An axiomatic foundation for epidemics on complex networks

Abstract

We provide a rigorous axiomatic framework to study epidemiology on complex networks. Our axioms apply to the epidemic spreading on complex networks in which there are explicit correlations among the degrees of connected vertices as described in [1]. We prove a necessary and sufficient condition for our epidemic model to have a nonzero stationary solution. We believe this is the first proof of such a general result. Moreover, under appropriate conditions we show that the time independent solution is the limit of a unique time dependent solution. We also provide a rigorous definition of the epidemic threshold, λc : = 1/ λ1 with λ1 denoting the largest positive eigenvalue of an operator T given in the axioms of our model. © 2009 Springer-Verlag Berlin Heidelberg.