Epidemic Spreading Over Multi-Layer Networks With Stubborn Agents

This brief studies the networked SIS (Susceptible-Infected-Susceptible) model coupled with the opinion dynamics, where the epidemic spread and the opinion evolution influence each other. We consider that the epidemic spreads over a contact network consisting of $n$ agents (i.e., communities), and the opinion evolves in an influence network with stubborn agents who do not change their opinions. To understand the effects of stubborn agents on the epidemic process, we derive the conditions for the stability, existence, and uniqueness of the healthy state and endemic equilibrium. These results show how stubborn agents' opinions affect the epidemic threshold. By the analysis of the endemic equilibrium, we find that agents' opinions can result in a lower size of the outbreak compared with the networked SIS model without considering the opinion, and show how to inhibit the endemic average level by reshaping stubborn agents' opinions.