A dynamical mathematical model for crime evolution based on a compartmental system with interactions

We use data on imprisonment in Spain to fit a system of three ordinary differential equations that describes the temporal evolution of three different groups in the country: offenders that are not in prison, offenders that are in prison, and the rest of people. These remaining people are considered as susceptible, who may become offenders by their relationships. That is, crime is regarded to behave as a social epidemic. We first investigate the dynamics of the model to find out when criminality becomes extinct or endemic in the long run, depending on the basic reproduction number. Then, we estimate the parameters of the model and conduct a sensitivity analysis. Finally, a random error is incorporated, and nonlinear regression is carried out to gather the unexplained variability of the data. Our results report a satisfactory model fitting to the crime data, closely delineating their dynamics.