Optimal control and stability analysis of an alcoholism model with treatment centers

Alcoholism affects individuals across all demographics and is a major global health challenge, contributing significantly to mortality rates. This study develops and analyzes a mathematical model of alcoholism, focusing on the dynamics of drinking behaviors within a population. The model identifies two equilibrium points: the non-endemic equilibrium and the endemic equilibrium, whose stability depends on the basic reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:math> . The non-endemic equilibrium is stable when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo><</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> , while the endemic equilibrium becomes stable when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo>></mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> . Sensitivity analysis highlights the critical role of the contact rate between at-risk individuals and moderate drinkers, as well as the rate of alcohol cessation among moderate drinkers. The study incorporates control strategies, including educational campaigns and government policy measures, to reduce the spread of alcoholism. Numerical simulations demonstrate the effectiveness of a combined approach in significantly lowering alcoholism prevalence and its social and economic impacts. This study offers practical insights for designing evidence-based policies to address this issue. Some key features of the proposed method include:•Utilizing the next-generation matrix (NGM) approach to calculate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> .•Conducting equilibrium point analysis to examine the stability of the system.•Applying Pontryagin's maximum principle to determine optimal control policies.