The susceptible--infected--susceptible (SIS) epidemic process on complex networks can show metastability, resembling an endemic equilibrium. In a general setting, the metastable state may involve a large portion of the network, or it can be localized on small subgraphs of the contact network. Localized infections are not interesting because a true outbreak concerns network--wide invasion of the contact graph rather than localized infection of certain sites within the contact network. Existing approaches to localization phenomenon suffer from a major drawback: they fully rely on the steady--state solution of mean--field approximate models in the neighborhood of their phase transition point, where their approximation accuracy is worst; as statistical physics tells us. We propose a dispersion entropy measure that quantifies the localization of infections in a generic contact graph. Formulating a maximum entropy problem, we find an upper bound for the dispersion entropy of the possible metastable state in the exact SIS process. As a result, we find sufficient conditions such that any initial infection over the network either dies out or reaches a localized metastable state. Unlike existing studies relying on the solution of mean--field approximate models, our investigation of epidemic localization is based on characteristics of exact SIS equations. Our proposed method offers a new paradigm in studying spreading processes over complex networks.
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