Washington (CNN) President Donald Trump was sitting for breakfast with American business leaders last week when the conversation turned to the dangerous new virus spreading from China .

Some of the assembled chief executives asked how Trump planned to confront the Wuhan coronavirus, which threatened to affect their global supply chains and spread into some of the world's most populous urban centers.

Trump shrugged off their concerns, according to a person familiar with the conversation, which occurred at the World Economic Forum in Davos, Switzerland. He told the group his administration had the situation under control. And he suggested the matter was a world away from the United States.

Since then, five cases of the Wuhan coronavirus have been confirmed in the US, while the death toll in China has topped 100 , with more than 4,500 cases in the mainland. Financial markets have tumbled amid concerns about the virus's spread. And Trump has changed his tone, tweeting the US was "strongly on watch" for more cases and offering to send additional assistance to China.

"Our experts are extraordinary!" he wrote.

President Donald Trump leaves after a news conference at the World Economic Forum in Davos, Switzerland.

For Trump's critics, a global pandemic has long loomed as the ultimate doomsday scenario, one that would strain a chaotic leader and expose the corners of government they claim have been hollowed out during his tenure.

That prediction, so far, hasn't materialized. But it will likely be tested as health experts prepare for more cases of the coronavirus in the US while Trump remains focused on his impeachment trial and his reelection campaign.

"The administration has been worryingly quiet how it is handling this global health crisis," said Tom Wright, director of the Center on the United States and Europe and a senior fellow in the Project on International Order and Strategy at the Brookings Institution. "The President's comment that he has it under control suggests a dangerous complacency at the highest levels. The national security adviser needs to speak on this and coordinate a whole of government response."

'Urgency, not panic'

Health and Human Services Secretary Alex Azar calls on a reporter during a press conference at the Department of Health.

On Tuesday, Health and Human Services Secretary Alex Azar told reporters that he and Trump were "speaking regularly" about the outbreak, and that he was in daily contact with White House officials.

"The President is highly engaged in this response and closely monitoring the work we're doing to keep Americans safe," Azar said.

A White House source described the attitude toward the coronavirus as a "sense of urgency, not panic."

Secretary of Commerce Wilbur Ross postponed a potential trip this week to China amid concerns over the spreading virus.

Inside the White House, the government's cross-agency response is being coordinated by the National Security Council, with Matt Pottinger, the deputy national security adviser, leading near-daily meetings, according to administration officials.

At a session Monday, the notion of travel restrictions to China was raised, according to a White House aide. Azar declined to rule out such a possibility on Tuesday, saying nothing was off the table.

"The full weight of the U.S. Government is working on this," a senior administration official said on Tuesday. "As with any interagency effort of this scale, the National Security Council works closely with the whole of government to ensure a coordinated and unified effort."

Trump has not, however, named a single official within the White House responsible for coordinating the administration's response. That has some wondering whether enough is being done in advance of a potential crisis, particularly since the role of the National Security Council under Trump has shifted away from leading a response to a health crisis to merely coordinating between agencies.

"There's a modest role within the NSC and the Cabinet just cross-talking, making sure everybody is plugged in and doing their part. But there isn't really a massive leadership role," said James Carafano, a national security expert at the Heritage Foundation. "Right now the White House might have a mild PR problem, as this is playing up a lot in the news and you're not really seeing the administration step out and say, 'This is all the things we're doing.' "

Who's in charge?

Chinese police officers stand in front of the Tiananmen Gate in Beijing, China.

In 2018, the National Security Council's top health and biodefense official, Tim Ziemer, left the council under then-national security adviser John Bolton. Ziemer's replacement, David Wade, holds a similar title, though officials said he carries less authority than his predecessor.

"One thing that's not clear is who ends up coordinating the interagency response," said Thomas Bollyky, director of the global health program at the Council on Foreign Relations. "It sounds like inside baseball, but it's a big deal -- responding to an outbreak of any kind requires coordination."

Wade reports to the head of the bureau responsible for countering weapons of mass destruction, Anthony Ruggiero, a specialist in terrorism and biodefense. Health issues were once "somebody's full-time job at Ruggiero's level," Bollyky said.

Tim Morrison, a former National Security Council official in the Trump administration who's now a senior fellow at the Hudson Institute, said the current structure had more to do with a shift in the council's role than a reduced emphasis on health security matters.

"The role of the NSC under the Trump administration has been restored to one of coordinating policy," Morrison said. "They are well and ably staffed to do everything we could ask them to do, to coordinate the interagency to deal with a coronavirus."

'Much more contagious'

A medical staff member takes the temperature of a man at the Wuhan Red Cross Hospital in Wuhan on January 25, 2020.

The administration has increased travel warnings for China and enacted screening measures at 20 airports for inbound flights from Wuhan. Roughly 2,400 people have been screened flying into US airports, the US Centers for Disease Control and Prevention said on Monday.

The State Department last week issued a Level 4 advisory for Wuhan, meaning "no American should travel to Wuhan while this virus continues to have impact," Vice President Mike Pence said Monday. Also on Monday, the State Department urged citizens to reconsider travel to China, raising the overall advisory for the country to Level 3.

The State Department also ordered non-emergency personnel working at the US Consulate General in Wuhan and their families to depart for the United States.

Still, the virus's spread has not stalled in China. Part of the problem is that it apparently can spread before symptoms appear, according to China's health minister, Ma Xiaowei. American health officials said they hadn't seen data to back up that claim.

"It means the infection is much more contagious than we originally thought," William Schaffner, a longtime adviser to the Centers for Disease Control and Prevention, told CNN. "This is worse than we anticipated."

Chinese women and children wear protective masks as they walk under decorations in a park after celebrations for the Chinese New Year and Spring Festival were cancelled by authorities on January 25, 2020 in Beijing, China.

Public health infrastructure in the US has improved after outbreaks of severe acute respiratory syndrome, Middle East respiratory syndrome and Ebola over the past two decades. The identification and testing procedures that led to the five US diagnoses of coronavirus have proceeded quickly, and experts have cited China's ability to sequence and share the virus's genome as a major step toward containing it.

But those previous outbreaks also demonstrated the importance of monitoring and responding to public perceptions and fear, something White House officials said Trump was now more attuned to after appearing to brush off concerns about the virus last week.

He wasn't the only senior official to dismiss concerns. Traveling in Florida last week, Defense Secretary Mark Esper had told reporters he wasn't "tracking" the virus.

'Unqualified to handle'

The Trump administration has already come under criticism from some Democrats, who allege it remains ill-equipped to confront a potential pandemic.

Former Vice President Joe Biden, who is vying for the Democratic presidential nomination, wrote in an op-ed published in USA Today that the prospect of a pandemic "is a challenge President Donald Trump is unqualified to handle as president" and "the steps he has taken as president have only weakened our capacity to respond."

Biden cited proposed cuts to the National Institutes of Health, the CDC and the US Agency for International Development that were included in versions of the President's budgets. While Trump has proposed cutting those agencies' funding, the federal funding package he signed in December increased the budget of NIH by billions of dollars.

In 2018, the CDC announced its Center for Global Health was discontinuing work in 39 out of 49 countries where it had helped prevent, detect and respond to dangerous infectious disease threats after funding allocated during the 2014 Ebola outbreak expired.

Some veterans of that crisis have suggested that Trump name a central figure at the White House to coordinate the government's response to the coronavirus.

A Chinese man wears a protective mask, goggles and coat as he stands in a nearly empty street during the Chinese New Year holiday in Beijing.

"I think it was a mistake for the President to say we've got it under control. We don't really know what we're dealing with yet," said Ron Klain, who was tasked by then-President Barack Obama with coordinating that administration's response to the Ebola outbreak. Klain is now an adviser to Biden's presidential campaign. "They need to put someone at the White House in charge of this. Someone needs to work on the all-of-government responses."

Obama named Klain the Ebola response coordinator after the virus infected three people in the United States, prompting widespread anxiety about the administration's handling of the crisis. Obama scrapped planned travel and meetings to oversee his government's response. During one meeting with Cabinet officials, Obama grew furious at his team's handling of the matter and insisted they be more proactive.

One of Obama's chief critics at the time: Donald Trump, who suggested the US government wasn't being forthright with the American people and offered suggestions widely rejected by health professionals.