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1.1- The Problem of Computer Viruses

Computer viruses were originally thought of as problematic primarily because of their ability to carry out directed attacks against isolated systems  . Their potential ability to flow from user to user in a system meant that attacks could reach parts of the system that had been thought to be more secure. In the last few years, however, as viruses have been written and released outside of controlled environments and into the world's computing community, their ability to spread between individual systems and thus affect a global collection of systems has proved to be of greater concern.

Since the first documented reports of microcomputer viruses  in the mid-1980's, they have spread throughout the world. At this date, we estimate very conservatively that there are many thousands of microcomputers with active virus infections, and this population of infected systems continues to spread the infection. The Internet Worm  infected at least hundreds and probably thousands of computers on the Internet in the space of a few hours in November, 1988, with the resultant loss of a substantial amount of service and many hours devoted to expurgating it from systems.

Our long-term goal is to develop and analyze quantitatively models which capture the spreading characteristics of computer viruses. The potential benefits of doing so can be divided into two major categories.

First, mathematical models could aid in the evaluation and development of general policies and heuristics for inhibiting the spread of viruses. Although it is well known that a general-purpose computing system need only satisfy minimal conditions to be capable in principle of being completely infiltrated by a virus  , the set of conditions under which this is likely to occur may be considerably more restricted. We wish to gain a quantitative understanding of the vulnerability of current systems to viral infections and to determine the effectiveness of proposed heuristics for inhibiting viral spread  . In a similar vein, mathematical modelling of the sort we describe could be helpful in the design of new systems -- allowing a reasonable tradeoff between the ease with which legitimate programs can flow and the ease with which viruses can spread.

A second major use for mathematical modelling, more in the spirit of biological epidemiology, is to apply it to a particular epidemic. In its more passive application, modelling can aid in predicting the course of a particular epidemic, so as to plan what resources will be needed to deal with the problem. A more aggressive role for modelling, which is gaining popularity in the biological realm, is to use it to determine the optimal policy for controlling the course of a particular epidemic by isolating or immunizing the population at appropriate times  .

In this paper, we shall deduce a number of general properties of the spread of computer viruses from simple models which capture some essential features of the networks in which they propagate. We believe that this work is an important first step towards a theory which ultimately will be sufficiently realistic to evaluate specific proposals for thwarting the spread of computer viruses.

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