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This paper considers the spread of worms in computer networks using insights from epidemiology and percolation theory. We provide three new results. The first result refines previous work showing that epidemics occur in scale-free graphs more easily because of their structure. We argue, using recent results from random graph theory that for scaling factors between 0 and ∼3.4875, any computer worm infection of a scale-free network will become an epidemic. Our second result uses this insight to provide a mathematical explanation for the empirical results of Chen and Carley, who demonstrate that the Countermeasure Competing strategy can be more effective for immunizing networks to viruses or worms than traditional approaches. Our third result uses random graph theory to contradict the current supposition that, for very large networks, monocultures are necessarily more susceptible than diverse networks to worm infections.