Modified Picard Iterative Method for epidemiological modeling and nonlinear dynamics

Mathematical models involving ordinary differential equations (ODEs) and partial differential equations (PDEs) are widely used to describe real-life dynamic systems such as disease transmission, biochemical reactions, population interactions, nonlinear growth, and reaction–diffusion processes. This thesis develops enhanced numerical schemes based on the classical Picard Iterative Method (PIM), namely the Multistage Picard Iterative Method (MPIM), the Picard Iterative Method for nth-order differential equations (PIMn), and the Multistage Picard Iterative Method for nth-order differential equations (MPIMn). The proposed methods are applied to selected nonlinear ODEs, systems of ODEs, higher-order differential equations, and PDEs. The study was motivated by two limitations of the classical PIM. First, although the classical PIM provides accurate approximations over short time intervals, it may exhibit increasing errors and reduced numerical stability over extended intervals when applied to nonlinear models. Second, the standard formulation of the classical PIM is developed for first-order initial value problems and requires transformation or reformulation when applied to higher-order differential equations. To address these limitations, the MPIM introduces a partitioned time-interval strategy, where Picard corrections are applied within each subinterval before the initial conditions are updated for the next subinterval, whereas PIMn and MPIMn are formulated to solve higher-order differential equations directly. All computations were implemented in Maple. For the system of ODE applications, the numerical performance of the PIM and MPIM was evaluated through absolute error analysis and comparison with the Adomian Decomposition Method (ADM), Multistage Adomian Decomposition Method (MADM), and the fourth-order Runge–Kutta (RK4) method. For the Monkeypox transmission model, the eighth-order Runge–Kutta (RK8) method was additionally included as a higher-order numerical reference. The results indicate that MPIM generally provides closer agreement with the reference numerical solutions than the classical PIM for the first-order nonlinear models considered, particularly over extended time intervals where PIM may accumulate larger errors and exhibit less stable numerical behaviour. The results also indicate that PIMn and MPIMn can solve higher-order differential equations directly without transforming them into first-order systems, with MPIMn generally producing more stable approximations and closer numerical agreement than PIMn. However, the improved numerical behaviour of MPIM and MPIMn requires additional computational effort because the multistage formulation divides the time domain into many subintervals, such as 12,000 subintervals in the long-term simulation. Therefore, the contribution of the proposed methods is not claimed in terms of computational efficiency, but rather in terms of improved long-term numerical accuracy and stability compared with the classical PIM and PIMn.