In many epidemic models the initial infection rate, suitably defined, plays a fundamental role in determining the probability of a major epidemic. An estimate for this rate is suggested on the basis of least squares and maximum likelihood methods. The model used to arrive at the estimate is a Galton-Watson process modified by letting the offspring distribution change from generaton to generation in a way as to approximate to an epidemic process. The estimates of the parameter and its associated variance are easily computed and compare well with other computationally tedious methods in an application to [human] smallpox data.