In times of crisis, when the federal government seems to be lagging or confused in its message, people often look to alternative leaders for reassurance and guidance, as they did with Rudolph W. Giuliani, then mayor of New York, after the 9/11 attacks. (Mr. Giuliani is now Mr. Trump’s personal lawyer.) That was what Senator Bernie Sanders of Vermont was trying to convey in his speech about the gravity of the coronavirus threat on Thursday. And it was what Mr. Biden meant to do in his address on Thursday — to hold himself up as a reassuring, steady alternative to Mr. Trump, to show that he was competent, sympathetic, forceful and presidential. “I’ll always tell you the truth,” he said. “This is the responsibility of a president.”

Recent American history has certainly provided enough disasters that required presidents to rise to an occasion: not just the 9/11 attacks but also the 2008 economic crisis, countless gun massacres, Hurricane Katrina in 2005 and Hurricane Sandy in 2012, for starters.

Presidents need to be straightforward and candid from the beginning, said David M. Litt, who worked as a speechwriter in the Obama White House. They need to admit the early shortcomings in their response, lay out what steps they will take to improve, and then provide regular updates.

“The number one thing you have to do is figure out how to address the problem — and to make it clear where you can and can’t solve it,” Mr. Litt said. Dissembling about it — as in Mr. Trump’s assertion, earlier in the crisis, that coronavirus would just “go away,” or his claim that it was just like the flu, or his blatant contradiction of his own experts on how long it would take to come up with a vaccine — is never helpful, Mr. Litt said.

“You have to be honest about it, that you can’t bring the number of cases down to zero,” he added. “A good response would be where you say that people will suffer and in some cases die — and that’s awful and graphic — but the least you can do is explain the situation.”

That is what the current British prime minister, Boris Johnson, did in a recent news conference. Until recently, he has tended to play down the threat as much as Mr. Trump has; on Thursday, his expression somber and his voice grave, Mr. Johnson said the coronavirus was “the worst public health crisis for a generation.”