This article presents a new model explaining acquired immunodeficiency syndrome (AIDS) transmission dynamics amongst heterosexually active couples. It covers the assumptions made, the variables analysed, the model's sensitivity and the ordinary differential equations and control strategies used. The information was obtained from the Colombian state Statistics Department (DANE) and applied to different simulations in the system (with and without control), using the MAPLE programme code. Simulation results led to concluding that control using condoms was relevant in the model. It was not important if control were applied in women or men, nor was the percentage of sexually-active men and women.

Hashimoto's Thyroiditis (HT) is a pathology which often causes a gradual thyroid insufficiency in affected patients due to the autoimmune destruction of this gland. The cellular immune response mediated by T helper lymphocytes TH1 and TH17 can induce the HT disease. In this pathologic condition, there is an imbalance between the TH17 and Treg lymphocytes as well as a gut microbiota dysfunction. The objective of this work was to describe the interactions of the cell subpopulations that participate in HT. To achieve this goal, we generated a mathematical model that allowed the simulation of different scenarios for the dynamic interaction between thyroid cells, the immune system, and the gut microbiota. We used a hypothetical-deductive design of mathematical modeling based on a system of ordinary differential equations, where the state variables are the TH1, TH17 and Treg lymphocytes, the thyrocytes and the bacteria from gut microbiota. This work generated a compartmental model of the cellular immune response occurring in the thyroid gland. It was observed that TH1 and TH17 lymphocytes could increase the immune cells activity, as well as activate effector cells directly and trigger the apoptosis and inflammation processes of healthy thyrocytes indirectly. Likewise, the model showed that a reduction in Treg lymphocytes could increase the activity of TH17 lymphocytes when an imbalance of the gut microbiota composition occurred. The numerical results highlight the TH1, TH17 and bacterial balance of the gut microbiota activities as important factors for the development of Hashimoto’s Autoimmune Thyroiditis disease.

Generally, the temporary discretization of the Navier-Stokes equations despise the convective term, and consider a boundary ∂Ω = . This paper introduces a Galerkin scheme designed to solve by means of the finite element method for the Oseen problem in three dimensions written in terms of speed, vorticity and pressure, in a viscous and incompressible fluid that flows through a porous medium, this problem is obtained from a temporary discretization of the Navier-Stokes equations and we consider a partitioned boundary ∂Ω = Γ1 ∪ Γ2 and disjoint, that is, the velocity is of the homogeneous Dirichlet type on Γ1, while the tangential velocity and pressure They are of the non-homogeneous Dirichlet type on Γ2. In the variational formulation the speed is completely decoupled, which allows you to approximate the vorticity and pressure independently. The speed is recovered from the vorticity and pressure. Galerkin’s scheme is based on Nédélec finite elements and continuous polynomials to pieces of the same order, for vorticity and pressure, respectively. Likewise, convergence rates are obtained for vorticity, speed and pressure in natural norms. Finally, a numerical example is provided that illustrates the behavior of the model.

This work approach the analysis of the problems caused by the incidence of allergic diseases, using an impulsive simulation model as a methodological approach, in this, the dynamics of the model is governed by two time scales, one discrete and the other continuous which represents the interaction dynamics among pollen allergens, immune system and intestinal microbiota, evidencing the variable regulation of the Thymus by sending mature T lymphocytes into the bloodstream as a pulse. Due to the complexity of the interacting systems, scientific computing is used to represents a model at the microscopic level where the particles obey a simplified dynamic. To carry out the simulations, the values of parameters used are taken from secondary sources. The scenarios described in the simulations show the behavior of helper T cells, these being an indicator of allergic reactions. The model shows that the intestinal microbiota helps Treg cells in the inhibition of the immune response, also showing that if there is a high concentration of bacteria produced by the ingestion of functional foods, it is possible to control the allergic symptoms.