This PhD thesis consists of two parts, both of them related to the modeling of biophysical phenomena.
In the first part we study a differential equation system designed to model the dynamics of the human immunodeficiency virus (HIV) within the host organism. This model generalizes a number of other models that have been extensively used to describe the HIV dynamics, including antigenic variation and antiretroviral therapy. Based on global stability properties of this model, we analyze the influence of the antiretroviral therapy in the long term dynamics. We characterize the outcome steady-state as a function of the antiretroviral efficacy. Additionally, we performed a multiscale analysis using Tikhonov’s theorem, in order to deal with the two intrinsic time scales of the model. This analysis leads to a way of approximating the solutions of the system by a lower dimensional nonlinear model. This reduced system is faster to evaluate numerically, and is globally asymptotically stable, as we have shown by using Lyapunov’s stability theory. We also introduce a method to estimate parameters of the HIV dynamics by comparing clinical data in the chronic stage with predicted equilibrium points of a well accepted mathematical model. We apply this method to estimate two parameters, using clinical data.
In the second part, our focus is the study of the dynamics of residential burglaries. Aiming to explain the presence of agglomerations (hotspots) in this dynamics, many theories have been raised. In order to investigate such theories, we analyze real data of residential burglaries of a Brazilian city with respect to the point pattern. Specifically, we analyze the data regarding the spatial, temporal, and spatio-temporal agglomerations. The main tool used in the analysis was the measure of homogeneity given by the Ripley’s K function. The analysis shows that, on a small scale, the dynamics of residential burglaries looks like a homogeneous Poisson process. On the other hand, on a larger scale, such dynamics cannot be explain as a homogeneous Poisson process, since the intensity of burglaries varies significantly along the regions of the city.