Global dynamics of epidemic spread over complex networks

In this paper we study the global dynamics of epidemic spread over complex networks for both discrete-time and continuous-time models. In this setting, the state of the system at any given time is the vector obtained from the marginal probability of infection of each of the nodes in the network at that time. Convergence to the origin means that the epidemic eventually dies out. Linearizing the model around the origin yields a system whose state is an upper bound on the true state. As a result, whenever the linearized model is locally stable, the original model is globally stable, with the origin being its fixed point. When the linearized model is unstable the origin is not a stable fixed point and we show the existence of a unique second fixed point. In the continuous-time model, this second fixed point attracts all points in the state space other than the origin. In the discrete-time setting we consider two models. In the first model, we show that the second fixed point attracts all points in the state space other than the origin. In the second model, however, we show this need not be the case. We therefore give conditions under which the second fixed point attracts all non-origin points and show that for random Erdös-Rényi graphs this happens with high probability.