Threshold conditions for SIS epidemic models on edge-weighted networks

• A pairwise model of SIS epidemic models with a general transmission rate is formulated.
• We obtained the epidemic threshold by the linear stability analysis.
• We developed a method to judge the sign of the leading eigenvalue of a matrix.
• The difference between SIS and SIR models on weighted networks is shown.

We consider the disease dynamics of a susceptible–infected–susceptible model in weighted random and regular networks. By using the pairwise approximation, we build an edge-based compartment model, from which the condition of epidemic outbreak is obtained. Our results suggest that there exists a remarkable difference between the linear and nonlinear transmission rate. For a linear transmission rate, the epidemic threshold is completely determined by the mean weight, which is different from the susceptible–infected–recovered model framework. While for a nonlinear transmission rate, the epidemic threshold is not only related to the mean weight, but also closely related to the heterogeneity of weight distribution.