The Broadway cast was less than thrilled when it found out who one of their fellow performers would be. It made them squeamish — not because of who it was but because of what it was.

They would be sharing the stage, it turned out, with a live rat.

“The idea of a rat was not exactly familiar to me,” said Alex Sharp, an actor who plays the leading role. “It was just a thing you see in the subway that has diseases.”

But Toby, the name of the rat kept by the teenager with autism at the center of the show, “The Curious Incident of the Dog in the Night-Time,” has managed to win over the affection of audiences and the cast — so much so that the rodent role has been expanded.

“She’s a special rat,” said Benjamin Klein, the associate director of the play, which opened in October to critical acclaim.