On the mixing time of the SIS Markov chain model for epidemic spread

Abstract

In this paper we study the mixing time of the Markov chain model for epidemic spread over a given complex network. Each node is either susceptible (healthy) or infected at any given time. In this setting, the state of the system is one of 2n where n is the number of nodes. The Markov chain has a unique stationary distribution where all nodes are healthy with probability 1, and all initial probability distributions converge to the stationary distribution. In other words, epidemic eventually dies out after long enough time passes. Since the number of states, 2n, increases exponentially as the number of nodes, n, increases, it is difficult to analyze this model. We study the relationship between the Markov chain model and n-dimensional nonlinear dynamical system which approximates the evolution of the marginal probability that each node is infected. Our main result shows that this approximated model gives an upper bound on the true marginal probability as given by the Markov chain model. As an application of this, we show that when the origin, the all-healthy-state in the n-dimensional system, is globally stable then the Markov chain mixes fast, i.e. has an O(logn) mixing time. We farther show that the linear system obtained from linearizing around the origin also provides the same upper bound on the mixing time. For this, we give an alternative proof using linear programming.