The boy's illness occurs fairly late in the book, right after the man shoots the flare gun after coming across the wrecked ship. The boy tells his dad that he doesn't feel so good and vomits shortly after. He is also running a very high fever, and we are specifically told early in the illness that the man is "terrified." He's an adult. He knows the situation and how bad an illness is with the current state of the world. Food is nearly impossible to come by. Clean water is an issue. Doctors and medications are an impossibility. The man has a first aid kit with some antibiotics in it, but he knows the shelf life of those is short. Basically, the man knows that he has nothing to help the boy with, and the boy is so sick that he won't even eat to keep his strength up or drink to maintain proper hydration levels. The man is incredibly worried about his son's life:
He held him all night, dozing off and waking in terror, feeling for the boy's heart.
As for the boy, he spends most of his time unconscious. When he is awake, the boy is barely able to function and put together a coherent thought. The one thing that alerts the reader to the boy's fear is his continued insistence that his father not leave him.
Don't go away, the boy said.
Of course I wont go away.
Even for just a little while.
No. I'm right here.