ALCOHOL ADDICTION USING A NOVEL FRACTIONAL AND FRACTAL-FRACTIONAL MODELLING APPROACH

Abstract

In recent years, alcohol addiction has become a major public health concern and a global threat due to its potential negative health and social impacts. Beyond the health consequences, the detrimental consumption of alcohol results in substantial social and economic burdens on both individuals and society as a whole. Therefore, a proper understanding and effective control of the spread of alcohol addictive behaviour has become an appealing global issue to be solved. In this study, we develop a new mathematical model of alcohol addiction with treatment class. We analyze the dynamics of the alcohol addiction model for the first time using advanced operators known as fractai-fractional operators, which incorporate two distinct fractal and fractional orders with the well-known Caputo derivative based on power law kernels. The existence and uniqueness of the newly developed fractal- fractional alcohol addiction model are shown using the Picard-Lindelof and fixed point theories. Initially, a comprehensive qualitative analysis of the alcohol addiction fractional model is presented. The possible equilibria of the model and the threshold parameter called the reproduction number are evaluated theoretically and numerically. The boundedness and biologically feasible region for the model are derived. To assess the stability of the proposed model, the Ulam-Hyers coupled with the Ulam-Hyers-Rassias stability criteria are employed. Moreover, utilizing effecting numerical schemes,the models are solved numerically and a detailed simulation and discussion are presented. The model global dynamics are shown graphically for various values of fractional and fractal dimensions. The present study aims to provide valuable insights for the understanding the dynamics and control of alcohol addiction within a community.

Specification

PREAMBLE TO THE DESCRIPTION
THE FIELD OF INVENTION
The invention is within the field of mathematical modeling, particularly focusing on the global dynamics and computational approaches for analyzing and controlling alcohol addiction. It introduces a novel fractional and fractal-fractional modeling approach to better understand and predict the behavior of alcohol addiction within a population.
BACKGROUND OF THE INVENTION
Alcohol addiction is a major public health concern worldwide due to its severe health, social, and economic impacts. Traditional methods of analyzing alcohol addiction have limitations, particularly in capturing the complex, non-linear dynamics involved in addiction spread and control. The background of this invention involves the need for a more sophisticated and accurate mathematical model that incorporates memory effects and self-similar patterns observed in real-life phenomena, which are not well represented in classical models.
SUMMARY OF THE INVENTION
The invention introduces a new mathematical model for alcohol addiction that utilizes advanced operators known as fractal-fractional operators. These operators include both fractal and fractional dimensions, allowing for a more detailed and accurate representation of the addiction dynamics. The model accounts for different classes of individuals (e.g., potential drinkers, moderate drinkers, heavy drinkers) and incorporates treatment dynamics. The invention also provides a comprehensive analysis of the model's stability and solutions using various numerical schemes, offering valuable insights into the control and eradication of alcohol addiction.
COMPLETE SPECIFICATION
Specifications *'
o Focus: The model provides insights for public health interventions to control and possibly eradicate alcohol addiction.
o Biological Control Methods: The model is first constructed using classical integerorder differential equations that describe the dynamics of each population class over time. The classical model is extended to include fractional differential equations using the Caputo derivative, which accounts for memory effects in the dynamics of alcohol addiction.
o Incorporation of Fractional Calculus: The model incorporates fractal-fractional operators based on the power-law kernel, enabling the analysis of systems with self-similar patterns and complex behaviors.
DESCRIPTION
• Objective: The primary aim of this invention is to develop a sophisticated mathematical model that effectively analyzes and controls the dynamics of alcohol addiction. This model aims to provide deeper insights into the spread and control of alcohol addiction within a population by incorporating advanced mathematical tools, such as fractional and fractal-fractional operators.
• Novel Perspective: Unlike traditional models that use integer-order derivatives, this invention employs fractional and fractal-fractional operators, specifically the Caputo derivative. These operators allow the model to account for memory effects and complex, non-linear behaviors that are characteristic of alcohol addiction. This is particularly important for capturing the long-term dependencies and anomalous diffusion processes seen in addictive behaviors.
We Claim
1. Understanding Alcohol Addiction Dynamics: To comprehensively understand the various factors influencing the spread of alcohol addiction and to predict the long-term behavior of different population groups (e.g., potential drinkers, heavy drinkers, and those undergoing treatment).
2. Control and Mitigation: To explore and develop effective strategies for controlling and mitigating alcohol addiction, including interventions such as treatment programs, public health policies, and social support systems.
3. Enhanced Accuracy: To improve the accuracy of predictions related to alcohol addiction dynamics by incorporating memory effects and self-similar behaviors observed in real-world phenomena, which are not well captured by traditional models.
4. Incorporation of Self-Similar Patterns: The model recognizes and incorporates self-similar patterns that are often present in the spread and persistence of alcohol addiction. By using fractal-fractional operators, the model can more accurately reflect the recursive and scale invariant nature of these patterns, providing a more realistic representation of the addiction dynamics

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