Nearly 4,000 people in Massachusetts who have been fully vaccinated against COVID-19 have tested positive for the disease, adding to the growing number of breakthrough cases nationwide.

According to the Massachusetts Department of Public Health, as of June 12, there were 3,791 infections among the 3.7 million fully vaccinated people in the state, or about one out of every 1,000 individuals.

“We’re learning that many of the breakthrough infections are asymptomatic or they’re very mild and brief in duration,” said Boston University infectious diseases specialist Davidson Hamer, the Boston Herald reported. “The viral load is not very high.”

“Breakthroughs are expected, and we need to better understand who’s at risk and whether people who have a breakthrough can transmit the virus to others,” Hamer said. “In some cases, they’ll be shedding such low levels of the virus and won’t be transmitting to others.”

So-called breakthrough cases refer to cases appearing two or more weeks after a person’s final shot, that is, the second Pfizer or Moderna dose or the single-shot Johnson & Johnson vaccine.