Second-order mean-field susceptible-infected-susceptible epidemic threshold

Given the adjacency matrix A of a network, we present a second-order mean-field expansion that improves on the first-order N-intertwined susceptible-infected-susceptible (SIS) epidemic model. Unexpectedly, we found that, in contrast to first-order, second-order mean-field theory is not always possible: the network size N should be large enough. Under the assumption of large N, we show that the crucial and characterizing quantity, the SIS epidemic threshold ₂, obeys an eigenvalue equation, more complex than the one in the first-order N-intertwined model. However, the resulting epidemic threshold is more accurate: ₂^ (2) =₂^ (1) +O ({₂^ (1) }N), where the first-order epidemic threshold is ₂^ (1) =1{₁ (A) } and where ₁ (A) is the spectral radius of the adjacency matrix A.