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1.4- Modelling Viral Epidemics on Directed Graphs

We account for the heterogeneous communication pattern among individual computer systems in a new and general way: by representing an individual system (a microcomputer, for instance) as a node in a graph. Directed edges from a given node j to other nodes represent the set of individuals that can be infected by j. A rate of infection is associated with each edge. Similarly, a rate at which infection can be detected and ``cured'' is associated with each node.

Throughout this paper, we shall study one of the simplest of the standard epidemiological models, the SIS (susceptible infected susceptible) model, on these graphs. In the SIS model, individuals immediately become susceptible once they are cured of an infection. In our case, this represents an extreme in which users do not become more vigilant after having experienced a viral infection. Our emphasis will be on determining the probability that an infection becomes extinct in a specified population. Under the conditions in which an epidemic is viable, we characterize the expected number of infections as a function of time, particularly equilibria and fluctuations about them. We shall recover some of the well-known results for homogeneous interactions as limiting cases of our more general results.

In the next section, we discuss this model on random graphs. In doing so, we gain a good deal of insight into the relationship between homogeneous and graph models of epidemics and develop a number of useful analytical techniques and approximations. Then, in sections 3 and 4, we investigate the model on hierarchical graphs and N-dimensional cartesian lattices. We shall conclude in section 5 with a summary of our findings, their potential implications for hindering the spread of computer viruses, and recommendations for future work in this area.

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