Stability Analysis of Parkinson's Disease Model with Multiple Delay Differential Equations using Laplace Transform Method

In this paper, we investigate the stability analysis of Parkinson's disease model multiple delay differential equations (DDEs) utilizing the Laplace transform method. Delay differential equations are often encountered in a wide range of scientific and engineering applications, such as signal processing, control systems, and population dynamics. These equations are formulated using delayed arguments. The analysis and solution of these equations are frequently made more difficult by the existence of delays. Here, we simplify the process of locating explicit solutions by converting the DDEs into algebraic equations using the Laplace transform technique. The stability characteristics of the solutions to the DDE are crucial for understanding the progression of Parkinson's disease and the effectiveness of treatment strategies. Stable and asymptotically stable solutions are associated with better control and management of the disease, while instability suggests a potential for rapid deterioration, requiring more intensive intervention. Understanding the impact of different delays $\tau_{1}$ and $\tau_{2}$ and coefficients on the stability can help in designing better therapeutic protocols, and potentially developing new treatments that target the specific dynamics of the disease as modeled by the DDE. According to our findings, the Laplace transform method offers a methodical and effective way to solve complicated delay differential equations, revealing important information and having potential uses in both theoretical and practical fields. Additionally, a spatiotemporal model of dopamine concentration in Parkinson's disease is developed using the Laplace transform method, demonstrating its potential to predict symptom fluctuations, treatment strategies, and improve disease understanding, ultimately enhancing patient management and quality of life.