Got the Flu (or Mumps)? Check the Eigenvalue!

For a given, arbitrary graph, what is the epidemic threshold? That is, under what conditions will a virus result in an epidemic? We provide the super-model theorem, which generalizes older results in two important, orthogonal dimensions. The theorem shows that (a) for a wide range of virus propagation models (VPM) that include all virus propagation models in standard literature (say, [8][5]), and (b) for any contact graph, the answer always depends on the first eigenvalue of the connectivity matrix. We give the proof of the theorem, arithmetic examples for popular VPMs, like flu (SIS), mumps (SIR), SIRS and more. We also show the implications of our discovery: easy (although sometimes counter-intuitive) answers to ‘what-if’ questions; easier design and evaluation of immunization policies, and significantly faster agent-based simulations. ∗[email protected] †[email protected] ‡[email protected] [email protected] ¶[email protected] 1 ar X iv :1 00 4. 00 60 v1 [ ph ys ic s. so cph ] 1 A pr 2 01 0