A fixed point theorem for a general epidemic model

Epidemic type models typically undergo a phase transition when the infection rate surpasses the epidemic threshold. However for networks having degree–degree correlations, the epidemic threshold has never formally been defined and there is a shortage of rigorous mathematics describing epidemic phase transitions. In the context of disease spreading on top of a complex network, this paper provides a set of necessary and sufficient conditions for the occurrence of a persistent infected state. As a proof of principle we demonstrate that the percolation and SIS/SIR epidemic models on complex correlated networks satisfy the assumptions necessary for a single phase transition. This paper attempts to highlight commonalities in a variety of different interacting particle systems.