The U.S. will see a significant reduction in new coronavirus infections in the coming weeks, Dr. Scott Gottlieb predicted Monday on CNBC.

“I think we’re going to start to see the pandemic roll over in the United States, in terms of cases coming down,” Gottlieb said in an interview on “Squawk Box.”

However, the former Food and Drug Administration chief cautioned that, even if the top-line number of new infections falls, “we’re still going to have outbreaks in some parts of the country.”

“We’re never going to virtually eliminate this virus,” said Gottlieb, who led the regulatory agency from 2017 to 2019 in the Trump administration, reiterating a concern he raised Friday when he warned that vaccine hesitancy in the U.S. makes it unlikely that the nation will eradicate Covid like it’s done with other diseases, such as polio and smallpox.

“But I think you’re going to start to see cases come down quite dramatically as we get into May,” said Gottlieb, who serves on the board of Pfizer, which makes one of the three Covid vaccines cleared for emergency use in the U.S. Moderna makes the other two-shot vaccine. Johnson & Johnson’s one-dose vaccine has been paused by the FDA due to cases of rare but severe blood clotting issues.

On Monday, the seven-day average of daily new coronavirus cases in the U.S. was about 67,400, according to a CNBC analysis of data compiled by Johns Hopkins University. That figure is down slightly from one week ago, although it represents an increase from levels seen in late March and on par with last summer’s surge.

Deaths from the disease have fallen more considerably in the U.S. According to CNBC’s analysis of Hopkins’ data, the seven-day average of daily new Covid deaths on Monday was 723, which is down 25% compared with one week ago.

Disclosure: Scott Gottlieb is a CNBC contributor and is a member of the boards of Pfizer, genetic testing start-up Tempus, health-care tech company Aetion Inc. and biotech company Illumina. He also serves as co-chair of Norwegian Cruise Line Holdings′ and Royal Caribbean‘s “Healthy Sail Panel.” The Associated Press contributed to this report.