The Terminal Experiment

Dr. Peter Hobson invents a machine that detects a brain pattern that leaves the body after death, a pattern many believe is a soul. In order to test their theories on immortality and life after death, Hobson and his friend Sarkar Muhammed create three electronic simulations of Hobson's own personality. When people Hobson had a grudge against begin to die, he and Sarkar must try to find out which is responsible. But all three simulations – two modified, one a "control" – escape Sarkar's computer, into the Internet and the World Wide Web.

The book won the 1995 Nebula Award for Best Novel, and was nominated for the Hugo Award for Best Novel in 1996. Sawyer received a writer's reserve grant from the Ontario Arts Council in 1993 in support of his writing the novel.

Wikipedia

Synopsis from Open Library.