In this paper, we consider the inverse problem of derivative order estimation in a fractional logistic model. In order to solve the direct problem, we use the Grünwald-Letnikov fractional derivative, then the inverse problem is tackled within a Bayesian perspective. To construct the likelihood function, we propose an explicit numerical scheme based on the truncated series of the derivative definition. By MCMC samples of the marginal posterior distributions, we estimate the order of the derivative and the growth rate parameter in the dynamic model, as well as the noise in the observations. To evaluate the methodology, a simulation was performed using synthetic data, where the bias and mean square error are calculated, the results give evidence of the effectiveness for the method and the suitable performance of the proposed model. Moreover, an example with real data is presented as evidence of the relevance of using a fractional model.

We implement the Bayesian statistical inversion theory to obtain the solution for an inverse problem of growth data, using a fractional population growth model. We estimate the parameters in the model and we make a comparison between this model and an exponential one, based on an approximation of Bayes factor. A simulation study is carried out to show the performance of the estimators and the Bayes factor. Finally, we present a real data example to illustrate the effectiveness of the method proposed here and the pertinence of using a fractional model.

In this work, we propose an efficient multi-stage homotopy perturbation method to find an analytic solution to the fractional Lotka-Volterra model. We obtain its order of accuracy, and we study the stability of the system. Moreover, we present several examples to show of the effectiveness of this method, and we conclude that the value of the derivative order plays an important role in the trajectories velocity.

El objetivo de esta investigación es proponer un modelo empírico que explique las relaciones entre factores o constructos (variables no observadas directamente) y variables observadas, que influyen en el desempeño académico de alumnos de bachillerato. Se utilizó la información contenida en el cuestionario de contexto del Centro Nacional de Evaluación para la Educación Superior (CENEVAL) de México, y los resultados del examen CENEVAL y de trayectoria escolar del bachillerato universitario de Guerrero. Se aplicó el método Stepwise en regresión múltiple multivariada para seleccionar las variables que explican el desempeño académico como un constructo, y con las variables seleccionadas se propuso un modelo empírico de relaciones que incluye los factores personal, escolar, socioeconómico, cultural y servicios disponibles en casa. Para validar la propuesta se utilizó el modelo de ecuaciones estructurales. Se concluyó que el desempeño académico está determinado principalmente por características personales del alumno y aquéllas que caracterizan el proceso escolar.The objective of this investigation is to propose an empirical model capable of explaining the relationships between factors, or constructs (variables not observed directly), and observed variables, which influence the academic performance of high school students. The conclusions were drawn from the information contained in the context questionnaire of the National Evaluation Center for Higher Education (Centro Nacional de Evaluación para la Educación Superior - CENEVAL) of Mexico, along with the results from the CENEVAL examination and the school career of the Baccalaureate University of Guerrero. The ‘Stepwise’ method was used in multiple multivariate regression to select the variables that explain the academic performance as a construct, and with the selected variables an empirical model of relationships was proposed, including personal, school, socioeconomic and cultural factors, along with the available services within the home. To validate the proposal, a structural equation model was used. The conclusion was that academic performance is determined mainly by the personal characteristics of the student and the things that characterize the scholarly process