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Simple Summary: Using tools from both mathematics (especially wavelet theory) and computer science (machine learning), we present a general new method for modelling the evolution of epidemics which is not restricted to human populations. A crucial novel feature of our approach is that it significantly takes into account that an epidemic may take place in certain types of waves which cannot only be of a global as well as local nature, but can also occur at multiple different times and locations. In the particular case of the current Covid-19 pandemic, based on recent figures from the Johns Hopkins database we apply our model to France, Germany, Italy, the Czech Republic, as well as the US federal states New York and Florida, and compare it and its predictions to established as well as other recently developed forecasting methods and techniques. We introduce the concept of epidemic-fitted wavelets which comprise, in particular, as special cases the number I (t) of infectious individuals at time t in classical SIR models and their derivatives. We present a novel method for modelling epidemic dynamics by a model selection method using wavelet theory and, for its applications, machine learning-based curve fitting techniques. Our universal models are functions that are finite linear combinations of epidemic-fitted wavelets. We apply our method by modelling and forecasting, based on the Johns Hopkins University dataset, the spread of the current Covid-19 (SARS-CoV-2) epidemic in France, Germany, Italy and the Czech Republic, as well as in the US federal states New York and Florida. [ABSTRACT FROM AUTHOR]