This might mean that the proper study of virists is virology, but that make problems, since we already have viruses (viri?) to complicate things.

The study of viruses should be iology (virus and Gk ios both being from PIE *wisos 'poison').

The study of men is andrology. The study of virists, if by that you mean to study virists, would be viristology, whatever a virist is.

If virist involves vir 'man' then its feminine counterpart could be muliebrist.

Endless possibilities.