Numerical Method for Solving Differential Equations for Epidemiological and Biological Models

This paper presents a novel block hybrid method designed to improve accuracy and efficiency in solving differential equations, with specific applications in epidemiological and biological models. The new method were derived using a power series polynomial via interpolation and collocation procedure. The basic properties of the new method was analyzed numerically and it is obvious that the method is of uniform order nine, consistency, zero-stability, convergent. We further obtain the absolute stability through the stability polynomial showing to be A-stable. Numerical experiments demonstrate the method’s effectiveness, with smaller absolute errors than existing methods across various models, including the SIR model and growth models in population dynamics. The results affirm the potential of the new method for high-precision applications in epidemiology, biology and related fields, marking an advancement in differential equation modeling.