Like real-life viruses, Pokerus can spread amongst your Pokemon. More specifically, after a battle (or escaping a battle), Pokerus can spread between adjacent Pokemon in your battle party, provided they aren’t immune to Pokerus.
Furthermore, Pokemon will automatically become cured of Pokerus over time. When this occurs, a little smiley face icon will appear below their picture. Cured Pokemon will become immune to Pokerus but will retain all the benefits of Pokerus.