MODELLI MATEMATICI NON LINEARI PER LA PREVISIONE ED IL CONTROLLO DELLE ATTIVITA’ CRIMINALI

This thesis work moves within the so-called Crime Modeling and aims to describe the emergence and spread of criminal activity in a given territory, characterizing its space-time dynamics. The approach is modeling and has its strength in the qualitative characterization of the various mechanisms that describe the phenomenon, in order to outline its impact on the dynamics and guide any decision-making actions to combat crime. The thesis work is divided into four chapters. In the first chapter the theoretical background of reference is outlined by introducing some basic notions in the field of the theory of nonlinear dynamic systems and of the theory of bifurcations. Starting from the assumption that one of the approaches used in Crime Modeling has borrowed techniques and models from mathematical epidemiology, in chapter 2 the main mathematical models born in the epidemic field are presented and discussed. Chapter 3 describes some approaches used in crime modeling, focusing on differential modeling. We also focused on the role played by periodic processes, such as seasonality, and on the related impact at the level of dynamics, taking as a starting point what emerged in the field of mathematical epidemiology. Finally, in chapter 4, we wanted to investigate the role played by some behavioral mechanisms for the spread of criminal behaviors. In particular, we wanted to evaluate the impact on the dynamics of the system of the interaction between the role played by the environment and the individual propensity to crime. This objective was pursued by means of a low-dimensional non-linear differential model which had the advantage of allowing an analytical study of the system equilibria and related stability properties as a function of the various parameters and also discussing various long-term scenarios. The impact on the dynamics of the ego system was also analyzed of periodic preventive measures to contain the individual propensity to crime.