Mathematical Model of Alcoholism Incorporating Treatment: A Case Study in Kenya

Abstract

Alcoholism is characterized by persistent and uncontrollable consumption of alcoholic beverages, which poses significant risks to physical, psychological, and emotional health, including conditions such as liver cirrhosis, epilepsy, cancer, hypertension, and diabetes. Furthermore, it contributes to substantial social and economic challenges, including road accidents, domestic abuse, unemployment, and elevated crime rates. In Kenya, 12.2% of the population engages in alcohol abuse, where 10.4% are afflicted by alcohol-use disorders, thereby constituting a pressing public health concern. This study introduces a deterministic mathematical model describing alcoholism, which integrates treatment and counselling components and is articulated through the Ordinary Differential Equations (ODEs) framework. The model evaluates the ramifications of treatment for alcoholism, with stability analysis executed through the Jacobian matrix methodology and sensitivity analysis employing normalized forward sensitivity techniques. The reproduction number R0 was ascertained utilizing the next-generation matrix approach, wherein R0 > 0 indicates ongoing alcohol misuse within the susceptible population. Global stability analysis conducted through the Quadratic- Lyapunov method indicates that the Alcohol-Free Equilibrium (AFE) and Alcoholic Equilibrium (AE) are globally asymptotically unstable. Numerical simulations were done to forecast the impact of critical parameters, with these simulations underscoring the necessity of enhancing recruitment into the treatment compartment and minimizing relapse rates to render alcoholism manageable. Effective intervention strategies encompass public awareness initiatives, reduction of stigma, provision of incentives for treatment engagement, enhancement of treatment services, and the utilization of technological advancements for ongoing support. This research bears significant implications for the legislators and the Ministry of Health in formulating policies that establish a robust foundation for future endeavours aimed at controlling alcohol addiction.

Keywords

• Alcohol abuse
• Treatment
• Alcohol Free Equilibrium
• Reproduction number
• Alcoholic equilibrium