• The CDC doesn't have enough resources to properly track breakthrough infections in real time, Dr. Scott Gottlieb told CNBC's "Squawk Box" on Wednesday.
• Currently, the CDC website indicated that 5,914 Americans have either died or been hospitalized with Covid breakthrough infections.
• "The breakthrough infections, as rare as they are, have the potential to forward transmit with the same capacity as an unvaccinated person," CDC Director Walensky said.

The Centers for Disease Control and Prevention doesn't have enough resources to properly track Covid-19 "breakthrough cases," Dr. Scott Gottlieb told CNBC's "Squawk Box" on Wednesday.

The CDC stopped tracking every breakthrough case in the U.S. on May 1, focusing just on those that result in hospitalization or death, a move physicians and scientists are increasingly criticizing.

"They don't have good real-time reporting," Gottlieb said. "We need to fix this, and this can be fixed. I mean we can properly resource them and build out better capabilities there."

With cases surging across the U.S. and new research showing that fully vaccinated people can transmit the virus as asymptomatic carriers, the scientific community is increasingly calling on the agency to track every breakthrough case, which is when fully vaccinated people test positive for Covid.

More than 5,900 fully vaccinated Americans have either died or been hospitalized with Covid breakthrough infections through July 19, according to the CDC's most recent data. The website also notes that 1,821 of those cases were either "asymptomatic or not related to Covid-19."

While none of the vaccines are 100% effective, they have shown strong results in protecting against getting severely sick or dying from the virus, health officials say. So the vast majority of breakthrough infections are generally mild or asymptomatic, according to the CDC and the World Health Organization.

People who are vaccinated and are exposed to the delta variant are seven times less likely to develop symptoms if infected than unvaccinated people, and 20 times less likely to experience hospitalization or death than someone who is unvaccinated, CDC Director Dr. Rochelle Walensky said in a briefing Tuesday.

Before the delta variant dominated the U.S., the alpha variant first found in the U.K. made up the majority of cases. People infected with the alpha strain weren't a big source of transmission, unlike delta, which was a driving factor in the CDC's decision to reverse its position on masks in areas with low vaccination rates where Covid cases are spiking.

"We felt it important for people to understand that they have the potential to transmit virus to others," Walensky said.

There is varying data on the effectiveness of vaccines against the delta strain, but if mRNA vaccines are 90% to 95% effective in general, then someone who is fully vaccinated and exposed to the delta strain has a 1-in-10 or 1-in-20 chance of getting a breakthrough infection.

"That's why we're saying in areas of substantial or high transmission, even if you are vaccinated, that we believe it's important to wear a mask in those settings," Walensky said.

People infected with the delta variant carry up to 1,000 times more virus in their nasal passages than other strains, resulting in higher transmissibility. New CDC data indicates that people who are vaccinated can carry the same amount of virus as someone who is unvaccinated.

"The breakthrough infections, as rare as they are, have the potential to forward transmit with the same capacity as an unvaccinated person," Walensky said.

Asymptomatic breakthrough infections are not the only thing that the CDC doesn't have real-time data on, Gottlieb also said that the agency doesn't have the resources to properly track flu infections or flu deaths each year either.

"They actually derive it off a model," Gottlieb said. "So the confidence interval is very wide."

Gottlieb expects the CDC to improve its tracking of the delta variant in the coming months.

"They'll come out with a very good analysis of this delta wave, who is getting infected and how probably in about four months, maybe a little longer than that," he said.