In recent decades, various methodologies have been proposed to model the complexities of challenging global problems across different domains. One such challenge involves understanding multi-step behaviors observed in certain situations. Newly proposed piecewise derivatives are known to address these issues. This study utilizes a mathematical model to examine the spread of a social issue of divorce among married couples resulting from extramarital affairs, using piecewise derivatives. Initially, we develop the model with Caputo fractional derivative and conduct some basic mathematical computations. Furthermore, the model is explored within the framework of the piecewise operator, incorporating both classical and Caputo operators. Within this framework, the study presents the existence and uniqueness of the solution using the fixed-point results. To analyze the behavior of the considered model, the Newton polynomial interpolation method is employed. The findings are subsequently illustrated through graphical representations, considering various values of fractional order.