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• A model for the expansion of COVID-19 in Italy, Spain and the USA is considered. • Predictions on cumulative deaths due to COVID-19 are provided. • The effect of favourable seasonal conditions is assessed in the expansion of COVID-19 in different countries. • The effect of the relaxation of lockdown is assessed in the expansion of COVID- 19 in different countries. It is well-known that the classical SIR model is unable to make accurate predictions on the course of illnesses such as COVID-19. In this paper, we show that the official data released by the authorities of several countries (Italy, Spain and The USA) regarding the expansion of COVID-19 are compatible with a non-autonomous SIR type model with vital dynamics and non-constant population, calibrated according to exponentially decaying infection and death rates. Using this calibration we construct a model whose outcomes for most relevant epidemiological paramenters, such as the number of active cases, cumulative deaths, daily new deaths and daily new cases (among others) fit available real data about the first and successive waves of COVID-19. In addition to this, we also provide predictions on the evolution of this pandemic in Italy and the USA in several plausible scenarios. [ABSTRACT FROM AUTHOR]