A great variety of complex physical, natural and artificial systems are governed by statistical distributions, which often follow a standard exponential function in the bulk, while their tail obeys the Pareto power law. The recently introduced k-statistics framework predicts distribution functions with this feature. A growing number of applications in different fields of investigation are beginning to prove the relevance and effectiveness of k-statistics in fitting empirical data. In this paper, we use k-statistics to formulate a statistical approach for epidemiological analysis. We validate the theoretical results by fitting the derived k-Weibull distributions with data from the plague pandemic of 1417 in Florence as well as partial data (until April 16, 2020) from the COVID-19 pandemic in China. The fact that both the approximate dataset of the Florence plague and the partial data of the Covid-19 pandemic in China are well described by means of the proposed model suggests that the k-deformed Weibull model is relevant and that both datasets faithfully represent the spreading of the epidemics.