Existence of global attractor in reaction–diffusion model of obesity-induced Alzheimer’s disease and its control strategies

Abstract

Evidence suggests that obesity, diabetes, and aging notably increase susceptibility to dementia-related conditions such as Alzheimer’s disease (AD). This article explores the correlations between obesity, diabetes, and AD. It introduces a diffusion-driven model encompassing variables like glucose dynamics, insulin levels, beta cells, microglia, cytokines, amyloid-$\beta$ plaques, neurofibrillary tangles ($\tau$ plaques), neurodegeneration, and cognitive decline. The study includes stability analysis (local and global), examining boundedness and long-time behavior via showing the existence of a global attractor for the diffusion-driven model. A global sensitivity analysis, utilizing the Partial Rank Correlation Coefficient (PRCC), identifies factors sensitively impacting $A_{\beta }$ plaque growth, $\tau$ plaques, and neurodegeneration. The deterministic model solution illustrates spatiotemporal dynamics, revealing a link between obesity and Alzheimer’s, which is characterized by distinct patchy patterns. While Alzheimer’s has no cure, employing optimal control techniques can help alleviate its effects and enhance affected individuals’ quality of life. An optimal control problem for AD management is developed, optimizing multiple aspects of disease management. The study highlights the efficacy of long-term healthy lifestyle practices and customized anti-amyloid therapy in significantly delaying obesity-induced AD progression. This research sheds light on the connection between obesity and Alzheimer’s, underscoring the negative impact of pro-inflammatory microglia on cognitive decline while proposing control strategies.