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 [Phys. Rev. E 89, 052802 (2014)PLEEE81539-375510.1103/PhysRevE.89.052802] stated that the correlation of infection at the same time between any pair of nodes in a network is non-negative for the Markovian susceptible-infected-susceptible (SIS) and susceptible-infected-removed (SIR) epidemic models. The arguments used to obtain this result rely strongly on the graphical construction of the stochastic process, as well as the Fortuin, Kasteleyn, and Ginibre (FKG) inequality. In this Comment, 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 a simple graph for which the nodal infection in the SIR Markovian model is negatively correlated.

Reply to “Comment on ‘Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated' ”

“…The counterexample in the Comment [1] seems to contradict positive correlations in SIS and SIR, but we argue that the counterexample is misleading. Rodriguez et al [1] consider the correlation between X t (i) and X t (j ) in the SIR model, where X t (i) ∈ {S, I, R} is the state of node i at time t and they choose S = 0 for the susceptible, I = 1 for the infected, and R = 2 for the removed state. However, it seems more natural as in Refs.…”

“…[2] also holds for the susceptible-infectedremoved (SIR) model. Rodriguez et al [1] rightly point out that the monotonicity, which is crucial in the proof in Ref. [2], does not hold for the SIR model, although monotonicity applies to the SIS model.…”

“…Cator and Van Mieghem are grateful to the authors of the Comment [1] on the paper [2], which focused mainly on the susceptible-infected-susceptible (SIS) model, for two reasons. First, Cator and Van Mieghem carelessly assumed that the proof in Ref.…”

“…[2], does not hold for the SIR model, although monotonicity applies to the SIS model. Second, Rodriguez et al [1,Ref. [2]] refer to a paper [3] by Donnelly, which was unfortunately overlooked in Ref.…”

“…The covariance between X t (i) and X t (j ) in Ref. [1] is difficult to physically interpret, in contrast to Y i (t ) and Y j (s), and, moreover, depends on the somewhat arbitrary coding of the states. Indeed, by choosing the states X t (i) ∈ {0, 1, −1}, a different sign in the covariance cov(X t (i), X t (j )) can be found.…”