Analysis of Vulnerable and Recovered Population Dynamics in the SAHTR Model Based on Variations in the Spread and Rehabilitation Rates

Drug abuse continues to be a significant public health issue, especially in developing countries such as Indonesia, with increasing prevalence among the productive age group. This study develops and analyzes a compartmental mathematical model, SAHTR (Susceptible, Addicted-Lighter, Addicted-Heavier, Treatment, Recovery), using a nonlinear system of differential equations to simulate the dynamics of drug abuse spread and recovery. The model integrates an optimal control approach to evaluate the effectiveness of two intervention strategies: anti-drug campaigns and psychological counseling. Numerical simulations were carried out using the Runge-Kutta 4th order method, focusing on the sensitivity analysis of two key parameters: the rate of social transmission (β) and rehabilitation success rate (γ₂). The results show that while the β parameter influences the initial decline of the susceptible population, its long-term impact is relatively minimal. In contrast, the γ₂ parameter significantly affects the dynamics of the recovery compartment, indicating that improving rehabilitation efficiency can substantially reduce system burden and accelerate the return of individuals to a drug-free state. These findings suggest that intervention strategies should prioritize enhancing the quality and accessibility of rehabilitation programs, including structured therapy and post-recovery support. Moreover, social influence control alone is insufficient without being complemented by preventive and community-based approaches. Overall, this study contributes to evidence-based policymaking by demonstrating the value of mathematical modeling in understanding and combating drug abuse effectively.