I'd just assume that when the city began to fall, and people with bites came in for help, the hospital got overrun by the walkers that followed people who fled them and the casualties from the inside, such as the bite victims. After the doctors/police/military, whoever was still inside and trying to hold the hospital, realized that it was too late they fled to the roof, got the ladder, and painted the sign to warn other survivors that the hospital was overrun and they would be walking into a building full of walkers. They may have starved to death on the roof waiting for rescue, died during an escape attempt down the fire stairs, or perhaps made it to Crawford or out of the city.
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