Generating function approach to the effective degree SIR model.

The effective degree SIR model describes the dynamics of diseases with lifetime acquired immunity on a static random contact network. It is typically modeled as a system of ordinary differential equations describing the probability distribution of the infection status of neighbors of a susceptible node. Such a construct may not be used to study networks with an infinite degree distribution, such as an infinite scale-free network. We propose a new generating function approach to rewrite the effective degree SIR model as a nonlinear transport type partial differential equation. We show the existence and uniqueness of the solutions the are biologically relevant. In addition we show how this model may be reduced to the Volz model with the assumption that the infection statuses of the neighbors of an susceptible node are initially independent to each other. This paper paves the way to study the stability of the disease-free steady state and the disease threshold of the infinite dimensional effective degree SIR models.