Serum (Blood Draw) taken from a patient that has completely recovered from an infectious disease is called Convalescent Serum. In which the bodies own immune system created Antibodies against the disease.
The Antibodies are detectable in a lab, even though the patient has completely recovered and shows no symptoms. Various tests such as IgG and IgM are used.
Also see: Acute-Phase Serum (active viral/bacterial serum)