The Doctor
The Doctor is the main character in the story and it is told by him and from his perspective. We assume it is also an autobiographical story that he is recalling from his memory. The Doctor is a respected professional whose reputation is good and who is conscientious enough to make house calls. He feels that he can make good diagnoses on what he first sees when he sees a patient in their own environment. He seems to be a personable professional who is able to put parents at their ease, even when they are worried about the health of their child.
The Doctor is very hands on in a general sense but the strange way in which he claims to have fallen in love with this particular child takes this hands on treatment to a new and disturbing level. The Doctor seems at once uneasy with the feelings that have taken him over, but at the same time accepting of them. He does not seem to have experienced feelings like this about a child before. The end of the story leaves us feeling very uneasy because his examination of the little girl seems to be taking on a darker and more inappropriate form.
The Child
The Child is the patient whom the Doctor has been summoned to see. She is feeling very unwell and her parents are at their wits' end, calling the doctor to see if he can diagnose what is wrong with her. She is fractious and difficult, possibly because she is feeling so sick. She becomes more and more uncontrollable and this does not seem to be her characteristic behavior because her parents seem shocked and embarrassed, rather than accustomed to her increasingly aggressive reaction to the Doctor. It is also possible that she is feeling strange about the doctor because she can sense something changing in his demeanor as he deals with her. As she gets more defensive she also becomes more "savage". It is suspected but never confirmed that she is suffering from diphtheria.
The Parents
The parents of the child have called the doctor to their home because they are worried about their sick child. They are not well-off - the doctor remarks that the home is poor - but they are loving and responsible. They are not sure how they feel about the doctor, or about doctors in general. They know that they need his expertise and knowledge if they are to find out what is wrong with their daughter. They do not wholly trust him, but we are not sure if they are distrustful of all doctors, or if they are getting some kind of sixth sense about this doctor in particular. They also seem to resent the fact that they cannot help their own daughter and are powerless to do anything without the doctor's help.
They are very embarrassed by their daughter's behavior and do not seem either familiar with it or accustomed to it. They don't know how to calm her or restrain her from fighting the doctor off. It is a strange dynamic, but one that they do not seem to know how to deal with, and it is clear they feel helpless in the face of an unfamiliar situation.