On the approaches of step functions of some growth models

This thesis examines the mathematical relationship between sigmoid and step functions, commonly encountered in mathematical and biological systems, and growth models. Sigmoid functions play a significant role as activation functions in biological and neural network models. This study investigates how step functions can approximate various sigmoid models and explores the integration of these functions through the Hausdorff distance. Different types of sigmoid functions, such as logistic, Gompertz, Rayleigh, Lindley, and Burr, are analyzed, and the distance bounds between these functions and step functions are addressed using numerical and theoretical methods. The findings shed light on modeling methods employed in various fields such as engineering, artificial intelligence, and biomedical sciences. The study demonstrates the broader applicability of Hausdorff continuity in both applied fields and theoretical mathematics.