This paper develops the exponentiated Mfamily of continuous distributions, aiming to provide new statistical models for data fitting purposes . It stands out from the other families, as it depends on two baseline distributions, with the use of ratio and power transforms in the definition of the main cumulative distribution function . Thanks to the joint action of the possibly different baseline distributions, flexible statistical models can be created, motivating a complete study in this regard . Thus, we discuss the theoretical properties of the new family, with emphasis on those of potential interest to the overall probability and statistics . Then, a new three-parameter lifetime distribution is derived, with the choices of the inverse exponential and exponential distributions as baselines . After pointing out the great flexibility of the related model, we apply it to analyze an actual dataset of current interest: the daily COVID-19 cases observed in Pakistan from 21 March to 29 May 2020 (inclusive). As notable results, we demonstrate that the proposed model is the best among the 15 top ranked models in the literature, including the inverse exponential and exponential models, several modern extensions of them depending on more parameters, and the``unexponentiated"version of the proposed model as well . As future perspectives, the proposed model can be of interest to analyze data on COVID-19 cases in other countries, for possible comparison studies.