Estimation for Discrete Time Branching Processes with Application to Epidemics

Certain estimators for the mean of the offspring distribution of a Galton-Watson process are considered. The asymptotic behaviour of each of these estimators is studied when the true underlying model is in fact a multitype branching process or a branching process with a random environment. It is revealed which of the estimators remain consistent indicators of whether or not the process is subcritical, under these alternative underlying models. It is then indicated how this "robustness" result might influence the choice of an estimator by considering the problem of estimating the level of immunity required in a community in order to prevent major epidemics. The application is illustrated with references to smallpox using data from an outbreak in São Paulo, Brazil.