Gerardo García-Almeida

Understanding the fundamental theorem of calculus (FTC) is complex for university students. In this paper, an alternative approach to the FTC that relies on the use of the calculation of average values of a continuous function for a certain interval is shown. Likewise, researchers present the reflections of twelve advanced university mathematics students on solving five tasks with this approach, as well as the reflections of two professors of calculus and mathematical analysis about the alternative approach and the resolutions given by the students. Students’ reflections reveal that they found the approach to FTC generally appropriate, because it helped them mobilize their knowledge of calculus and supported their understanding of the theorem. The professors expressed in their reflections that the approach is pedagogically viable, and they seemed it a way of strengthening the students’ conceptual knowledge in calculus.

El objetivo de esta investigación fue analizar cómo el uso artículos de investigación puede ser una estrategia pedagógica de aprendizaje activo por indagación que ayude a los estudiantes de matemáticas en la construcción de pruebas. El estudio se realizó bajo un paradigma cualitativo con enfoque interpretativo. Los datos se recopilaron durante un curso de modelación matemática de un programa de formación de matemáticos en una universidad pública del sureste de México. Nueve estudiantes participaron en este estudio y el profesor del curso. Los datos fueron analizados con base en la codificación de los razonamientos y estrategias seguidos por los estudiantes en su proceso de construcción de pruebas en relación con las cinco componentes que estructuran una competencia matemática y el ciclo de los cuatro modos de aprendizaje experiencial. El uso de artículos de investigación tuvo un rol esencial en el razonamiento adaptativo, la generación de argumentos e integración de conocimientos de los estudiantes durante la construcción de sus pruebas matemáticas sobre la positividad de las soluciones de un sistema de ecuaciones diferenciales e incluso, ayudó a superar dificultades tales como la comprensión de algunos teoremas implicados en las pruebas.

There is a wide range of works that have proposed mathematical models to describe the spread of infectious diseases within human populations. Based on such models, researchers can evaluate the effect of applying different strategies for the treatment of diseases. In this article, we generalize previous models by studying an SIR epidemic model with a nonlinear incidence rate, saturated Holling type II treatment rate, and logistic growth. We compute the basic reproduction number and determine conditions for the local stability of equilibria and the existence of backward bifurcation and Hopf bifurcation. We also show that, when the disease transmission rate and treatment parameter are varied, our model undergoes a Bogdanov‐Takens bifurcation of codimension 2 or 3. Simulations of the solutions and numerical continuation of equilibria are carried out to generate 2D and 3D bifurcation diagrams, as well as several related phase portraits that illustrate our results. Our work shows that incorporating these factors into epidemic models can lead to very complex dynamics.

In this article, the clasical Bazykin model is modifed with Bedding–ton–DeAngelis functional response, subject to self and cross-diffusion, in order to study the spatial dynamics of the model.We perform a detailed stability and Hopf bifurcation analysis of the spatial model system and determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. We present some numerical simulations of time evolution of patterns to show the important role played by self and cross-diffusion as well as other parameters leading to complex patterns in the plane.

We study the dynamics of a SIR epidemic model with nonlinear incidence rate, vertical transmission vaccination for the newborns and the capacity of treatment, that takes into account the limitedness of the medical resources and the efficiency of the supply of available medical resources. Under some conditions we prove the existence of backward bifurcation, the stability and the direction of Hopf bifurcation. We also explore how the mechanism of backward bifurcation affects the control of the infectious disease. Numerical simulations are presented to illustrate the theoretical findings.