“Long Covid” is a poorly understood set of symptoms that can plague people for several months after an active infection has ended. While these symptoms eventually go away in many patients, “there is this subset of people who have long had Covid who just cannot recover,” said Akiko Iwasaki, an immunologist at Yale University.

Only a few small studies have looked at how common or how severely Covid can occur after breakthrough infections. It’s likely rare, said Dr. Iwasaki, as breakthrough infections are unusual in the beginning and of shorter duration.

In a study in Israel, about seven out of 36 people with breakthrough infections had symptoms that lasted more than six weeks. And in a survey of Covid-19 survivors, 24 out of 44 people with a symptomatic breakthrough infection reported persistent problems.

“We really need a broader national or even international survey,” said Dr. Iwasaki.

If you can survive a breakthrough infection relatively unscathed, you will likely get away with more robust protection against variants. Essentially, the infection acts as a booster shot, researchers say, boosting your immune system’s ability to recognize and fight the virus.

Studies have shown that when people recovering from Covid-19 receive even one dose of a vaccine, their antibody levels skyrocket. “I assume similar things would happen if you had a breakthrough infection,” said Dr. Iwasaki.

The vaccines train the immune system to recognize a piece of the original virus, a strategy that could leave us vulnerable to future variants. But any exposure expands the immunity repertoire, said Dr. Mina.

Eventually, through booster vaccinations or through repeated infections, our bodies will gain sufficient training in the virus to face versions with new mutations, he said, adding, “But we’re not there yet.”