Exact Solutions of Epidemic Models on Networks

The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physical and mathematical literature. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/recovered models, and many of their generalizations, can be solved exactly on a wide variety of networks. Solutions are possible for cases with heterogeneous or correlated probabilities of transmission, cases in which a portion of the individuals in the network are vaccinated against the disease, either uniformly at random or in some correlated fashion, and cases in which the network has complex structure of various kinds, such as separation of the population in groups according to personal, geographic, or social characteristics. We confirm the correctness of our exact solutions by comparison with computer simulations of epidemics propagating on the corresponding networks.