Members included Vivek Murthy, a former surgeon general; Ezekiel J. Emanuel, a prominent oncologist and a vice provost at the University of Pennsylvania; and Lisa Monaco, who served as a homeland security and counterterrorism adviser to Mr. Obama. And when he spoke on Thursday, he did so against a backdrop of American flags, reading from teleprompters to the click of cameras and beginning with a nod to his “fellow Americans,” a setting reminiscent of a White House address.

“No president can promise to prevent future outbreaks,” he said. “But I can promise you this. When I’m president, we will be better prepared, respond better and recover better. We’ll lead with science. We’ll listen to the experts. We’ll heed their advice. And we’ll build American leadership and rebuild it to rally the world to meet the global threats.”

Throughout his remarks, Mr. Biden nodded — as he often does — to what he cast as the resiliency and potential of the American people.

Mr. Biden, who is 77, did not stop to take shouted questions about his own health.

He has previously expressed shock and frustration at Mr. Trump’s past skeptical remarks about the severity of the virus, and has sketched out other steps he would take as president to fight it, noting his work as vice president in combating Ebola. Ron Klain, who was Mr. Obama’s Ebola “czar,” is a top Biden adviser.

Mr. Trump’s own somber address Wednesday night, in which he announced he was blocking most travel from continental Europe and promised new aid for workers and businesses, was a break from his previous efforts to play down the effects of the outbreak. But he also mischaracterized some of his administration’s new travel policies and described the threat as a “foreign virus,” though Americans are infected along with many in other countries.

The Trump campaign quickly issued a response to Mr. Biden’s remarks on Thursday. “In times like this, America needs leadership and Biden has shown none,” said Tim Murtaugh, a campaign spokesman. “President Trump acted early and decisively and has put the United States on stronger footing than other nations. His every move has been aimed at keeping Americans safe, while Joe Biden has sought to capitalize politically and stoke citizens’ fears.”

As for Mr. Sanders, Mr. Murtaugh said in another statement, “He’s just another Democrat candidate for president trying to score political points by recklessly provoking anxiety and fear.” He also argued that the proposal from Mr. Sanders, the Vermont senator — who supports a sweeping single-payer system — would “drive doctors and other medical workers away from the profession, leaving America woefully unprepared for public health emergencies.”