Epidemiology is the medical discipline in charge of sudying the dynamics of a disease in a given population; particularly, we are interested in human infectious diseases. In the 18th century, with Daniel Bernouilli, a new branch appeared: mathematical epidemiology. During this project, we will first establish some basic concepts and a motivational introduction in order to present some common results on mathematical epidemiology. For that, it will be introduced some theory using the basic SIRS model so as to understand well the involved tools, such as Lyapunov theory on equilibrium points and their stability or the basic reproduction number R0, which will be crucial to determine the future of the disease. Finally, we will try to apply all the learned concepts to study a more complex mathematical model, still in developement, whose aim is trying to reproduce the behaviour of the well-known COVID-19; concretely, there will be shown some numerical results for its evolution in the Metropolitan Area of Barcelona.
