No, your Pokemon do not have Pokerus anymore, but still have the symptoms. Over time, the Pokerus gets cured, and the infection can no longer spread to other party members. However, they retain the infection's EV training buff even when the purple face replaces Pokerus, which could explain why the game thinks they have Pokerus during battles when this is not actually the case.
Once a Pokémon's immune system has fought off the virus, they cannot spread it further, nor can it be spread to them by other Pokémon. In Generations II and III, a dot will appear on the Pokémon's status screen to indicate that they have had the virus previously, while in Generation IV and onward, it will be a small face. Though the virus is gone, the Pokémon will still gain twice the EVs it would have gained before the virus.
Also, the amount of time it takes for Pokerus to stop becoming contagious varies, so double check to see if there are any Pokemon that still have it. If so, either keep them in a PC so the virus doesn't decay at all so you can save it for later, or spread it to as many other Pokemon as possible.
If you want to learn more about Pokerus go here.