<p>Effective public health policy for alcohol addiction requires a quantitative understanding of treatment efficacy. In this study, we introduce a parameter investigation of a mathematical model governed by a system of six non-linear ordinary differential equations and discuss the impact of public and private treatment. We solve the model using Chebyshev and Laguerre spectral methods. Application of these methods convert the system of differential equations into a system of algebraic equations. We compare our numerical results with the results of the Runge–Kutta method and observe that they agree very well. We further validate the results by calculating residual errors and show that both methods are accurate, efficient and reliable. We also observe that treatment accessibility (admission rates) and treatment quality (recovery efficacy) are the most important factors for reducing addiction in the long term. The numerical results demonstrate that structured care is important for combating alcohol addiction.</p>