The reality of Geostigma is that it was contracted by coming into contact with the tainted Lifestream, which was corrupted when Sephiroth's body was destroyed at the end of FFVII. When the Lifestream reinforced Holy at Midgar, the Spirit Energy carried this contamination, and those who came into contact with the Lifestream were afflicted with the disorder. Rufus, although he survived, was presumably still in Midgar during the Meteor/Holy/Lifestream incident, and thus came into contact with the Lifestream, setting the stage for the eventual emergence of his Geostigma. The same holds true for Denzel;
On the Way to a Smile explicitly reveals his contact with the Lifestream.

Originally Posted by
On the Way to a Smile, Denzel, 1-3
"Mrs. Levy!" Denzel pulled at the handle and tried to open the door.
"Denzel, stop it!"
"But—!" Denzel pulled the handle again.
Levy stood with her back to the door, legs and arms spread against the frame to keep him from coming out. "Close the door, Denzel!"
He could see several streams of light go through Levy's body, dancing and ricocheting off the walls, glowing snakes running riot in the room. This wasn't in the monster book. Have to run away and tell an adult. No…I'm the one who has to fight.
"Mrs. Levy!" As soon as he shouted, the light struck her. She groaned and the light shifted into a slender rope-like form that forced its way into the bedroom through the gap between Levy and the wall. He got a glimpse of her collapsing to the ground an instant before he was thrown back by the light and passed out.
While Geostigma can vary in how long it takes to appear (Mrs. Levy exhibits the symptoms immediately) and how long it takes to cause death (Mrs. Levy dies almost immediately, while Gaskin survives at least a month after the first evidence of Geostigma), it's clear that Rufus, Denzel, and the children contracted it by coming into contact with those glowing green tendrils of Spirit Energy.