Modelled Simulation of Epidemiological Differential Equations Using New Block Method

Abstract

This study presents the development and application of a New Block Method (NBM) for the numerical simulation of first-order differential equations arising in epidemiological and real-life dynamical systems. The method is formulated using a power series polynomial approach combined with interpolation and collocation techniques to generate a high-order implicit block scheme. The resulting method is applied to three test problems, namely the susceptible-infected-recovered (SIR) model, a mixture (tank) model, and a logistic growth model. These models represent key applications in epidemiology, chemical engineering, and population dynamics, respectively. The performance of the proposed method is evaluated by comparing the computed solutions with exact analytical solutions across selected evaluation points. The results demonstrate that the NBM produces highly accurate approximations that closely match the exact solutions, as also illustrated by the graphical comparisons. The study further confirms that the method is stable, convergent, and efficient for solving nonlinear and linear differential equations. The NBM is shown to be a reliable and effective computational tool for modelling complex dynamical systems.