Mathematical models for epidemics, particularly COVID-19. Developing simulation codes

• Autores: Alejandra María Wyss Porras
• Directores de la Tesis: Lourdes Tello del Castillo (dir. tes.)
• Lectura: En la Universidad Politécnica de Madrid ( España ) en 2025
• Idioma: inglés
• Tribunal Calificador de la Tesis: Ángel Manuel Ramos del Olmo (presid.) , Juan Francisco Padial (secret.) , Olindo Zanotti (voc.) , Ángela Jiménez Casas (voc.) , Ana Isabel Muñoz Montalvo (voc.)
• Programa de doctorado: Programa de Doctorado en Ingeniería Matemática, Estadística e Investigación Operativa por la Universidad Complutense de Madrid y la Universidad Politécnica de Madrid
• Menciones:
  • Internacional
• Calificación: Sobresaliente Cum Laude
• Enlaces
  • Tesis en acceso abierto en: Archivo Digital UPM
• Resumen

  • This research proposes, analyzes, and simulates compartmental epidemiological models that reflect key epidemic dynamics with relevance to public health decision-making. The population is divided into four classes: susceptible (S), vaccinated (V), infected (I), and recovered (R), forming the SVIR structure used in both the ODE and PDE formulations. These models represent the evolution of each of these populations and incorporate reinfection, immunity loss, vaccination, and spatial spread, supported by theoretical analysis and simulation. Two mathematical settings are considered: ordinary differential equations (ODEs) for well-mixed populations and partial differential equations (PDEs) with linear and nonlinear diffusion in one dimension.

    A central contribution is the introduction of a time- and state-dependent infection rate function, $\kappa(t,I,S)$, which evolves based on both temporal progression and current epidemiological state. This formulation captures regulatory interventions and behavioral responses, enabling adaptive transmission dynamics and self-sustained epidemic waves. The SVIR model with recruitment and birth is analyzed in the ODE setting, with results including a basic reproduction number, a disease-free equilibrium characterization, and local stability analysis. Simulations explore how initial conditions, vaccination timing, and contact rate variations shape epidemic trajectories.

    Neural network methods are used to estimate the time-varying contact rate $\beta(t)$. LSTM models trained on regional COVID-19 data from Spain provide adaptive estimates that address data sparsity and reporting inconsistencies. Results show that machine learning can improve parameter adaptability, though performance depends on data quality and preprocessing.

    The PDE framework extends the SVIR model with one-dimensional nonlinear diffusion to study spatial dynamics. A detailed analytical study is conducted for the birth-extended version, establishing properties such as non-negativity, boundedness, and solution existence. A positive invariant region is constructed and validated numerically. Simulations examine how spatial heterogeneity, initial infection levels, and parameter variation influence the emergence and persistence of epidemic waves.

    The results suggest that combining temporal and spatial dynamics, nonlinear transmission, and data-driven estimation enhances the descriptive power of epidemic models. Analytical findings ensure biological interpretation, while simulations highlight the impact of key mechanisms on epidemic trends. This integration of theory, simulation, and estimation provides a flexible modeling framework applicable to more complex scenarios, including multi-strain dynamics or preliminary policy analysis.

    Despite its contributions, the framework has limitations. Estimation accuracy depends on data quality, and the infection rate uses empirically defined thresholds. The PDE model is restricted to one spatial dimension. Future extensions may include two-dimensional spatial modeling, nonlocal formulations of the infection rate, and hybrid parameter estimation methods combining machine learning with epidemiological constraints.