I checked the Firefly wiki for this one.
In the entry on Reavers, it says, under Origins and Infection:
Reavers originated from one of the outermost planets in the solar system named Miranda. The Alliance had been testing behavioral modification on this planet through a chemical known as "Pax." Pax being Latin for peace. The chemical was designed to inhibit aggression and anger, and was administered through the air recycling system on the planet. A tenth of a percent of the population had the opposite intended reaction, with their aggression and primal instincts boosted beyond simple madness and into the realm of animalism, robbing them of their humanity.
Reaverdom does not seem to be exclusively chemical in origin. Some victims of Reaver attacks are left alive and forced by Reavers to view the tortures they inflict on others. When this happens, the person's mind is so damaged by what they were shown, that the only way to cope with it is to begin to act as a Reaver does, eventually becoming one themselves. This was witnessed on one occassion by the crew of the Serenity when they found one such victim on a derelict ship (Bushwhacked). (Mal's knowledge of this indicates that he either has heard of such things before or witnessed it himself.)
So the actual origin is not a virus, it's chemical and the way it "spreads," in the few times when it does, is by people being driven insane by the torture from Reavers.
(Note: For those who saw this when I first posted it, it's been edited so the links now go to the Firefly Wiki at Wikia, which is up and working, so there's no need for the Google cache.)