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

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.