Abstract
Recently, the notion of a reinfection threshold in epidemiological models of only partial immunity has been debated in the literature. We present a rigorous analysis of a model of reinfection which shows a clear threshold behaviour at the parameter point where the reinfection threshold was originally described. Furthermore, we demonstrate that this threshold is the mean field version of a transition in corresponding spatial models of immunization. The reinfection threshold corresponds to the transition between annular growth of an epidemics spreading into a susceptible area leaving recovered behind and compact growth of a susceptible-infected-susceptible region growing into a susceptible area. This transition between annular growth and compact growth was described in the physics literature long before the reinfection threshold debate broke out in the theoretical biology literature.

Abstract
We prove a one-to-one correspondence between C1+ conjugacy classes of diffeomorphisms with hyperbolic sets contained in surfaces and stable and unstable pairs of one-dimensional C1+ self-renormalizable structures.

Abstract
We analyse the effect of the regulatory T cells (Tregs) in the local control of the immune responses by T cells. We obtain an explicit formula for the level of antigenic stimulation of T cells as a function of the concentration of T cells and the parameters of the model. The relation between the concentration of the T cells and the antigenic stimulation of T cells is an hysteresis, that is unfold for some parameter values. We study the appearance of autoimmunity from cross-reactivity between a pathogen and a self antigen or from bystander proliferation. We also study an asymmetry in the death rates. With this asymmetry we show that the antigenic stimulation of the Tregs is able to control locally the population size of Tregs. Other effects of this asymmetry are a faster immune response and an improvement in the simulations of the bystander proliferation. The rate of variation of the levels of antigenic stimulation determines if the outcome is an immune response or if Tregs are able to maintain control due to the presence of a transcritical bifurcation for some tuning between the antigenic stimuli of T cells and Tregs. This behavior is explained by the presence of a transcritical bifurcation.

Abstract
We study the SIS and SIRI epidemic models discussing different approaches to compute the thresholds that determine the appearance of an epidemic disease. The stochastic SIS model is a well known mathematical model, studied in several contexts. Here, we present recursively derivations of the dynamic equations for all the moments and we derive the stationary states of the state variables using the moment closure method. We observe that the steady states give a good approximation of the quasi-stationary states of the SIS model. We present the relation between the SIS stochastic model and the contact process introducing creation and annihilation operators. For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we present the phase transition lines using the mean field and the pair approximation for the moments. We use a scaling argument that allow us to determine analytically an explicit formula for the phase transition lines in pair approximation.

Abstract
We introduce a model of tourism choice where we consider that the choice of a tourism resort by a tourist, depends not only on the characteristics of the product offered by the resort but depends also on certain characteristics -crowding types- of the other tourists that have chosen the same resort. To get insights about the effect of crowding types in the allocation of tourists across resorts we exploit a club formation approach and model the framework by means of a Nash game. We establish existence of strategic equilibrium and characterize relevant equilibria.

Abstract
Background: Body fat is related to changes in lipid profile, blood pressure and metabolism of insulin and glucose, known as the metabolic syndrome (MS). The aim of this study was to estimate the prevalence of metabolic syndrome (MS) and its components among overweight and obese Portuguese schoolchildren, and to identify associated clinical and biochemical characteristics. Methods: A total of 82 children (14 overweight and 68 obese; 40 boys and 42 girls) aged 7-9 years, underwent anthropometric measurements. A blood sample was obtained to assess biochemical parameters. Insulin resistance (IR) was determined by the Homeostasis Model Assessment of Insulin Resistance (HOMA-IR). MS was defined by the National Cholesterol Education Program Adult Treatment Panel III criteria modified by Cook. Results: The prevalence of MS was 15.8%. Abdominal obesity was present in all children. Frequency of elevated blood pressure, low HDL-cholesterol and elevated triglyceride concentrations were 62.6%, 13.4% and 11.0%, respectively. None of the children presented impaired fasting glucose, however hyperinsulinemia (7.3%) and IR (8.5%) were observed. The number of components of MS was higher in children with higher z-BMI (rho = 0.411; p < 0.001). MS was associated with higher leptin concentrations. No association was found with adiponectin or ghrelin levels. Leptin correlated positively with obesity, glucose metabolism, lipid profile, hepatic function and C-reactive protein, and negatively with HDL and Apolipoprotein A-I/B ratio. Conclusions: This study shows a significant prevalence of MS among obese schoolchildren. Abdominal obesity and elevated blood pressure were the most frequent components of this syndrome. Dyslipidemia, IR and high levels of leptin were also associated with MS in this young group.