Gilberto González-Parra

A nonstandard numerical scheme of predictor-corrector type for epidemic models

In this paper we construct and develop a competitive nonstandard finite difference numerical scheme of predictor-corrector type for the classical SIR epidemic model. This proposed scheme is designed with the aim of obtaining dynamical consistency between the discrete solution and the solution of the continuous model. The nonstandard finite difference scheme with Conservation Law (NSFDCL) developed here satisfies some important properties associated with the considered SIR epidemic model, such as positivity, boundedness, monotonicity, stability and conservation of frequency of the oscillations. Numerical comparisons between the NSFDCL numerical scheme developed here and Runge-Kutta type schemes show its effectiveness.

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.

Combination of nonstandard schemes and Richardson's extrapolation to improve the numerical solution of population models

In this paper we combine nonstandard finite-difference (NSFD) schemes and Richardsons extrapolation method to obtain numerical solutions of two biological systems. The first biological system deals with the dynamics of phytoplankton-nutrient interaction under nutrient recycling and the second one deals with the modeling of whooping cough in the human population. Since both models requires positive solutions, the numerical solutions need to satisfy this property. In addition, it is necessary in some cases that numerical solutions reproduce correctly the dynamical behavior while in other cases it is necessary just to find the steady state. NSFD schemes can do this. In this paper Richardsons extrapolation is applied directly to the NSFD solution to increase the order of accuracy of the numerical solutions of these biological systems. Numerical results show that Richardsons extrapolation method improves accuracy.

Stochastic modeling of the transmission of respiratory syncytial virus (RSV) in the region of Valencia, Spain

In this paper, we study the dynamics of the transmission of respiratory syncytial virus (RSV) in the population using stochastic models. The stochastic models are developed introducing stochastic perturbations on the demographic parameter as well as on the transmission rate of the RSV. Numerical simulations of the deterministic and stochastic models are performed in order to understand the effect of fluctuating birth rate and transmission rate of the RSV on the population dynamics. The numerical solutions of stochastic models are calculated using Euler-Maruyama and Milstein schemes, and confidence intervals for stochastic solutions are given using Monte-Carlo method. Analysis of the numerical results reveals that perturbations on the transmission rate are more decisive in the dynamics of RSV than perturbations on demographic parameters. In addition, the stochastic models show the advantage of reproducing more effectively the noisy RSV hospitalization data. It is concluded that these stochastic models are a viable option to provide a realistic modeling of the RSV dynamics on the population.

Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order

In this paper we construct nonstandard finite difference (NSFD) schemes to obtain numerical solutions of the susceptible-infected (SI) and susceptible-infected-recovered (SIR) fractional-order epidemic models. In order to deal with fractional derivatives we apply the Caputo operator and use the Grunwald-Letnikov method to approximate the fractional derivatives in the numerical simulations. According to the principles of dynamic consistency we construct NSFD schemes to numerically integrate the fractional-order epidemic models. These NSFD schemes preserve the positivity that other classical methods do not guarantee. Additionally, the NSFD schemes hold other conservation properties of the solution corresponding to the continuous epidemic model. We run some numerical comparisons with classical methods to test the behavior of the NSFD schemes using the short memory principle. We conclude that the NSFD schemes, which are explicit and computationally inexpensive, are reliable methods to obtain realistic positive numerical solutions of the SI and SIR fractional-order epidemic models.

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.

Modeling the epidemic waves of AH1N1/09 influenza around the world

The 2009 swine flu pandemic was a global outbreak of a new strain of H1N1 influenza virus and there are more than 14,000 confirmed deaths worldwide. The aim of this paper is to propose new mathematical models to study different dynamics of H1N1 influenza virus spread in selected regions around the world. Spatial and temporal elements are included in these models to reproduce the dynamics of AH1N1/09 virus. Different models are used since H1N1 influenza virus spread in regions with different contact structures are not the same. We rely on time series notifications of individuals to estimate some of the parameters of the models. We find that, in order to reproduce the time series data and the spread of the disease, it is convenient to suggest spatio-temporal models. Regions with only one wave are modeled with the classical SEIR model and regions with multiple waves using models with spatio-temporal elements. These results help to explain and understand about potential mechanisms behind the spread of AH1N1 influenza virus in different regions around the world.

Polynomial Chaos for random fractional order differential equations

Fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models, where such effects are difficult to incorporate and often neglected. Also, many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations. Thus, to be able to deal with many real life problems, it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty. The aim of this paper is to study the application of the generalized Polynomial Chaos (gPC) to random fractional ordinary differential equations. The method of Polynomial Chaos has played an increasingly important role when dealing with uncertainties. The main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses. However, to apply Polynomial Chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations. In order to avoid more complex numerical computations and obtain accurate solutions we rely on Richardson extrapolation. It is shown that the application of generalized Polynomial Chaos method in conjunction with Richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations.

Dynamical analysis of the transmission of seasonal diseases using the differential transformation method

The aim of this paper is to apply the differential transformation method (DTM) to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models where the solutions exhibit periodic behavior due to the seasonal transmission rate. These models describe the dynamics of the different classes of the populations. Here the concept of DTM is introduced and then it is employed to derive a set of difference equations for this kind of epidemic models. The DTM is used here as an algorithm for approximating the solutions of the epidemic models in a sequence of time intervals. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge-Kutta method solutions. Numerical comparisons show that the DTM is accurate, easy to apply and the calculated solutions preserve the properties of the continuous models, such as the periodic behavior. Furthermore, it is showed that the DTM avoids large computational work and symbolic computation.