Both are modern methods of diagnosis.
$PCR$: It helps in detecting very low concentration of bacteria or virus at the time when the symptoms of the cdisease are not visible by amplification of their nucleic acid.
$ELISA$: Infection by pathogen can be detected by the presence of antigens or by detecting the antibodies synthesized against the pathogen.
In serum analysis early detection is not possible as presence of a pathogen is normally suspected only when the pathogen has produced a disease symptoms and by this time concentration of pathogen is already very high in the body.