’s relative dearth of cases, Mr. Meitzner added, “We don’t know for sure if we’re lucky or what.”

The scenarios do not estimate what effect individual local actions will have on the outbreak. But if the country as a whole does not take collective action, they suggest that Sedgwick County could see a significant proportion of its population — hundreds of thousands of people — infected with the virus in the next two months. If the country, as a whole, can reduce rates of transmission by 50 percent, Sedgwick County could instead see fewer than 1,000 cases in that time.

Several researchers not involved in the Columbia University analysis who reviewed the findings said they made sense and broadly agreed with their own thinking. Even in places that now appear to have few cases, they said, officials should act sooner rather than later.

“You have to think of this as an insurance for the future: The earlier you do it, the greater effect you have on the virus,” said Alessandro Vespignani, director of the Network Science Institute at Northeastern University, who said the estimates were in line with his own projections. “It’s better to take excessive precautions than not.”

The economic challenges of that loom.

“We’re staying open until someone forces us to shut down,” Beau McKee, the general manager at Not Watson’s Kitchen and Bar in Knoxville, said in a telephone interview, as dishes clattered in the background. “I feel like those people that are vulnerable know who they are and they should be careful, and we should continue testing. But small businesses are getting crushed.”

Mr. McKee said he has grown increasingly worried about how the virus could affect his own father, in his 70s, but that there were other worries too. “At least we can offer a handful of people good-paying jobs and when we get out of this and we’re on the other side, we’ll hopefully have gained more clientele by doing so because we kind of stuck it out and didn’t wave the white flag,” he said.

The Columbia researchers’ model works by observing the behavior of the outbreak in the United States up until March 13, based on case records compiled by The New York Times, commuting patterns and other data.

The researchers use those observations to infer key features of the outbreak. One is how many people each infected person has tended to infect so far, about 2.2. Another is how many people may pass on the virus without knowing that they have it. The disease is spreading far too fast to be explained by known cases alone, and only about 1 in 11 infections have been reported, they found.

Those factors allow the researchers to simulate the spread of a virus in the future. They adjust those simulations under different scenarios in which the nation imposes a range of control measures to stop the spread.

The estimates are imperfect, but they are consistent with the available data. It is impossible to know the exact number of cases a week ago or to predict the future. It is also impossible to model the precise impact of unprecedented measures that America has already put in place to control the outbreak.

But the research estimates that if measures to slow the disease are not effective, the virus could sicken millions of people or more, losing steam on the coasts in May before spreading to the rest of the country over the summer.

A worst-case scenario is unlikely to play out. A handful of countries, including China and South Korea, have managed to slow down their own fast-moving outbreaks through tracking and isolating sick people and reducing social contact. It is not known if measures imposed in the last week have started slowing the outbreak in the United States.

But the researchers’ estimates show the enormity of the task. As in China, most of those who have been infected but not tested are likely to have mild symptoms, or none at all, the researchers say. That means they can spread the virus without being aware they are infected.

Several researchers said that the slow pace of testing in the United States is one major reason the proportion of unreported infections is so high here. Despite a ramping up of testing in recent days, the United States still lags both Italy and South Korea in the total number of people tested and in testing per-capita, figures show.

This map shows how many people the model estimates could get the virus in each county in the United States by two months from now.

One scenario — which is unlikely — simulates what could happen without any intervention. A second scenario envisions what would happen with some control measures, such as partial adherance to social distancing guidlines and a patchwork of government-imposed restrictions on work, travel, and dining out. A third envisions severe control measures: strict adherance across the country to social distancing, working remotely, closing schools and restaurants and banning large gatherings.