Our boy Francone had the hiccups. Not knowing what to do, I tried every type of medicine, eventually giving him a shot of Vancomycin.

Then I got the "Gravely Ill" trophy right after the shot. He still had the hiccups. Eventually I figured out what to do with the hiccups. But was that a bug? or we had a deadly case of the hiccups?

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No... it was the shot that did it. You gave them a med for something they didn't have and in turn made them sick.

The trophy wouldn't have been awarded if the child had not needed vancomycin. As the game tip says, "If someone is sick a lot, consult a doctor." Unless a doctor's consultation had been tried, the underlying infection would remain undiscovered, although the game was aware of the condition and acted accordingly. A peep with an underlying infection will be repeatedly ill, possibly with many different symptoms, which also need to be treated, Administering antibiotics always has side effects, but it looks like they were needed here.