While it is easy to forget that Schatz's mother is in the story. Remember "He came into the room to shut the window while we were still in bed." This implies that the narrator, who is clearly Schatz's father, is in bed with someone. The most reasonable inference to make would be that the someone was Schatz's mom. The most reasonable explanation for why the mother was not more involved might simply be that as a mother, she would probably been more attuned to Schatz's thinking and emotions. She would have realized sooner that something more than illness was wrong with him. Now why would the father be the one taking care of Schatz? The most logical conclusion would be to infer the mother, herself, is ill with the influenza that is going around.
Enotes has some great resources.