President Trump tapped Health and Human Services Secretary Alex Azar in January to head up the coronavirus response task force. Then, after the White House repeatedly denied reporting that he might select a coronavirus "czar," the president announced in a Wednesday night press conference he was placing Vice President Mike Pence in charge of the administration's coronavirus response.

But the president claimed Pence isn't a "czar," and insisted the vice president isn't replacing Azar.

"Mike is going to be in charge, and Mike will report back to me," the president said late Wednesday.

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In another twist on Thursday, Pence's office announced Ambassador Debbie Birx, a global health official and physician, will be the administration's new coronavirus response coordinator in Pence's office. Pence's office also named a handful of other Trump administration officials will join the coronavirus task force Azar is tasked with leading.

CBS News has requested more clarity on exactly what roles each of the three is playing. But so far, it's unclear who is in charge of the response.

Vice President Mike Pence, during a coronavirus task force meeting on Thursday afternoon, didn't provide much clarity on the matter. Asked by a reporter whether he's in charge of the task force of Azar is, Pence responded, "Uh, the president has asked me to lead our administration's effort in response to the coronavirus."

Asked again if he's leading the task force, Pence responded, "I'm leading the task force, will continue to rely on the secretary's role as chairman of the task force, and the leader of Health and Human Services. But we've known each other for many years, and worked together very closely over the years, and the president has every confidence in the secretary, as I do But the president wanted to make it clear to the American people that we're going to bring a whole of government approach to this."

Meanwhile, response to the coronavirus outbreak ramped up after Wednesday's Centers for Disease Control announcement about a confirmed coronavirus patient in California who did not come into contact with someone who recently traveled from an infected area, or travel herself. On Thursday, California Governor Gavin Newsom announced state health officials are monitoring 8,400 people for the virus, and 33 have tested positive for the virus.

Mr. Trump took reporters' questions for the first time ever in the White House briefing room Wednesday night, flanked by members of the coronavirus task force, a public showing that his administration is doing everything in their power to handle the global public health concern.

The administration has been criticized not calling for enough funding to address the spread of the virus, although Mr. Trump told reporters Wednesday night he'll approve whatever is appropriate.

The White House established the coronavirus task force on January 29. As of Thursday, here are the members:

Secretary Alex Azar, Department of Health and Human Services

Robert O'Brien, Assistant to the President for National Security Affairs

Dr. Robert Redfield, Director of the Centers for Disease Control and Prevention

Dr. Anthony Fauci, Director of the National Institute of Allergy and Infectious Diseases at the National Institutes of Health

Deputy Secretary Stephen Biegun, Department of State

Ken Cuccinelli, acting Deputy Secretary, Department of Homeland Security

Joel Szabat, acting Under Secretary for Policy, Department of Transportation

Matthew Pottinger, Assistant to the President and Deputy National Security Advisor

Rob Blair, assistant to the President and Senior Advisor to the Chief of Staff

Joseph Grogan, Assistant to the President and Director of the Domestic Policy Council

Christopher Liddell, Assistant to the President and Deputy Chief of Staff for Policy Coordination

Derek Kan, Executive Associate Director, Office of Management and Budget

Steven Mnuchin, Secretary of the Treasury

Dr. Jerome Adams, Surgeon General of the United States

Larry Kudlow, Director of the National Economic Council

— Sara Cook contributed to this report