I'd guess so. The doctor has to pronounce on whether it's a "grave" infection. I certainly wouldn't recommend giving the peeps vancomycin or even penicillin without a doctor's prescription; in fact, IIRC, there's a warning about it in the game.

Although 58 is fairly young, it's not an unusual age for the peeps to die. You should be able to choose the next generation from all the peeps' children; that is, those still in the house and those who have gone to college.