We present a model for the COVID-19 epidemic that offers analytical expressions for the newly registered and latent cases. This model is based on an epidemic branching process with latency that is greatly simplified when the bare memory kernel is given by an exponential function as observed in this pandemic. We expose the futility of the concept of “reaching the peak” of the epidemic as long as the number of latent cases is not depleted. Our model offers the possibility of laying out different scenarios for the evolution of the epidemic in different countries based on the most recent observations and in terms of only two constants obtained from clinical trials. Furthermore, by analyzing the number of registered new deaths, our model suggests that the recent surge in new COVID-19 cases in the USA is a consequence of an increase in testing, but only up to the second week of June of 2020.