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This study aims to analyze the effects of allergen immunotherapy, used to treat allergic symptoms such as pollen allergy. Mathematical models are used as a methodological approach to simulate from a system of impulsive differential equations the dynamics of the model. Immunotherapy is based of supplying small amounts of pollen to the patient, which leads to minimizing severe allergic symptoms when patients are subsequently exposed to higher amounts of pollen in the environment. Lymphocyte concentrations are considered state variables, allowing the behavior and efficacy of allergen immunotherapy to be identified. The manuscript proposes a method that allows to model mixed systems. Phenomena that present continuous times in some instants and discrete times in others, these are phenomena that are frequently found in the field of physics. Allergen immunotherapy is most effective when a treatment is created with pollen dose increments in a linear form.

Determining the response to the forces applied to an elastic solid containing an ideal fluid with constant density is essential in the engineering and biomedical fields. Therefore this paper aims to present and analyze a mixed finite element method for an interaction problem solid-fluid that contributes to understanding these areas. It is assumed transmission conditions are maintained at the fluid boundary and are given by the balance of forces and the equality of normal displacements. The mixed variational formulation that avoids the locking phenomenon, for the coupled problem is in terms of displacement, stress tensor, and rotation in the solid and by pressure and scalar potential in the fluid, the main contribution of this work. The first transmission condition is imposed in the definition of the space and the rest of the conditions appear naturally, which means Lagrange multipliers are not needed at the coupling border. The unknowns for the fluid and the solid are approximated by finite element subspaces of Lagrange and Arnold-Falk-Winther of order 1, which lead to a Galerkin scheme for the continuous problem. Also, the resulting Galerkin scheme is convergent and derives optimal convergence rates. Finally, the model is illustrated using a numerical example.