From the SIR/SEIR/SVEIR Model to a Chemotactic System: Analysis of Spatiotemporal Mechanisms of Infection Spread within the Einstein Paradigm

RESUMO

This work investigates the class of SIR/SEIR models and their extension to an ODE-based SVEIR system, whose parameters are calibrated and validated using real influenza outbreak data. The theoretical part establishes existence and uniqueness of solutions for the SVEIR model via the Picard-Lindelof theorem and analyzes invariance of the admissible region. To describe spatiotemporal infection spread in an urban setting, a diffusion-chemotaxis extension of the SIR framework is formulated as a PDE system; an additional term is introduced to represent fear/avoidance effects in the system dynamics. A numerical workflow is proposed, including grid discretization, Neumann boundary conditions enforced by mirror extension, a semi-implicit IMEX time-stepping scheme, and sparse linear solves; computational correctness is monitored, in particular, by checking conservation of the total population.