Fractional-Order Modeling and Analysis of Secondhand Smoking with Sensitivity and Control Strategies

Abstract

This paper develops a fractional-order mathematical model to describe the transmission dynamics of secondhand smoking, employing Caputo derivatives to capture memory and hereditary effects in smoking behavior. The model incorporates initiation, cessation, relapse, and awareness processes, with analytical results establishing positivity, boundedness, and stability of equilibria. The basic reproduction number $R_0$, obtained through the next-generation matrix approach, serves as a key threshold for disease-free and smoking-present states. Sensitivity analysis highlights the contact and recruitment rates as dominant contributors to $R_0$, while cessation and awareness reduce its magnitude. An optimal control problem, formulated via Pontryagin’s Minimum Principle, evaluates intervention strategies for minimizing smoking exposure. Numerical simulations using the Adams--Bashforth--Moulton method confirm theoretical results. The study’s novelty lies in validating the fractional-order model with WHO and national Indian data (2000--2020), demonstrating that fractional dynamics ($\alpha = 0.9$) yield more realistic behavioral trends than classical integer-order models.