The understanding of the molecular mechanisms driving the fitness of the SARS-CoV-2 virus and its mutational evolution is still a critical issue. We built a simplified computational model, called SpikePro, to predict the SARS-CoV-2 fitness from the amino acid sequence and structure of the spike protein. It contains three contributions: the inter-human transmissibility of the virus predicted from the stability of the spike protein, the infectivity computed in terms of the affinity of the spike protein for the ACE2 receptor, and the ability of the virus to escape from the human immune response based on the binding affinity of the spike protein for a set of neutralizing antibodies. Our model reproduces well the available experimental, epidemiological and clinical data on the impact of variants on the biophysical characteristics of the virus. For example, it is able to identify circulating viral strains that, by increasing their fitness, recently became dominant at the population level. SpikePro is a useful, freely available instrument which predicts rapidly and with good accuracy the dangerousness of new viral strains. It can be integrated and play a fundamental role in the genomic surveillance programs of the SARS-CoV-2 virus that, despite all the efforts, remain time-consuming and expensive.