From exact stochastic to mean-field ODE models: a new approach to prove convergence results (Q2857689)

From the article: ``The present paper, in the case of a simple SIS model and a suitable class of networks, including completely connected and regular graphs, proposes a unifying framework that incorporates and discusses the details of two existing proofs and proposes a new ODE-based proof. This complements and offers an alternative to the existing proofs that are purely based on stochastic theory and partial differential equation (PDE) arguments.'' In fact: ``The main purpose of this paper is to show a new ODE-based approach. We will call this an elementary approach since a self-contained proof of the theorem will be shown without using a combination of highly specialist mathematical tools from different areas, the availability of which is beyond the opportunities of the average scientist working in\dots applied research areas.''NEWLINENEWLINEIndeed, the authors give a very intelligent proof of certain convergence (ODE-based approach) ``based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system'' and use tools of differential and integral calculus like Taylor's theorem.NEWLINENEWLINEAbout the other existing proofs, based on stochastic theory and partial differential equation (PDE) arguments, the main steps are presented.NEWLINENEWLINE``The paper is organized as follows. In Section 2, the model is formulated and we present the main result in general terms and discuss the three different approaches used to prove the convergence of exact stochastic to the mean-field model. In Section 3, we give the detailed proof based on PDE arguments, while in Section 4, the proof based on stochastic theory arguments is presented. Section 5 contains the new ODE-based approach, with pluses and minuses of the three different models included in the final section.''