Mrs. Slade and Mrs. Ansley have known each other for a long time. When they were young girls both competing for the attention of Delphin Slade, Mrs. Slade had hoped that Mrs. Ansley would catch "Roman Fever" which then meant a real illness like malaria.
Mrs. Ansley does catch the "Roman Fever"--both the real illness and the figurative one...the one that made her secretly meet Delphin one evening and conceive her daughter. So Roman Fever also means something like Spring Fever when love is in the air.
Later in life, as the two women and their families continue to meet and run in the same circles and see each other in Rome, "Roman Fever" means the jealously Mrs. Slade feels toward Mrs. Ansley since the Slade daughter is so much less captivating than the Ansley daughter. This fever comes to a head when Mrs. Ansley admits that her brilliant and vivacious daughter is the offspring of the late Mr. Delphin Slade.