Comment on "Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated''

TL;DR

This paper critically examines a previous claim about non-negative correlation of infection states in Markovian epidemic models on networks, showing that the claim does not hold for the SIR model and providing a counterexample.

Contribution

It clarifies the limitations of the original proof, demonstrating that the non-negative correlation does not extend to the SIR model and offering a counterexample.

Findings

The original non-negative correlation claim applies only to the SIS model.

The FKG inequality cannot be used for the SIR model due to lack of monotonicity.

A simple graph example shows negative correlation in the SIR model.

Abstract

Cator and Van Mieghem [Cator E, Van Mieghem P., Phys. Rev. E 89, 052802 (2014)] stated that the correlation of infection at the same time between any pair of nodes in a network is non-negative for the Markovian SIS and SIR epidemic models. The arguments used to obtain this result rely strongly on the graphical construction of the stochastic process, as well as the FKG inequality. In this note we show that although the approach used by the authors applies to the SIS model, it cannot be used for the SIR model as stated in their work. In particular, we observe that monotonicity in the process is crucial for invoking the FKG inequality. Moreover, we provide an example of simple graph for which the nodal infection in the SIR Markovian model is negatively correlated.