Despite serious concerns raised by the proven ability of computer viruses to spread between individual systems and establish themselves as a persistent infection in the computer population, there have been very few efforts to analyze their propagation theoretically. The strong analogy between biological viruses and their computational counterparts has motivated us to adapt the techniques of mathematical epidemiology to the study of computer virus propagation. In order to allow for the most general patterns of program sharing, we extend a standard epidemiological model by placing it on a directed graph and use a combination of analysis and simulation to study its behavior. We determine the conditions under which epidemics are likely to occur, and in cases where they do, we explore the dynamics of the expected number of infected individuals as a function of time. We conclude that an imperfect defense against computer viruses can still be highly effective in preventing their widespread proliferation, provided that the infection rate does not exceed a well-defined critical epidemic threshold.