Student Presentations

Nurdan Kar (Ankara University)

Reaction-Diffusion Formalism for Glioma Growth

More than 120 different types of tumors can occur in the human brain. In this talk, we discuss the modeling of glioma growth, a primary brain tumor that originates in the glial cells, using reaction-diffusion equations. We first provide a brief biological and mathematical foundation for glioma growth, then introduce a time-fractional partial differential equation for glioblastoma growth, the most aggressive subtype of glioma. We conclude with analysis of the fractional model’s dynamics.

Evaluation of the reproductive behavior of the mosquito Aedes aegypti incorporating polyandry

The mosquito Aedes aegypti is the primary vector responsible for transmitting diseases such as yellow fever, dengue, chikungunya, and zika. Traditionally, it has been believed, mainly due to laboratory observations, that females of this species mate only once in their lifetime. Nonetheless, recent research has demonstrated the occurrence of polyandry both in controlled environments and in natural populations, where a single female may produce offspring sired by multiple males. This reproductive behavior has significant consequences for vector control strategies like the Sterile/Incompatible Insect Technique SIT/IIT, which often assume monandry as a key premise and overlook the possibility of multiple matings. In this study, we present biomathematical models that incorporate polyandry into the reproductive dynamics of Aedes aegypti, aiming to more accurately represent its biology for the development and assessment of vector control approaches.

Jefferson Prada (Pontificia Universidad Católica de Chile)

On the Controllability of Systems Involving Chemotaxis Phenomena

Chemotaxis is a biological process in which organisms or cells move in response to chemical gradients. Mathematically, this behavior is often modeled by nonlinear partial differential equations, typically of parabolic-parabolic or parabolic-elliptic type. In this talk, we discuss recent results concerning the controllability of such systems as [3, 2, 5, 7, 1, 4], with a particular focus on the Keller Segel model from insensitizing approach, see [8]. We present strategies based on Carleman estimates [6] and duality techniques to address controllability problems and explore how the coupling and nonlinearity in uence the control mechanisms. These results are relevant for understanding population dynamics, cancer modeling, and other applications where chemotaxis e ects play a fundamental role.

This work is in collaboration with F.W. Chaves-Silva at Universidade Federal da Paraíba, Brasil.

References

[1] M. Bendahmane and F.W. Chaves-Silva.  Null controllability of a degenerated reaction di usion system in cardiac electro-physiology . In: Comptes Rendus. Mathématique 350.11-12 (2012), pp. 587 590.

[2] M. Bendahmane and F.W. Chaves-Silva.  Uniform null controllability for a degenerating reaction-diffusion system approximating a simpli ed cardiac model . In: SIAM Journal on Control and Optimization 53.6 (2015), pp. 3483 3502.

[3] P. Biler.  Local and global solvability of some parabolic systems modelling chemotaxis . In: Adv. Math. Sci. Appl. 8 (1998), pp. 715 743.

[4] F.W. Chaves-Silva and S. Guerrero.  A controllability result for a chemotaxis  uid model . In: Journal of Differential Equations 262.9 (2017), pp. 4863 4905.

[5] F.W. Chaves-Silva, S. Guerrero, and J.P. Puel.  Controllability of fast di usion coupled parabolic systems . In: Mathematical Control and Related Fields 4.4 (2014), pp. 465 479.

[6] A.V. Fursikov and O.-Y. Imanuvilov. Controllability of Evolution Equations. Tech. rep. Lecture Notes. Seoul National University, 1996.

[7] E.F. Keller and L.A. Segel.  Initiation of slime mold aggregation viewed as an instability . In: Journal of theoretical biology 26.3 (1970), pp. 399 415.

[8] J.-L. Lions.  Quelques notions dans l'analyse et le contrôle de systèmes à données incomplètes . In: Proceedings of the XIth Congress on Di erential Equations and Applications/First Congress on Applied Mathematics (1990), pp. 43 54.

Loidel Barrera Rodríguez (ICIMAF)

Solving Helmholtz equation for ultrasound propagation with Isogeometric analysis

High-intensity focused ultrasound has now been used for clinical treatment of a variety of solid malignant tumors, including those in the pancreas, liver, kidney, bone, prostate, and breast, as well as uterine fibroids and soft-tissue sarcomas. The ultrasound radiation problem is modeled mathematically by means of the Helmholtz equation with mixed boundary conditions. Isogeometric analysis (IgA) is applied to compute the acoustic pressure field produced by a curved ultrasound transducer in a 2D nonconvex physical domain.  A key step of the IgA approach is the parametrization of the domain, which plays in IgA a role similar to mesh generation in the classical Finite Element Method (FEM). The main goal of the research is to study the influence of the parametrization on the overall performance and accuracy of the numerical solution of Helmholtz equation with the Isogeometric method.

Sofía Albizu-Campos Rodríguez (Universidad de La Habana)

An isogeometric approach to the solution of the one-dimensional Westervelt equation

The Westervelt equation is widely used to model the nonlinear propagation of waves, and it is obtained by introducing a nonlinear perturbation to the standard wave equation. It is convenient to study the behavior of ultrasonic waves due to the use of high-frequency sources, both in the context of imaging and non-invasive medical treatments. This work aims to obtain the numerical solution of the Westervelt equation in a one-dimensional physical domain through Isogeometric Analysis (IgA) and compare the results and numerical difficulties with those founded via the finite element method (FEM) using linear and quadratic Lagrange basis.

Camilo Medina González (Universidad Nacional de Colombia)

Heated Pathways: Examining the Influence of Temperature Rise on Leaf Vein Patterns

The impact of temperature increases on the formation of venation architecture in leaves was investigated. A model of biological networks was employed, establishing a constant initial configuration for the networks and edges, and varying a metabolic coefficient as an equivalent to temperature variation to assess its impact. The results demonstrate an inversely proportional relationship between the vein conductivity and the metabolic coefficient value c0. The network structure remained unchanged, maintaining the same functional veins but varying in size, reducing as the metabolic coefficient increased.