Relations between Markov chain models and differential equation models for infectious diseases near the infection-free state are derived. Approximation of the Markov chain model by a multitype branching process leads to an estimate of the probability of disease extinction. We summarize some extinction results for multi-patch, multi-group, and multi-stage models of infectious diseases for epidemics and within-host models. The successful invasion of a pathogen often depends on the conditions of the environment at a specific time and location.