Separatrices between healthy and endemic states in an adaptive epidemic model

The separatrices between the healthy and endemic states in the bistable regime and issues related to determining the critical infected fraction i(crit) in a recently proposed epidemic model on an adaptive network are analyzed. The epidemic follows the susceptible-infected-susceptible (SIS) process and the network adapts by breaking connections between healthy and infected individuals and rewiring to other healthy individuals. Using a set of mean field equations, the separatrix can be found for a given set of system parameters. The unstable fixed point is shown to correspond to a highly atypical network configuration and thus using it as an estimate of i(crit) may be erroneous. The characteristics of initial configurations leading to either a healthy or an endemic final state are investigated, and how the configurations determine i(crit) is discussed. Results of numerical simulations confirm that the unstable fixed point does not in general give a good estimate of i(crit). While our discussion focuses on an adaptive SIS model, the approach in determining i(crit) and the conclusion are general, and they can be applied to other co-evolving dynamical systems.