A markov chain approach to calculate r(0) in stochastic epidemic models

R(0) has been defined as "The expected number of secondary infections originated by a "typical" infective individual when introduced into a population of susceptibles", and it is perhaps the single most important parameter in epidemic models. A general framework to calculate R(0) that can be applied to complicated stochastic epidemic models that may include demography, several strains, latent or carrier-like states, with or without density-dependent parameters is introduced. This framework helps us to understand the concept of a "typical" infective individual used in the deterministic definition of R(0). The method is illustrated with applications to several epidemic models, including some in which it has been found that the disease may persist even if R(0)<1. It is shown that although the probability of extinction is difficult to calculate in these latter cases, it is possible to give general conditions on the parameters under which eventual extinction is certain.