The role of chemotaxis in acute inflammation

In this work, we investigate the role of chemotaxis in acute inflammation, with a particular focus on mathematical modeling approach. Starting from standard diffusion models in biomedical fields, we firstly discuss the classical Keller-Segel system and its recent modifications, incorporating reaction terms and logistic type in order to account for environmental constraints and also prevent the non-physiscal blow-up of solutions. As an immediate generalization, we present an attraction-repulsion chemotaxis model for Alzheimer's disease (AD) and investigate conditions leading to aggregation of microglia and formation of accumulations of chemical observed in (AD) senile plaques. Next we investigate two recent reaction-diffusion-chemotaxis systems that describe the immune response during an inflammatory attack. We carry out a linear stability analysis of such reaction-diffusion type compartmental models, with multiple interacting species, in both parabolic and hyperbolic frameworks, for some medical applications, just ranging from AD to acute inflammations. Besides a linear stability analysis, we also employ a generalized energy method to obtain decay bounds in the model. We then generalize these models to describe macrophage-driven inflammatory responses, introducing additional chemokine dynamics and studying the emergence of stationary and traveling wave solutions. The final part of this thesis explores the connection between chemotaxis and neurodegenerative processes, specifically Alzheimer's disease, where microglial activation follows chemotactic principles. We suggest modifications to existing models, incorporating memory effects (via the Cattaneo correction) and nonlocal interactions to better capture the complex interplay of cellular and chemical species. Our findings provide new insights into the mathematical structures underlying inflammation and neurodegeneration, with potential implications for therapeutic strategies.