A Complex Delay Differential Equations Model for Acute Lymphoblastic Leukemia

In this paper, we introduce an innovative mathematical model designed to capture the dynamics of Acute Lymphoblastic Leukemia (ALL) under therapeutic interventions, employing delay-differential equations to account for the time delays inherent in biological processes. The model consists of 13 delay-differential equations, incorporating six distinct delays to represent various time-dependent factors such as drug effects, immune responses, and tumor growth cycles. To facilitate the analysis, we first identified the equilibrium points, which serve as critical benchmarks for understanding the system’s behavior under steady-state conditions, followed by a detailed stability analysis to assess the robustness of these points against perturbations. Utilizing the critical case theorem, we translated the system by shifting the equilibrium point to zero, simplifying the stability examination. A series of transformations were applied to aid this process, allowing for deeper insights into the dynamics of ALL under treatment. Our findings contribute to understanding treatment efficacy and tumor progression, offering a mathematical framework that not only highlights the complex interplay between treatment, tumor dynamics, and time delays but also provides a foundation for future research aimed at optimizing therapeutic strategies for ALL management.