The COVID-19 pandemic has had worldwide devastating effects on human lives, highlighting the need for tools to predict its development. The dynamics of such public-health threats can often be efficiently analyzed through simple models that help to make quantitative timely policy decisions. We benchmark a minimal version of a Susceptible-Infected-Removed model for infectious diseases (SIR) coupled with a simple least-squares Statistical Heuristic Regression (SHR) based on a lognormal distribution. We derive the three free parameters for both models in several cases and test them against the amount of data needed to bring accuracy in predictions. The SHR model is (Formula presented.) accurate about 20 days past the second inflexion point in the daily curve of cases, while the SIR model reaches a similar accuracy a fortnight before. All the analyzed cases assert the utility of SHR and SIR approximants as a valuable tool to forecast the disease’s evolution. Finally, we have studied simulated stochastic individual-based SIR dynamics, which yields a detailed spatial and temporal view of the disease that cannot be given by SIR or SHR methods.