This article presents a novel mathematical model to describe the spread of an infectious disease in the presence of social and health events: it uses 15 compartments, 7 convolution integrals and 4 types of infected individuals, asymptomatic, mild, moderate and severe. A unique feature of this work is that the convolutions and the compartments have been selected to maximize the number of independent input parameters, leading to a 56-parameter model where only one had to evolve over time. The results show that 1) the proposed mathematical model is flexible and robust enough to describe the complex dynamic of the pandemic during the first three waves of the COVID-19 spread in the region of Madrid (Spain) and 2) the proposed model allows us to calculate the number of asymptomatic individuals and the number of persons who presented antibodies during the first waves. The study shows that the following results are compatible with the reported data: close to 28% of the infected individuals were asymptomatic during the three waves, close to 29% of asymptomatic individuals were detected during the subsequent waves and close to 26% of the Madrid population had antibodies at the end of the third wave. This calculated number of persons with antibodies is in great agreement with four direct measurements obtained from an independent sero-epidemiological research. In addition, six calculated curves (total number of confirmed cases, asymptomatic who are confirmed as positive, hospital admissions and discharges and intensive care units admissions) show good agreement with data from an epidemiological surveillance database.