Epidemics and graphs

Abstract

Random graphs provide us with a useful tool in understanding the structure of stochastic epidemic models. By representing the individuals in a population by vertices and transmission links by arrows between these vertices, we obtain a graph that contains information on many important characteristics, such as the final epidemic size, the basic reproduction number and the probability of a large outbreak. The connection between SIR epidemics and random graphs was observed by Ludwig (1974), and has then been more fully exploited by von Bahr and Martin-Löf (1980), Ball and Barbour (1990) and Barbour and Mollison (1990). In the first two sections of this chapter the random graph interpretation of the standard SIR epidemic model will be given; we also show that the special case of aconstantinfectious period yields a particularly nice class of random graphs.