Epidemic spread in human networks

One of the popular dynamics on complex networks is the epidemic spreading. An epidemic model describes how infections spread throughout a network. Among the compartmental models used to describe epidemics, the Susceptible-Infected-Susceptible (SIS) model has been widely used. In the SIS model, each node can be susceptible, become infected with a given infection rate, and become again susceptible with a given curing rate. In this paper, we add a new compartment to the classic SIS model to account for human response to epidemic spread. In our model, each individual can be infected, susceptible, or alert. Susceptible individuals can become alert with an alerting rate if infected individuals exist in their neighborhood. Due to a newly adopted cautious behavior, an individual in the alert state is less probable to become infected. The problem is modeled as a continuous-time Markov process on a generic graph and then formulated as a set of ordinary differential equations. The model is then studied using results from spectral graph theory and center manifold theorem. We analytically show that our model exhibits two distinct thresholds in the dynamics of epidemic spread. Below the first threshold, infection dies out exponentially. Beyond the second threshold, infection persists in the steady state. Between the two thresholds, infection spreads at the first stage but then dies out asymptotically as the result of increased alertness in the network. Finally, simulations are provided to support our findings.