Management and control of biological invasions and emerging infectious diseases are leading topics of research in theoretical and applied ecology. When the initial number of potentially invasive or infectious individuals is small, demographic stochasticity can lead to rapid extinction, suggesting that the transient dynamics of establishment for invasive species and epidemics should be modeled as a stochastic process. Quantitative risk assessment can exploit this conceptualization to calculate risk metrics such as the chance of invasion or epidemic and to study the potential effectiveness of rapid response interventions. Here I review some simple establishment models and demonstrate how understanding the underlying stochastic processes can lead to more effective policies for risk management. As examples, I study the dynamics of feral nutria (Myocastor coypus) in East Anglia, UK, and infection by bovine tuberculosis (Mycobacterium bovis) in various species. Nutria are fur-bearing aquatic mammals that have been introduced around the world intentionally and through escape from farms and cause severe damage to marsh vegetation. Bovine tuberculosis is a chronic wasting disease of diverse mammal species and results in long term emaciation and decline in animal fitness. I find that both nutria and bovine tuberculosis exhibit high intrinsic rates of increase as measured by their increase in abundance when the population or epidemic is small. While Allee effects may affect the chance of establishment for nutria, analysis of additional data will be required to reject the hypothesis that nutria dynamics follow the simple Ricker growth model. The distinction is important, however, as the estimated chance of establishment under the Allee effect model is considerably less than under the Ricker model for introduced populations less than about 850 individuals. The chance of bovine tuberculosis epidemic for introductions of small numbers of individuals infectious with bovine tuberculosis in populations of brushtail possums (Trichosurus vulpecula) or badgers (Meles meles) is high. Thus, if surveillance programs are not almost completely effective at preventing introductions of infectious individuals, periodic outbreaks are likely to occur. Epidemics in these species might be controlled by culling.