I got it right now.
A Ditto I traded over GTS to my HG cartridge have it. Afraid of the Ditto's origin (you know, GTS stuff are hacked "sometimes") I passed the Virus to a few of my pokemon and then released the Ditto. Saved them on the PC, and there they are, waiting for the next move. Never had it before, but heard about it.
By the way. The Pokerus doesn't desapear if the Pokemon infected (the one with pink banner on the "status" page) is at the PC when the DS clock hits midnight. If its not, the Virus benefits will still be active, but it won't spread it to other pokemons anymore.
As for spreading, I'm not sure, actually. I know that if the infected Poke is in your team, slowly the other members will be getting infected.