A continuity bound for the expected number of connected components of a random graph: a model for epidemics

We consider a stochastic network model for epidemics, based on a random graph proposed by Ross [Journal of Applied Probability, 18, 309-315 (1981)]. Members of a population occupy nodes of the graph, with each member being in contact with those who occupy nodes which are connected to his or her node via edges. We prove that the expected number of people who need to be infected initially in order for the epidemic to spread to the entire population, which is given by the expected number of connected components of the random graph, is Lipschitz continuous in the underlying probability distribution of the random graph. We also obtain explicit bounds on the associated Lipschitz constant. We prove this continuity bound via a technique called majorization flow, which provides a general way to obtain tight continuity bounds for Schur concave functions. To establish bounds on the optimal Lipschitz constant we employ properties of the Mills ratio.