  We investigate the effect of degree correlation on a
susceptible-infected-susceptible (SIS) model with a nonlinear cooperative
effect (synergy) in infectious transmissions. In a mean-field treatment of the
synergistic SIS model on a bimodal network with tunable degree correlation, we
identify a discontinuous transition that is independent of the degree
correlation strength unless the synergy is absent or extremely weak. Regardless
of synergy (absent or present), a positive and negative degree correlation in
the model reduces and raises the epidemic threshold, respectively. For networks
with a strongly positive degree correlation, the mean-field treatment predicts
the emergence of two discontinuous jumps in the steady-state infected density.
To test the mean-field treatment, we provide approximate master equations of
the present model. We quantitatively confirm that the approximate master
equations agree with not only all qualitative predictions of the mean-field
treatment but also corresponding Monte-Carlo simulations.
