Epidemics on general graphs

Abstract

Introduction In this chapter we investigate the behaviour of two classical epidemic models on general graphs. We consider a closed population of n individuals, connected by a neighbourhood structure that is represented by an undirected, labelled graph G = ( V, E ) with node set V = {1, …, n } and edge set E . Each node can be in one of three possible states: susceptible (S), infective (I) or removed (R). The initial set of infectives at time 0 is assumed to be non empty, and all other nodes are assumed to be susceptible at time 0. We will focus on two classical epidemic models: the susceptible-infected-removed (SIR) and susceptible-infected-susceptible (SIS) epidemic processes. In what follows we represent the graph by means of its adjacency matrix A , i.e. a ij = 1 if ( i, j ) ∈ E and a ij = 0 otherwise. Since the graph G is undirected, A is a symmetric, non-negative matrix, all its eigenvalues are real, the eigenvalue with the largest absolute value ρ is positive and its associated eigenvector has non-negative entries (by the Perron–Frobenius theorem). The value ρ is called the spectral radius . If the graph is connected, as we shall assume, then this eigenvalue has multiplicity one, the corresponding eigenvector is strictly positive and is the only one with all elements non-negative.