The president has theorized that the virus is likely to dissipate during warm weather. “A lot of people think that goes away in April with the heat — as the heat comes in,” he said. “Typically, that will go away in April.” Many health experts say that it is true of some strains of influenza and may be possible in this case, but that too little is known about the coronavirus for such predictions.

At a news conference last week, the president spoke with an optimism that even then appeared misguided. Asked whether schools should prepare for the virus’s wider spread, Mr. Trump played down the prospect. “I don’t think it’s going to come to that,” he said, “especially with the fact that we’re going down, not up. We’re going very substantially down, not up.”

“We have it so well under control,” he added.

More recently, Mr. Trump has taken to suggesting that drug makers are making rapid progress toward a vaccine. “They’re moving along very quickly. All of the pharmaceutical companies are moving along very quickly,” he said on Saturday and then repeated similar sentiments on Monday after meeting with drug company executives.

But health officials caution that, even under the best circumstances, a vaccine would most likely not be available for widespread use before mid- to late-2021. Dr. Fauci made a point of emphasizing that more than once on Tuesday, using the slide show at the National Institutes of Health to demonstrate that while safety trials on a vaccine will begin soon and progress has been strikingly fast compared with past health crises, it will take considerable time to move to mass production if it proves safe.

“Very soon, we’ll be sticking the first person with the vaccine,” Dr. Fauci said at a later appearance with the president after landing by helicopter back at the White House. “But I want to caution everybody: That’s only the first stage of the development of the vaccine.” Over all, he said, “the whole process is going to take a year, a year and a half at least.”

Mr. Trump did not quarrel with that, but urged Dr. Fauci to tell reporters about the progress toward “therapeutics” that will combat the infection once a person has it.

Dr. Fauci noted that two large trials for such medication are underway in China and the United States that could move more quickly than a vaccine. “That may take three months. That may take five months. I don’t know,” he said. “When the trial is over and they evaluate the data, if the drug works, then you’ll be able to apply it.”