Global stability for epidemic models on multiplex networks

In this work, we consider an epidemic model in a two-layer network in which the dynamics of susceptible–infected–susceptible process in the physical layer coexists with that of a cyclic process of unaware–aware–unaware in the virtual layer. For such multiplex network, we shall define the basic reproduction number R 0 V in the virtual layer, which is similar to the basic reproduction number R 0 P defined in the physical layer. We show analytically that if R 0 P ≤ 1 and R 0 V ≤ 1 , then the disease and information free equilibrium is globally stable and if R 0 P ≤ 1 and R 0 V > 1 , then the disease free and information saturated equilibrium is globally stable for all initial conditions except at the origin. In the case of R 0 P > 1 , whether the disease dies out or not depends on the competition between how well the information is transmitted in the virtual layer and how contagious the disease is in the physical layer. In particular, it is numerically demonstrated that if the difference in R 0 V and R 0 P is greater than the product of R 0 P , the deviation of R 0 V from 1 and the relative infection rate for an aware susceptible individual, then the disease dies out. Otherwise, the disease breaks out.