Mathematical Modelling: Deterministic, Stochastic and Machine Learning Methods.

This project falls within the field of mathematical modeling using both stochastic and deterministic models, with applications in biology, biomedicine, and social sciences. We are a group of experts in partial differential equations, stochastic calculus, and machine learning, combining these areas to tackle complex problems. The project continues a previous one, now including new members to face additional challenges. To facilitate the presentation, the project is structured in three main blocks, although they are not separate blocks but deeply interconnected: Mathematical Modeling and Population Dynamics. A general framework for physiologically structured populations will be developed using PDEs and delay equations, studying the behavior of their solutions. Cancer-immune interaction models will be analyzed to determine conditions for tumor clearance, persistence, or oscillatory regimes. The efficacy of phage therapy will be evaluated using deterministic, stochastic, and machine learning models. The basic and dynamic reproduction numbers will be computed for periodic population models, and selection-mutation equations combining phenotypic structure and spatial distribution will be studied. Additionally, discretized linear inverse problems in ECGI and tomography will be analyzed, including uncertainty-aware regularization and variability compensation. Stochastic Analysis Models. Weak approximations for SPDEs with general diffusion coefficients will be studied, including the one-dimensional stochastic heat equation driven by spacetime white noise, and weak approximation methods for parabolic SPDEs driven by cylindrical Wiener processes. Almost sure convergence to the Brownian sheet and convergence in law of multi-parameter stochastic integrals will be investigated. Positivity and stability of stochastic differential equations driven by fractional Brownian motion, including delay and reflection, will also be analyzed. Stochastic variability will be incorporated into parameters of SIR-type and multi-class models, integrating supervised classification to identify epidemic dynamic regimes. Statistical Methods and Machine Learning Models. Robust statistical tests will be developed to evaluate prediction models, along with feature-aware data coherence measures and probabilistic explainability via Shapley values. The MAP criterion will be applied to ordinal classification, and classifier ensembles for ordinal tasks will be developed. Finally, methods will be applied to practical problems and case studies.