# On bounding exact models of epidemic spread on networks

## TL;DR

This paper rigorously proves that the N-Intertwined Mean-Field Approximation (NIMFA) provides an upper bound on the true epidemic spread on networks, using comparison theorems and negative correlation properties.

## Contribution

It introduces a rigorous proof that NIMFA overestimates the epidemic process and extends the framework to alternative mean-field closures for weighted and directed networks.

## Key findings

- NIMFA provides an upper estimate of the epidemic process.
- The epidemic process exhibits negative correlation.
- Results apply to arbitrary weighted and directed networks.

## Abstract

In this paper we use comparison theorems from classical ODE theory in order to rigorously show that the N-Intertwined Mean-Field Approximation (NIMFA) model provides an upper estimate on the exact stochastic process. The proof of the results relies on the observation that the epidemic process is negatively correlated (in the sense that the probability of an edge being in the susceptible-infected state is smaller than the product of the probabilities of the nodes being in the susceptible and infected states, respectively), which we also prove rigorously. The results in the paper hold for arbitrary weighted and directed networks. We cast the results in a more general framework where alternative closures, other than that assuming the independence of nodes connected by an edge, are possible and provide a succinct summary of the stability analysis of the resulting more general mean-field models.