If the doctor says he's okay, he'll get better with time. You can help this by giving him health enhancing objects as dansmot listed as well as the hand gel if you happen to run into it. Keep the fridge well-stocked since they can starve if it runs out of food and get even weaker.

The only problem is if he gets sick while the game is closed, his health will start to drop even more so just keep a good watch over him and I'm sure he'll be okay!