The virus moved under the radar swiftly in February and March, doctors and researchers said, because few cities or states had adequate surveillance systems in place. And testing, if it was being done at all, was haphazard. Emergency rooms were busy preparing for the predicted onslaught and likely missed some early virus-related deaths, and did not have the time or tools to verify infections on the fly, experts said.

It was mid-March before teams at N.Y.U. and Mount Sinai began taking samples for testing in New York. On Thursday, Gov. Andrew M. Cuomo announced results from antibody testing of grocery store shoppers— which researchers warned were preliminary and could change — that suggests one of every five New Yorkers may have been infected.

The new findings from the model produce a range of possible outcomes for when the virus may have infected 10 people in each city. In New York, for example, the model shows that the first 10 infected people could have been walking the streets of the city as early as the last week in January, or as late as the middle of February. From there, the infections in the centers of the outbreak grew exponentially.

Trevor Bedford, an associate professor at the Fred Hutchinson Cancer Research Center and the University of Washington in Seattle, said it became clear in late February that “community transmission” — an infectious outbreak — was probably silently underway in Washington after a single test result came back positive for someone who had no symptoms.

Whatever the precise scale of the initial outbreak, that same dynamic will accelerate once measures to mitigate the spread are relaxed without other public health measures in place, Dr. Burke said. “When you take away social distancing, everything will go right through the roof,” he said.