Nearly 4,000 Massachusetts residents who were fully vaccinated against COVID-19 have tested positive for the virus.

These so-called ‘breakthrough’ cases occur when people contract the disease 14 days or more after receiving their second dose of the Pfizer-BioNTech or Moderna vaccine or the Johnson & Johnson one-shot jab, and which officials say are not surprising.

With more than 3.7 million people in Massachusetts who have completed their vaccine series, this means breakthrough infections have occurred in about one out of every 1,000 – or 0.001 percent – according to the state’s Department of Public Health (DPH).

This is roughly in line with mid-May figures from the Centers for Disease Control and Prevention (CDC) – the latest for which data is available – which showed 9,245 out of at least 95 million Americans, or 0.009 percent, later tested positive for the virus

Experts say the data show how well the vaccines work in real life and, although the shots are not 100 percent foolproof, suggest that developing the virus after being fully vaccinated is very rare.

According to health department data, the number of breakthrough infections have slowly been rising.

On May 17, 3,083 fully vaccinated people later contracted COVID-19. By June 5, that number increased to 3,641, an 18 percent increase.

The latest update from the health department shows that the number then rose by 4.1 percent to 3,791.

Experts have warned that breakthrough cases will continue to occur as tens of thousands of people are vaccinated every day across the country.

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