The idea, all three said in interviews, is to have a centralized command structure, anchored inside the White House, to manage production and distribution. Ventilators and other equipment would be shipped across the country based on data about where outbreaks were emerging. If, for example, Des Moines had 5,000 ventilators but few cases of coronavirus, those machines could be sent to New York, with the promise that New York or other cities would supply Des Moines when that city was in need.

“We need a way to get out of the mess we’re currently in, and I think this is part of the solution,” Dr. Cassidy said in an interview. He also outlined his idea for the immunity registry, which he said would be akin to “vaccine registries” maintained by schools.

Not all of Congress’ doctors, however, are so actively engaged. In a brief interview in the Capitol before his diagnosis became public, Mr. Paul insisted that he had little relevant expertise: “I am a physician,” he said. “I’m aware of science, I’m able to read scientific papers and I think, make judgments. But I don’t put myself out as an expert.”

Others, like Dr. Schrier, the only female doctor in Congress, have simply been trying to keep their colleagues and constituents well. Dr. Schrier recounted how, after coronavirus cases first appeared in the United States but before social distancing became a household phrase, she watched in horror as her colleagues were still coughing into their hands and sharing fruit and other food, including with Representative John Lewis, the civil rights icon who is batting pancreatic cancer.

“They did a birthday party for John Lewis with a cake,” she said, sounding incredulous. “I pulled my friends aside and said, ‘You can’t do this with an 80-year-old whose immune system is already compromised.’”

There are 17 doctors in Congress — 14 in the House and three in the Senate — as well as three dentists, two nurses, a pharmacist and a former health secretary, Representative Donna E. Shalala, Democrat of Florida, who served under President Bill Clinton. The group is overwhelmingly white, male and Republican. Only the two nurses and three of the doctors are Democrats. The Republican doctors have their own group, the G.O.P. Doctors Caucus.