Social Reinforcement in Age-Structured Smoking Dynamics: The Role of Education and the Allee Effect

We develop a two-age smoking-dynamics model for youth and adult groups that incorporates education acquisition, aging, cessation and relapse, disease progression, age-dependent social mixing, and a weak Allee effect in smoking initiation. Education is modeled as a protective status acquired through schooling and aging transitions, while initiation depends on both education status and prevalence-dependent social reinforcement. We establish the well-posedness of the system, derive the smoking-free equilibrium in closed form, and obtain the compact age-structured threshold R0age=ρdiag(gY,gA)C, where C is the age-mixing matrix and ga summarizes the within-age smoking-invasion potential. Using center-manifold analysis, we derive conditions under which Allee-type social reinforcement can generate a backward bifurcation, implying that reducing R0age below one may not always be sufficient for elimination when endemic prevalence is high. We also analyze the impact of cross-age mixing on the threshold and use a quasi-steady-state approximation to characterize the quitting–relapse loop while preserving the threshold structure. Numerical simulations illustrate baseline youth and adult prevalence trends, identify youth initiation, relapse, cessation, and education protection as dominant drivers of threshold sensitivity, and show that education-based interventions are most effective when they directly reduce the susceptibility of educated youths to smoking initiation.