Solving a Medical Model Using the Triple Laplace -Adomian Decomposition Method with Algorithmic Parameter Tuning

This paper develops a numerical analytical framework for solving the interaction and diffusion system from nonlinear partial differential equations of a medical model. It describes the concentration evolution of two drugs and the dynamics of two cell populations: the infected group (p) and the sensitive group (q). This framework utilizes a triplet Laplace transform with enhanced Adomian analysis to derive approximate analytical solutions for the nonlinear system without requiring linear equations. The Laplace transform handles the time derivatives and transforms the system into a suitable algebraic form. The nonlinear boundaries are then decomposed using Adomian multiples to obtain an asymptotic series of the solution. The method is advantageous because it does not require small approximation assumptions or numerical networks. It also demonstrates high efficiency in reducing computational effort compared to traditional numerical methods. The solution is retrieved in the time domain using the inverse Laplace transform. The results show that the proposed method is an effective and reliable tool for modeling nonlinear medical systems. To ensure therapeutic safety, the Safety Tune algorithm was used for precise calibration, known as "safety tuning." All symbolic and numerical calculations were performed in the Maple environment.