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

TL;DR

This paper establishes a Lipschitz continuity bound for the expected number of initially infected individuals needed for an epidemic to infect an entire population in a random graph model, using majorization flow and properties of the Mills ratio.

Contribution

It introduces a novel continuity bound for the expected number of initial infections in a stochastic epidemic model, employing majorization flow and explicit bounds on the Lipschitz constant.

Findings

Proves Lipschitz continuity of the expected initial infections in the model.

Provides explicit bounds on the Lipschitz constant.

Demonstrates the effectiveness of majorization flow in this context.

Abstract

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.