Though breakthrough COVID-19 cases have much lower hospitalization and mortality rates than infections among the unvaccinated, the Delta variant has led to more cases than anticipated. In late July, a leaked document from the CDC revealed there were around 35,000 symptomatic COVID infections per week among the fully vaccinated (though the agency has not yet confirmed that data). “Our vaccines are working exceptionally well. They continue to work well for Delta—with regard to severe illness and death, they prevent it,” CDC Director Rochelle Walensky told CNN on Aug. 5. “But what they can’t do anymore is prevent transmission.” Doctors and scientists are still studying the three vaccines in the U.S.—Moderna, Pfizer, and Johnson & Johnson—to learn about their efficacy against the new and more contagious Delta variant and the mechanisms through which breakthrough infections occur. Now, a new study out of Israel, which has not yet been peer-reviewed, found that people given the Pfizer vaccine are more likely to test positive for COVID after a certain amount of time. RELATED: Leaked CDC Data Causes Concern About Breakthrough Infections. Researchers from Leumit Health Services and the Shamir Medical Center Institutional Review Board in Israel studied a group of 33,943 fully vaccinated adults who were given the Pfizer vaccine. They broke the patients into three age groups: 60 or above, between 40 and 59, and between 18 and 39 years old. The study followed the vaccinated patients over the course of several months and tested them for breakthrough cases of COVID-19. Overall, around 1.8 percent of patients had breakthrough cases, indicating it’s still quite rare to get the virus after being vaccinated. But the odds, regardless of age, for testing positive were higher among people who had their last vaccine dose more than five months ago. Among patients 60 and older, researchers found that the odds of testing positive for a breakthrough case were three times higher after five months had passed.

RELATED: For more up-to-date information, sign up for our daily newsletter. The study’s results align with what Pfizer CEO Albert Bourla said about the vaccine earlier this year, though Pfizer’s internal numbers differ. Bourla acknowledged in July that the company’s vaccine efficacy drops to around 84 percent after four to six months. A company-funded study, which has also not yet been peer reviewed, found that the vaccine was strongest between one week and two months after receiving the second dose. It then declined in efficacy an average of 6 percent every two months.ae0fcc31ae342fd3a1346ebb1f342fcb Appearing on CNBC’s The Exchange in late July, Bourla said the data suggested a strong need for a booster dose. “The good news is that we are very, very confident that a third dose, a booster, will take up the immune response to levels that will be enough to protect against the delta variant,” he said. Last week, the U.S. Food and Drug Administration (FDA) and CDC approved a booster shot for those with weak or compromised immune systems, but wider use of booster shots has not yet been mandated. RELATED: The FDA Is Not Authorizing a Booster for This One Vaccine. In the meanwhile, the CDC recommends that vaccinated people remain vigilant about social distancing and masking up—both inside and out, given the increased infectiousness of the Delta variant. “If you’re going home to somebody who has not been vaccinated, to somebody who can’t get vaccinated, somebody who might be immunosuppressed or a little bit frail, somebody who has comorbidities that put them at high risk, I would suggest you wear a mask in public indoor settings,” Walensky said on CNN in early August. RELATED: Moderna Says These 3 Things Will Cause More Vaccinated People to Get COVID