Martingale methods for the analysis of epidemic data

After explaining why martingale methods play an important role in statistical inference for parameters of epidemic models, we give a tutorial introduction to these methods in the more familiar context of data on independent and identically distributed survival times. In this simpler setting we introduce requisite results from martingale theory and demonstrate that martingale methods simply lead to well known estimates and their standard errors. We then turn to the context of epidemics, and illustrate how martingale methods can be used to derive a method of inference for the infection potential in a simple model with removal of infectives. The resulting method involves only simple computations and we demonstrate that the method applies under much more general assumptions. There follows a critical review of several applications of martingale methods for the analysis of infectious disease data. It emerges that the approach provides simple methods of inference in some situations where standard methods of inference are not available, or are too cumbersome. The range of applications seems limited, but new applications continue to be found. Little has been done to confirm high efficiency of martingale methods in epidemic applications.