Stability analysis of fast food consumption dynamics using a fractional-order framework

In this research, we propose a novel fractional-order model, referred to as \( PLSCQ \), which is designed to capture the dynamics of fast food consumption and its impact on both health and social behavior. The model incorporates the Caputo derivative and classifies individuals into five distinct categories: potential fast food consumers \( P \), moderate consumers \( L \), excessive consumers \( S \), individuals affected by obesity \( C \), and those who have ceased consuming fast food \( Q \). These categories encompass the full range of fast food consumption behaviors, from those who may potentially consume it to those who have experienced its detrimental consequences, such as obesity, cardiovascular diseases, and other social problems associated with unhealthy eating habits. The study primarily examines the health risks and behavioral issues associated with fast food consumption. We demonstrate the existence of unique, non-negative solutions and compute the basic reproduction number \( R_0 \). Sensitivity analysis highlights the factors that most significantly influence \( R_0 \), while stability analysis reveals that the system is both locally and globally stable at the equilibrium point with no fast food consumption (\( E_0 \)) when \( R_0 \leq 1 \), and stable with ongoing consumption (\( E^* \)) when \( R_0 > 1 \). Finally, we present numerical simulations to corroborate the theoretical findings, showing how the order of the partial derivative affects the system's dynamics under varying parameter conditions.