Rafael J. Villanueva

Mathematical modelling of social obesity epidemic in the region of Valencia, Spain

In this article, we analyse the incidence of excess weight in 24- to 65-year-old residents in the region of Valencia, Spain, and predict its behaviour in the coming years. In addition, we present some possible strategies to prevent the spread of the obesity epidemic. We use classical logistic regression analysis to find out that a sedentary lifestyle and unhealthy nutritional habits are the most important causes of obesity in the 24- to 65-year-old population in Valencia. We propose a new mathematical model of epidemiological type to predict the incidence of excess weight in this population in the coming years. Based on the mathematical model sensitivity analysis, some possible general strategies to reverse the increasing trend of obesity are suggested. The obese population in the region of Valencia is increasing (11.6% in 2000 and 13.48% in 2005) and the future is worrisome. Our model predicts that 15.52% of the population in Valencia will be obese by 2011. Model sensitivity analysis suggests that obesity prevention strategies (healthy advertising campaigns) are more effective than obesity treatment strategies (physical activity) involving the obese and overweight subpopulation in controlling the increase of adulthood obesity in the region of Valencia.

A Nonstandard Dynamically Consistent Numerical Scheme Applied to Obesity Dynamics

The obesity epidemic is considered a health concern of paramount importance in modern society. In this work, a nonstandard finite difference scheme has been developed with the aim to solve numerically a mathematical model for obesity population dynamics. This interacting population model represented as a system of coupled nonlinear ordinary differential equations is used to analyze, understand, and predict the dynamics of obesity populations. The construction of the proposed discrete scheme is developed such that it is dynamically consistent with the original differential equations model. Since the total population in this mathematical model is assumed constant, the proposed scheme has been constructed to satisfy the associated conservation law and positivity condition. Numerical comparisons between the competitive nonstandard scheme developed here and Eulers method show the effectiveness of the proposed nonstandard numerical scheme. Numerical examples show that the nonstandard difference scheme methodology is a good option to solve numerically different mathematical models where essential properties of the populations need to be satisfied in order to simulate the real world.

Mathematical modelling of respiratory syncytial virus (RSV): vaccination strategies and budget applications.

We propose an age-structured mathematical model for respiratory syncytial virus in which children aged <1 year are especially considered. Real data on hospitalized children in the Spanish region of Valencia were used in order to determine some seasonal parameters of the model. Weekly predictions of the number of children aged <1 year that will be hospitalized in the following years in Valencia are presented using this model. Results are applied to estimate the regional cost of paediatric hospitalizations and to perform a cost-effectiveness analysis of possible vaccination strategies.

Modeling the spread of seasonal epidemiological diseases: Theory and applications

In this paper we study the existence and uniqueness of periodic solutions of seasonal epidemiological disease, by using a continuation theorem based on coincidence degree theory. We obtain criteria for the existence, uniqueness and global asymptotic stability of the periodic solution of the system. Examples related to the transmission of respiratory syncytial virus are included.

A mathematical model of the pressure of an extreme ideology on a society

Extreme behavior is produced by small groups, but affects a large amount of people. The fear, the usual strategy of these groups, influences the decisions of the whole population. In this paper we propose a dynamical mathematical model to study the ideological evolution of the population in a region where some groups want to get political goals through violence. After a classification of the subpopulations using data from votes obtained by political parties in general elections, model parameters have been estimated and future tendencies are studied.

Computing mean square approximations of random diffusion models with source term

This paper presents a discrete numerical method for computing mean square approximations of random diffusion models. Mean square consistency of the random difference scheme is established. Sufficient conditions for the mean square stability of the proposed numerical solution are given.

Matrix Cubic Splines for Progressive 3D Imaging

Mathematical theory of matrix cubic splines is introduced, then adapted for progressive rendering of images. 2D subsets of a 3D digital object are transmitted progressively under some ordering scheme, and subsequent reconstructions using the matrix cubic spline algorithm provide an evolving 3D rendering. The process can be an effective tool for browsing three dimensional objects, and effectiveness is illustrated with a test data set consisting of 93 CT slices of a human head. The procedure has been implemented on a single processor PC system, to provide a platform for full 3D experimentation; performance is discussed. A web address for the complete, documented Mathematica code is given.

Piecewise finite series solution of nonlinear initial value differential problem

In this paper, we apply a piecewise finite series as a hybrid analytical-numerical technique for solving some nonlinear systems of ordinary differential equations. The finite series is generated by using the Adomian decomposition method, which is an analytical method that gives the solution based on a power series and has been successfully used in a wide range of problems in applied mathematics. We study the influence of the step size and the truncation order of the piecewise finite series Adomian (PFSA) method on the accuracy of the solutions when applied to nonlinear ODEs. Numerical comparisons between the PFSA method with different time steps and truncation orders against Runge-Kutta type methods are presented. Based on the numerical results we propose a low value truncation order approach with small time step size. The numerical results show that the PFSA method is accurate and easy to implement with the proposed approach.

Progressive transmission of images: PC-based computations, using orthogonal matrix polynomials

Two methods for reconstructing a 3-D image as its 2-D parallel slices are transmitted progressively, in some order, are presented and analyzed. In the originating data base, an ordered set of 2-D slices could represent computer tomography (CT), magnetic resonance images (MRI), or cryosection cross-sections of a 3-D object, for example. With this digital formulation, matrix interpolating polynomial machinery renders a progressively-improving image as slices are received. A piecewise matrix polynomial reconstruction is also considered for reducing computational needs.

Analysing the effect of public health campaigns on reducing excess weight: A modelling approach for the Spanish Autonomous Region of the Community of Valencia

Excess weight is fast becoming a serious health concern in the developed and developing world. The concern of the public health sector has lead to the development of public health campaigns, focusing on two-fold goals: to inform the public as to the health risks inherent in being overweight, and the benefits of a change in nutritional behaviour. Recent studies indicate that the effects of the average public health campaign on the target community is around 5%. In this study we aim to quantify the effect of different public health campaigns on lifestyle behaviour in the target populations in order to bring about weightloss in a significant number of people over the next few years. This study is based on recent works that consider excess weight as a consequence of the transmission of unhealthy lifestyles from one individual to another. Following this point of view, first a mathematical model is presented. Then, policies based on public health campaigns addressed to stop people gaining weight (prevention; this type of policy acts on individuals in order to maintain their weight and to stop an increase in weight) and, policies addressed to overweight individuals to reduce their weight (treatment; these campaigns act on overweight and/or obese individuals in order to reduce their weight) are simulated in order to evaluate their effectiveness. The study concludes that combination of preventive plus treatment campaigns are more effective than considering them separately.