Stability Analysis of Fractional-Order Nonlinear Alcohol Consumption Model and Numerical Simulation

Abstract

This study explores the motivational factors behind alcohol consumption and its societal impacts. It identifies key reasons for drinking, such as improving sociability, boosting confidence, coping with challenges, participating in rituals, and seeking detoxification. The study highlights the substantial societal harm caused by the use of alcohol and drugs, with a particular emphasis on the increase in sexual violence, both within and outside families, as a result of impaired behavior. In this work, we present a newly developed mathematical model within the Caputo fractional framework, designed to reflect the lifestyle and behaviors of individuals in the alcohol model. We analyze the existence and uniqueness of solutions using the Lipschitz growth condition and fixed point theory. In addition, we explore equilibrium points, compute basic reproduction numbers, and explore the Hyers–Ulam stability of the alcohol model. Finally, we develop a numerical scheme for computational analysis of the proposed fractional model.

1. Introduction

In recent years, numerous challenges in the natural sciences, engineering, economics, and medicine have prompted the exploration of mathematical models that incorporate rates of change. These models are essential for analyzing problems where the behavior of a solution evolves over time [1,2]. Ordinary differential equations serve as a fundamental framework for such models, as they describe systems involving derivatives, applicable to functions where the concept of differentiation is well-defined. Differential equations are commonly used to represent dynamic systems, including physical processes like particle motion, chemical reactions, elastic vibrations, fluid dynamics, electromagnetic oscillations, and even financial phenomena. The study of these models constitutes a vital area of applied mathematics, often referred to as mathematical modeling [3,4,5]. This field highlights the power of mathematics in collecting, interpreting, and generalizing scientific data to understand and predict complex systems.

Alcohol consumption is a widespread behavior in many societies and often leads to addiction for a significant portion of the population. Alcohol addiction has been extensively studied due to its profound physical and psychological impacts. However, traditional models primarily focus on the mechanisms of addiction and strategies to control alcohol consumption, frequently overlooking the depressive tendencies that often accompany addiction. Depression, often co-occurring with addiction, intensifies its adverse effects on individuals’ well-being and further complicates treatment approaches. Understanding the relationship between alcohol addiction and depression is essential for developing comprehensive models that can inform more effective intervention strategies. This study seeks to address this gap by incorporating the effects of depression into models of alcohol addiction, offering a more realistic representation of the challenges faced by individuals dealing with both conditions. Alcohol abuse is on the rise, posing a growing threat to global communities. Among younger generations, the misuse of alcohol is escalating rapidly, often driven by the pursuit of fun and enjoyment. Over recent decades, the consequences of alcohol addiction have extended beyond financial costs, disrupting marriages, family life, and social structures. Alcohol addiction negatively impacts neighborhoods, schools, workplaces, healthcare systems, and society at large. It is a significant risk factor for numerous chronic illnesses and disorders, including certain types of cancer, psychiatric conditions, cardiovascular diseases, and digestive disorders. It also increases the likelihood of diabetes, stroke, and heart disease. In the past two decades, researchers have made considerable efforts to identify the factors contributing to the spread of these harmful effects within society.

Fractional-order mathematical models play a crucial role in representing the dynamic behavior of real-world problems and offer numerous advantages. Compared to conventional integer-order models, fractional-order models are more accurate and reliable as they incorporate hereditary and memory effects along with other properties. Fractional calculus, a generalization of traditional calculus, has been extensively studied over the past two decades. Its primary advantage lies in its ability to account for historical effects through memory, enhancing the analysis and study of real-world phenomena. Fractional calculus has been widely applied in various fields of engineering and science [6,7,8]. It extends the concept of differentiation and integration to fractional orders, enabling more flexible and precise modeling. For example, Zitane [9] proposed an innovative extension involving non-local and non-singular kernels in fractional calculus, introducing fractional operators where the order depends on variables such as time, space, or other factors. Applications of fractional-order models in real-world scenarios are diverse. Jan et al. [10] investigated plant infections and developed preventive strategies for the spread of the yellow virus in red chilli plants. Boulaaras et al. [11] analyzed the dynamics of typhoid fever infections, incorporating the effects of vaccination and carriers. Similarly, Bahi et al. [12] studied the transmission dynamics of sexually transmitted human papillomavirus (HPV) to provide insights for public health interventions. These studies highlight the versatility and effectiveness of fractional calculus in addressing complex problems across various disciplines.

In recent years, fractional analysis has revolutionized problem-solving with mathematical models, offering highly efficient methods that have revitalized mathematics and applied sciences. Fractional analysis provides greater effectiveness than classical techniques, particularly in addressing processes or problems that traditional methods struggle to describe adequately. Developing novel derivative and integral operators has significantly advanced this field, making it a focal point of intensive research [13]. With its wide range of applications, fractional calculus is rapidly gaining popularity and has become the subject of numerous research initiatives. This area has drawn the attention of mathematicians across various disciplines [14]. Previously, analytical and numerical methods were employed to solve problems within the Caputo fractional differential framework. Additionally, researchers have explored the use of fractional derivatives in control systems [15,16,17]. Recent advancements in fractional calculus have opened new avenues for capturing intricate dynamics in various fields. These include economic and environmental mathematical models [18], brain tumour modeling [19], and the transmission dynamics of Lassa fever [20]. Furthermore, researchers have investigated numerical solutions [21] and approximation methods [22] using fractional derivatives, highlighting the versatility and growing impact of this mathematical tool.

The Caputo fractional derivative is a widely used operator in fractional calculus, a branch of mathematical analysis that extends the concept of differentiation to non-integer orders. Unlike classical derivatives, which measure instantaneous rates of change, fractional derivatives incorporate memory effects, making them suitable for modeling complex systems with hereditary properties. The Caputo derivative is defined in terms of an integral operator and a traditional derivative, ensuring compatibility with initial conditions expressed in terms of integer-order derivatives [23,24].

The primary objective of functional equation theory is to establish stability. This field has gained significant attention due to its wide range of applications in both pure and applied mathematics. The foundational problems in this area were introduced by D. H. Hyers and S. M. Ulam in 1940 and later independently revisited by Th. M. Rassias in 1978. Subsequently, researchers identified similar types of stability issues in f-quadratic and f-Cauchy equations [25,26]. The stability problem of the linear functional equation with a scalar coefficient function can be traced back to Ulam [27]. Our study is driven by the motivation to explore Hyers–Ulam stability in mathematical analysis. This research provides a foundation for further investigations into solutions of the contractive functional equation. It is particularly significant as it establishes a connection between the solutions of functional equations and contractive difference equations. Moreover, our work contributes to constructing and analyzing the stability of differential equations associated with contractive functional difference equations, building on our notable success in solving the contractive difference equation.

This paper aims to utilize the Caputo fractional derivative to solve a novel alcohol addiction model and examine its existence, uniqueness, and various Hyers–Ulam stabilities of solutions. A key advantage of the proposed model is its ability to capture a broader range of families with depressive tendencies and addictions. This allows researchers to gain deeper insights into the underlying physical dynamics of the system and make more accurate predictions about its behavior. The study focuses on exploring the impact of alcohol addiction on individuals’ mental health, emphasizing its potential to cause depression at any stage of life. The novelty of this research lies in its detailed analysis of how alcohol addiction can lead to depression through various social and psychological factors. These include social comparison, increased feelings of loneliness, harassment, negative feedback from others, decreased motivation, elevated stress levels, reduced real-life social interactions, and relationship issues. By incorporating these factors into a fractional mathematical model, the study offers a comprehensive understanding of the interplay between alcohol addiction and depression. This relationship is visually represented in Figure 1, which depicts the alcohol model diagram. The proposed mathematical model for alcohol addiction is based on the following hypotheses:

• The alcohol addiction model is studied within a closed environment.
• Economic status and gender do not influence the risk of alcohol addiction.
• All participants interact with each other uniformly and to the same degree.
• The alcohol consumption of addicted individuals can influence those who are not yet addicted.
• Alcohol use plays a significant role in the spread of depression among addicted individuals, with personal differences either mitigating or intensifying this effect.

To the best of our knowledge, there is a gap in research investigating the Hyers–Ulam stability analysis of the alcohol model with fractional derivative operators. This lack of research highlights the need for more study. To address this gap, this study employs the fractional derivative in the Caputo sense to investigate the Hyers–Ulam stability analysis of the alcohol model with fractional derivative operators. This paper aims to address the gap in research based on the aforementioned facts and motivation. The primary contributions of this work are summarized below:

• Insufficient research has focused on the analysis of the Hyers–Ulam stability, which leads to a gap in the existing literature. Our primary objective is to examine the Hyers–Ulam stability of a fractional-order alcohol model.
• The methodology employs a fixed point approach to examine the existence and uniqueness solution, as well as investigate the Hyers–Ulam stability and highlight other interesting findings related to the stability of this system, which have not been previously explored in this context.
• In addition, this approach simplifies the calculation of approximate solutions for the given problem, enabling us to test and examine the theoretical results using real-world scenarios.

Our contributions seek to address this research gap and bring new perspectives to the field of fractional derivatives. We provide valuable theoretical and numerical results for studying fractional-order differential equations using Caputo sense.

The structure of this paper is organized as follows: Section 2 introduces the mathematical framework of the proposed model. Section 3 presents key definitions, associated concepts, and an overview of a basic computational approach. Section 4 identifies the equilibrium points of the model. Section 5 focuses on finding a unique solution involving the Caputo fractional derivative, while Section 6 demonstrates the Hyers–Ulam stability of the model. Section 7 involves numerical simulations are carried out to illustrate the effect of key parameters on the dynamics of the system using the Adams–Bashforth method. Finally, Section 8 provides a brief conclusion.

2. Model Description for the Alcohol Model

This model categorizes the population into five compartments: Susceptible individuals (S) are those who do not currently consume alcohol but are at risk of starting. Exposed individuals (E) are occasional alcohol users who are at risk of developing an addiction. Addictive individuals ($I_a$) are frequent alcohol users who are currently suffering from addiction. Depressed individuals ($I_d$) are individuals experiencing depression as a result of alcohol consumption. Recovered individuals (R) are those who have completed treatment and recovered from addiction. The total number of individuals (population), denoted by $\mathbb{N}$, is given as

$$\mathbb{N}=S+E+I_a+I_d+R.$$

Susceptible individuals are recruited into the population at a rate of $\Delta$. Peer pressure from addicted individuals influences these susceptible individuals to use alcohol at a contact rate of $\alpha$, with a possible transmission rate of $\beta$, leading them to join the exposed class. The exposed population, affected by alcohol use, transitions to the addicted class at a rate of $\gamma \mathsf{\Psi }$, while the remainder recovers through treatment at a rate of $(1-\gamma )\mathsf{\Psi }$. Addicted individuals become depressed at a rate of $\varrho$, and those who leave treatment or require a second dose also become depressed at a rate of $(1-\mathsf{\Phi })\lambda$. Depressed individuals either transition to the recovered class through education and/or treatment at a rate of $\mu$ or succumb to depression at a rate of $\sigma$. The probability of achieving recovery through treatment is $\lambda \mathsf{\Phi }$, and the overall population average death rate is $\rho$.

$$\left\{\begin{array}{c}{\displaystyle \frac{dS}{ds}}=\Delta +\mathsf{\Omega }\mathsf{\Phi }R-\alpha \beta I_{a}S-S\rho ,\hfill \\ {\displaystyle \frac{dE}{ds}}=\alpha \beta I_{a}S-(\mathsf{\Psi }+\rho )E,\hfill \\ {\displaystyle \frac{d{I}_a}{ds}}=\gamma \mathsf{\Psi }E-(\rho +\lambda +\varrho )I_a,\hfill \\ {\displaystyle \frac{d{I}_d}{ds}}=\varrho I_a+\lambda (1-\mathsf{\Phi })I_a-(\mu +\sigma +\rho )I_d,\hfill \\ {\displaystyle \frac{dR}{ds}}=(1-\gamma )\mathsf{\Psi }E+\mu I_d+\lambda \mathsf{\Phi }{I}_a-(\rho +\tau )R.\hfill \end{array}\right.$$

(1)

Next, we present a novel approach to the proposed model in fractional form (1) given as

$$\left\{\begin{array}{c}{}^{C}D^{n}S\left(s\right)=\Delta +\mathsf{\Omega }\mathsf{\Phi }R-\alpha \beta I_{a}S-S\rho ,\hfill \\ {}^{C}D^{n}E\left(s\right)=\alpha \beta I_{a}S-(\mathsf{\Psi }+\rho )E,\hfill \\ {}^{C}D^n{I}_a\left(s\right)=\gamma \mathsf{\Psi }E-(\rho +\lambda +\varrho )I_a,\hfill \\ {}^{C}D^n{I}_d\left(s\right)=\varrho I_a+\lambda (1-\mathsf{\Phi })I_a-(\mu +\sigma +\rho )I_d,\hfill \\ {}^{C}D^{n}R\left(s\right)=(1-\gamma )\mathsf{\Psi }E+\mu I_d+\lambda \mathsf{\Phi }{I}_a-(\rho +\tau )R,\hfill \end{array}\right.$$

(2)

along with the initial condition, $S\left(0\right)\ge 0,E\left(0\right)\ge 0,I_a\left(0\right)\ge 0,I_d\left(0\right)\ge 0$ and $R\left(0\right)\ge 0$.

3. Preliminaries

Here, we review some fundamental concepts of projection techniques that are critical to the depth of this study.

Definition 1 

([28]). The Caputo time-fractional derivative of order $n>0$ for a function $h\left(s\right)$ is defined as

$${}^{C}D_a^{n}h\left(s\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(m-n)}}{\int }_a^s{(s-\phi )}^{m-n-1}{h}^m\left(\phi \right)d\phi ,0<n\le 1.$$

Definition 2. 

From the above function, as discussed, the integral fractional operator in the Caputo fractional operator is displayed as

$${}^{C}I_a^{n}h\left(s\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(n\right)}}{\int }_a^{s}h\left(\phi \right){(s-\phi )}^{n-1}d\phi ,{}^{C}I_a^{0}h\left(s\right)=h\left(s\right).$$

Lemma 1 

([29]). Consider $h\left(s\right)\in C(0,T)$, then the solution of fractional differential equation

$$\left\{\begin{array}{c}{}^{C}D^{n}Y\left(\upsilon \right)=h\left(\upsilon \right),\upsilon \in J=[0,T],\hfill \\ Y\left(0\right)=h_0\hfill \end{array}\right.$$

such that

$$Y\left(s\right)=\sum _{i=0}^n{N}_i{\upsilon }^j+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(r\right)}}{\int }_0^s{(\upsilon -\phi )}^{r-1}h\left(\phi \right)d\phi .$$

Here $N_i\in R$, $i=0,1,...,n$.

4. Equilibrium Points of Alcohol Model

This section aims to determine the equilibrium points of the alcohol model.

$$\left\{\begin{array}{c}\Delta +\mathsf{\Omega }\mathsf{\Phi }R-\alpha \beta I_{a}S-\rho S=0,\hfill \\ \alpha \beta I_{a}S-(\mathsf{\Psi }+\rho )E=0,\hfill \\ \gamma \mathsf{\Psi }E-(\rho +\lambda +\varrho )I_a=0,\hfill \\ \varrho I_a+\lambda (1-\mathsf{\Phi })I_a-(\mu +\sigma +\rho )I_d=0,\hfill \\ (1-\gamma )\mathsf{\Psi }E+\mu I_d+\lambda \mathsf{\Phi }{I}_a-(\rho +\tau )R=0.\hfill \end{array}\right.$$

Solving the above set of model (1) yields two equilibrium points in the feasible region. Note that we have

$$E_{q0}=(S^0,E^0,I_a^0,I_d^0,R^0)=\left({\displaystyle \frac{\Delta }{\rho }},0,0,0,0\right).$$

It is seen in the alcohol model that when $R_0<1$, there is no infection with the specified values

$$\begin{array}{cc}\hfill & E^1=-\left(\right(\rho +\sigma +\mu (\rho +\varrho +\lambda )(\rho +\tau )({\rho }^3+(\lambda +\mathsf{\Psi }+\varrho +\mathsf{\Psi }){\rho }^2+\left((\lambda +\varrho )\mathsf{\Psi }\right)+(\lambda +\varrho ))\rho \hfill \\ & -\mathsf{\Psi }(\Delta \alpha \beta \gamma -\varrho -\lambda )\left)\right)/\left(\beta \right({\rho }^4+(\tau +\lambda +\sigma \varrho +\mathsf{\Psi }+\mu ){\rho }^3+(\left((1+(\gamma -1)\mathsf{\Psi })\tau \right)+\lambda +\sigma +\varrho +\mu )\mathsf{\Psi }\hfill \\ & +(\lambda +\sigma +\sigma \varrho +\mathsf{\Psi }+\mu ){\rho }^3+(((1+(\gamma -1)\mathsf{\Psi })\tau +\lambda +\sigma +\varrho +\mu )\mathsf{\Psi }+(\lambda +\sigma +\varrho +\mu )\tau \hfill \\ & +(\sigma +\mu )(\lambda +\varrho )){\rho }^2+(\left(\left((1+(-1+(-\mathsf{\Phi }+1)\gamma )\mathsf{\Psi }+\sigma +\varrho +\mu )(1+(\gamma -1)\omega )\right)\tau \right)\hfill \\ & +(\sigma +\mu )(\lambda +\varrho ))\mathsf{\Psi }+\tau +\tau (\sigma +\mu \left)\right(\lambda +\varrho \left)\right)\rho -\mathsf{\Phi }\left(\right((-1+(1+(\mathsf{\Phi }-1)\gamma \left)\mathsf{\Psi }\right)\sigma +\mu (\mathsf{\Psi }-1))\lambda \hfill \\ & -\varrho \left(\right(1+(\gamma -1)\mathsf{\Omega }\sigma -\mu (\mathsf{\Omega }-1)\left)\mathsf{\Psi }\right)\alpha \gamma \mathsf{\Psi }).\hfill \end{array}$$

$$\begin{array}{cc}\hfill & I_a^1=-\left((\rho +\sigma +\mu )(\rho +\tau )({\rho }^3+(\lambda +\varrho ){\rho }^2+\left((\lambda +\varrho )\mathsf{\Psi }\right)+(\lambda +\varrho ))\rho \right)-\mathsf{\Psi }(\Delta \alpha \beta \gamma -\varrho -\lambda )\left)\right)/\hfill \\ & \left(\alpha \right({\rho }^4+(\tau +\lambda +\sigma +\varrho +\mathsf{\Psi }+\mu ){\rho }^3+(((1+(\gamma -1)\mathsf{\Omega })\tau +\lambda +\sigma +\mu )\mathsf{\Psi }+(\lambda +\sigma +\mu +\varrho )\tau \hfill \\ & +(\sigma +\mu )(\lambda +\varrho )){\rho }^2+\left(\right(((1+(-1+(-\mathsf{\Phi }+1)\gamma )\mathsf{\Psi })\lambda +(\sigma +\varrho +\mu )(1+\gamma \mathsf{\Omega }-\mathsf{\Omega }))\tau +(\sigma +\mu )\hfill \\ & (\lambda +\varrho ))\mathsf{\Psi }+\tau (\sigma +\mu \left)\right(\lambda +\varrho \left)\right)\rho -\tau \left(\right((-1+(1+(\mathsf{\Phi }-1)\gamma )\mathsf{\Psi }+\mu (\mathsf{\Omega }-1\left)\right)\lambda )-\varrho ((1+(\mathsf{\Phi }-1\left)\mathsf{\Omega }\right)\sigma \hfill \\ & -\mu (\mathsf{\Omega }-1)\left)\right)\left)\mathsf{\Psi }\beta \right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill & I_b^1=\left(((\mathsf{\Phi }-1)\lambda -\varrho )(\rho +\tau )({\rho }^3+(\lambda +\varrho +\mathsf{\Psi }){\rho }^2+((\lambda +\varrho )\mathsf{\Psi }+(\lambda +\varrho ))\rho -\mathsf{\Psi }(\Delta \alpha \beta \gamma -\varrho -\lambda ))\right)/\hfill \\ & \alpha ({\rho }^4+(\tau +\lambda +\varrho +\mathsf{\Psi }+\sigma +\mu ){\rho }^3+(((1+(\gamma -1)\mathsf{\Omega })\tau +\lambda +\mu +\varrho +\sigma )\mathsf{\Psi }+\lambda +\mu +\varrho +\mu )\tau \hfill \\ & +(\sigma +\mu )(\lambda +\varrho )){\rho }^2+(((1+(-1+(\mathsf{\Phi }+1)\gamma )\mathsf{\Omega })\lambda +(\sigma +\varrho +\mu )(1+(\mathsf{\Phi }-1)\gamma )\mathsf{\Omega })\sigma +\mu (\mathsf{\Omega }-1))\lambda \hfill \\ & -\varrho \left(\right(1+(\gamma -1)\mathsf{\Omega })\rho -\mu (\mathsf{\Omega }-1\left)\right)\mathsf{\Psi }\beta .\hfill \end{array}$$

$$\begin{array}{cc}\hfill & R^1=(({\rho }^3+(\lambda +\varrho +\mathsf{\Psi }){\rho }^2+(\lambda +\varrho )\mathsf{\Psi }+(\lambda +\varrho ))\rho -\mathsf{\Psi }(\Delta \alpha \beta \gamma -\varrho -\lambda ))((\gamma -1){\rho }^2\hfill \\ & +\left(\right(-1+(\mathsf{\Phi }+1)\gamma )\lambda +(\sigma +\varrho +\mu \left)\right(\gamma -1\left)\right)\rho +\left(\right(-1+(\mathsf{\Phi }+1)\gamma \left)\right)\sigma -\mu )\lambda +\varrho ((\gamma -1)\sigma -\mu \left)\right))\hfill \\ & /\left(\beta \right({\rho }^4+(\tau +\lambda +\sigma +\varrho +\mathsf{\Psi }+\mu ){\rho }^3+(\left((-\mathsf{\Omega }\gamma -\mathsf{\Omega }+1)\tau \right)+\lambda +\sigma +\varrho +\mu )\mathsf{\Psi }+(\lambda +\sigma +\varrho +\mu )\tau \hfill \\ & +(\sigma +\mu )(\varrho +\lambda )){\rho }^2+(((-\mathsf{\Omega }(\mathsf{\Phi }-1)\gamma -\mathsf{\Omega }+1)\lambda +(\sigma +\varrho +\mu )(\mathsf{\Omega }\gamma -\mathsf{\Omega }+1))\tau +(\sigma +\tau )(\lambda +\varrho ))\mathsf{\Psi }\hfill \\ & +\tau (\sigma +\mu )(\lambda +\varrho ))\rho -(\left(\right(\mathsf{\Omega }(\mathsf{\Phi }-1)\gamma +\mathsf{\Omega }-1)\sigma +\mu (\mathsf{\Omega }-1\left)\right)\lambda -\left(\right(\mathsf{\Omega }\gamma -\mathsf{\Omega }+1)\sigma \hfill \\ & -\mu (\mathsf{\Omega }-1)\left)\varrho \right)\mathsf{\Psi }\tau )\alpha \gamma .\hfill \end{array}$$

The reproduction number $R_0$ represents the expected number of secondary infections produced by a single infected individual during their infectious period. If $R_0<1$, the infection will not cause an epidemic within the population. Conversely, if $R_0>1$, the infection will persist and potentially spread throughout the population. The reproduction number $R_0$, provides a crucial measure for controlling the spread of the infection.

$$\left\{\begin{array}{c}{\displaystyle \frac{dS}{ds}}=\Delta +\mathsf{\Omega }\mathsf{\Phi }R-\alpha \beta I_{a}S,\hfill \\ {\displaystyle \frac{dE}{ds}}=\alpha \beta I_{a}S-(\mathsf{\Psi }+\rho )E,\hfill \\ {\displaystyle \frac{d{I}_a}{ds}}=\gamma \mathsf{\Psi }E-(\rho +\lambda +\varrho )I_a,\hfill \\ {\displaystyle \frac{d{I}_d}{ds}}=\varrho I_a+\lambda (1-\mathsf{\Phi })I_a-(\mu +\sigma +\rho )I_d,\hfill \\ {\displaystyle \frac{dR}{ds}}=(1-\gamma )\mathsf{\Psi }E+\mu I_d+\lambda \mathsf{\Phi }{I}_a-(\rho +\tau )R.\hfill \end{array}\right.$$

which implies that

$$F=\left(\begin{array}{c}\alpha \beta S{I}_a\\ 0\\ 0\\ 0\end{array}\right),V=\left(\begin{array}{c}(\mathsf{\Psi }+\rho )E\\ -\gamma \mathsf{\Psi }E+(\rho +\lambda +\varrho )I_a\\ -\varrho I_a-\lambda (1-\mathsf{\Phi })I_a+(\mu +\sigma +\rho )I_d\\ -(1-\gamma )\mathsf{\Psi }E-\mu I_d-\lambda \mathsf{\Phi }{I}_a+(\rho +\tau )R\end{array}\right)$$

which, with the help of the Jacobi matrices of F and V for the alcohol model, can be rewritten as

$$\mathbb{F}=\left(\begin{array}{cccc}0& {\displaystyle \frac{\Delta \alpha \beta }{\rho }}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\mathbb{V}=\left(\begin{array}{cccc}\mathsf{\Psi }+\rho & 0& 0& 0\\ -\mathsf{\Psi }\gamma & \rho +\lambda +\varrho & 0& 0\\ 0& -\varrho -\lambda (1-\mathsf{\Phi })& \mu +\sigma +\rho & 0\\ -(1-\gamma )\mathsf{\Psi }& -\lambda \mathsf{\Phi }& -\mu & \rho +\mathsf{\Phi }\end{array}\right).$$

Now the matrix $\mathbb{F}{\mathbb{V}}^{-1}$ is given by,

$$\mathbb{F}{\mathbb{V}}^{-1}=\left(\begin{array}{cccc}{\displaystyle \frac{\Delta \alpha \beta \gamma \mathsf{\Phi }}{\rho (\mathsf{\Phi }+\rho )(\rho +\lambda +\varrho )}}& {\displaystyle \frac{\Delta \alpha \beta }{\rho (\rho +\lambda +\varrho )}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

Therefore, $R_0$ is determined by selecting the dominant eigenvalue from the matrix $\mathbb{F}{\mathbb{V}}^{-1}$.

• Here, $R_0={\displaystyle \frac{\Delta \mathsf{\Phi }\alpha \beta \gamma }{(\rho +\lambda +\varrho )(\rho +\mathsf{\Phi })\rho }}$.

5. Existence and Uniqueness Results

This section aims to discuss the existence and uniqueness of solutions for the alcohol model using a fixed point method.

$$\left\{\begin{array}{c}{Y}_1(s,S,E,I_a,I_d,R)=\Delta +\mathsf{\Omega }\mathsf{\Phi }R-\alpha \beta I_{a}S-\rho S,\hfill \\ Y_2(s,S,E,I_a,I_d,R)=\alpha \beta I_{a}S-(\mathsf{\Psi }+\rho )E,\hfill \\ Y_3(s,S,E,I_a,I_d,R)=\gamma \mathsf{\Psi }E-(\rho +\lambda +\varrho )I_a,\hfill \\ Y_4(s,S,E,I_a,I_d,R)=\varrho I_a+\lambda (1-\mathsf{\Phi })I_a-(\mu +\sigma +\rho )I_d,\hfill \\ Y_5(s,S,E,I_a,I_d,R)=(1-\gamma )\mathsf{\Psi }E+\mu I_d+\lambda \mathsf{\Phi }{I}_a-(\rho +\tau )R.\hfill \end{array}\right.$$

(3)

Suppose that we have a Banach space $B=CJ$ in which the norm is defined as,

$$\left|\right|Y\left(s\right)\left|\right|=su{p}_{s\in J}\left[\left|S\left(s\right)\right|+\left|E\left(s\right)\right|+|I_a\left(s\right)|+|I_d\left(s\right)|+|R\left(s\right)|\right]$$

where

$$Y\left(s\right)=\left\{\begin{array}{c}S\left(s\right)\hfill \\ E\left(s\right)\hfill \\ I_a\left(s\right)\hfill \\ I_d\left(s\right)\hfill \\ R\left(s\right).\hfill \end{array}\right.,Y_0\left(s\right)=\left\{\begin{array}{c}S\left(s\right)\hfill \\ E\left(s\right)\hfill \\ I_a\left(s\right)\hfill \\ I_d\left(s\right)\hfill \\ R\left(s\right).\hfill \end{array}\right.,O(s,Y\left(s\right))=\left\{\begin{array}{c}{Y}_1(s,S,E,I_a,I_d,R)\hfill \\ Y_2(s,S,E,I_a,I_d,R)\hfill \\ Y_3(s,S,E,I_a,I_d,R)\hfill \\ Y_4(s,S,E,I_a,I_d,R)\hfill \\ Y_5(s,S,E,I_a,I_d,R).\hfill \end{array}\right.$$

(4)

If we employ the last relation, then we can write Equation (4) by using (2)

$${}^{C}D^{n}Y\left(s\right)=\nabla (s,Y\left(s\right)),s\in J,Y\left(0\right)=Y_0.$$

(5)

Thanks to Lemma 1 and the solution of Equation (5), this can be expressed in the following form

$$Y\left(s\right)=Y_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(n\right)}}{\int }_0^s{(s-\upsilon )}^{n-1}\nabla (\upsilon ,Y\left(\upsilon \right))d\upsilon ,s\in J.$$

(6)

We now provide some interesting facts for existence: we shall assume that $\left(A_1\right)$ and $\left(A_2\right)$ are true

• (A1) For the constants $\omega ,U$, such that

  $$|\nabla (s,Y\left(s\right)))|\le {\omega \left|Y\right|}^q+U$$
• (A2) For the constants £, with $Y,\overline{Y}$ then,

  $$|\nabla (s,Y)-\nabla (s,\overline{Y})|\le \pounds ||Y-\overline{Y}\left|\right|$$

The observability of the operator $T:B\to B$ as

$$TY\left(s\right)=Y_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(n\right)}}{\int }_0^s{(s-\upsilon )}^{n-1}\nabla (\upsilon ,Y\left(\upsilon \right))d\upsilon ,s\in J.$$

(7)

Theorem 1. 

If the inequalities $\left(A1\right)$ and $\left(A2\right)$ hold true, then the system given in $\left(2\right)$ has at least one solution.

Proof. 

System $\left(5\right)$ provides at least one fixed point in the solution representation of the system. The projected problem outcome applies Kransoselki’s fixed point theorem. We split the proof into the following steps:

• Step 1: In this step, we shall prove that the state variables can be displayed in terms of the continuity of the operator T as follows: Suppose that $Y_n\to Y$, for $Y_n$, $Y\in B$.

  $$\begin{array}{cc}\hfill \left|\right|T{Y}_n-TY\left|\right|& =ma{x}_{s\in J}|{\displaystyle \frac{1}{\mathsf{\Gamma }\left(l\right)}}{\int }_0^s{(s-\upsilon )}^{l-1}{\nabla }_n(\upsilon ,Y_n\left(\upsilon \right))d\upsilon -{\displaystyle \frac{1}{\mathsf{\Gamma }\left(l\right)}}{\int }_0^s{(s-\upsilon )}^{l-1}{\nabla }_n(\upsilon ,Y_n\left(\upsilon \right))d\upsilon |\hfill \\ & \le ma{x}_{s\in J}{\int }_0^s\left|{\displaystyle \frac{{(1-\upsilon )}^{l-1}}{\mathsf{\Gamma }\left(l\right)}}\right||{\nabla }_n(\upsilon ,Y_n\left(\upsilon \right))-\nabla (\upsilon ,Y\left(\upsilon \right))|\hfill \\ & \le {\displaystyle \frac{T^s}{\mathsf{\Gamma }(l+1)}}\left|\right|Y_n-Y\left|\right|\to 0asn\to \infty .\hfill \end{array}$$

  (8)
  As a result, the continuity ∇, with the $T{Y}_n\to TY$, consequently T is continuous.
• Step 2: In this step, we verify that this requirement for T is bounded. Assume that $Y\in B$, the operator T, which satisfies the growth conditions as follows:

  $$\begin{array}{cc}\hfill \left|\right|TY\left|\right|=& ma{x}_{s\in J}|Y_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(l\right)}}{\int }_0^s{(s-\upsilon )}^{l-1}\nabla (\upsilon ,Y\left(\upsilon \right))d\upsilon |,\hfill \\ & \le |Y_0|+ma{x}_{s\in J}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(l\right)}}{\int }_0^s\left|{(s-\upsilon )}^{l-1}\right||\nabla (\upsilon ,Y\left(\upsilon \right))|d\upsilon ,\hfill \\ & \le |Y_0|+{\displaystyle \frac{T^l}{\mathsf{\Gamma }(l+1)}}{\left[\omega \right|\left|Y\right||}^q+U].\hfill \end{array}$$

  (9)
  We characterize $\mathbb{S}$ to be a bounded subset of B and there exist $K_q\ge 0$ we have $\left|\right|Y\left|\right|\le K_q$. With the help of the growth condition, it follows that $Y_n,Y\to B$, in the same manner, such that

  $$\begin{array}{cc}\hfill \left|\right|TY\left|\right|& \le |Y_0|+{\displaystyle \frac{T^l}{\mathsf{\Gamma }(l+1)}}{\left[\omega \right|\left|Y\right||}^q+U]\hfill \\ & \le |Y_0|+{\displaystyle \frac{T^l}{\mathsf{\Gamma }(l+1)}}[K.K_q+U].\hfill \end{array}$$

  (10)
  So, we reach this goal as $T\left(\mathbb{S}\right)$ is bounded.
• Step 3: In addition, we prove that the continuity of T is continuous. For this purpose, let $s_1,s_2\in J$, such that $s_1\ge s_2$, we have

  $$\begin{array}{cc}\hfill |TY\left(s_1\right)-TY\left(s_2\right)|& =|{\displaystyle \frac{1}{\mathsf{\Gamma }\left(l\right)}}{\int }_0^s{(s_1-\upsilon )}^{l-1}\nabla (\upsilon ,Y\left(\upsilon \right))d\upsilon -{\displaystyle \frac{1}{\mathsf{\Gamma }\left(l\right)}}{\int }_0^s{(s_2-\upsilon )}^{l-1}\nabla (\upsilon ,Y\left(\upsilon \right))d\upsilon |\hfill \\ & \le |{\displaystyle \frac{1}{\mathsf{\Gamma }\left(l\right)}}{\int }_0^s{(s_1-\upsilon )}^{l-1}-{\displaystyle \frac{1}{\mathsf{\Gamma }\left(l\right)}}{\int }_0^s{(s_2-\upsilon )}^{l-1}\left)\right|\nabla (\upsilon ,Y\left(\upsilon \right))|d\upsilon \hfill \\ & \le {\displaystyle \frac{T^l}{\left|\mathsf{\Gamma }\right(l+1)}}{\left[\omega \right|\left|Y\right||}^q+U\left]\right|s_2-s_1|.\hfill \end{array}$$

  (11)
• Step 4: Suppose that $Y\in E=\{Y\in B:Y={\iota }_{1}TY,{\iota }_1\in (0,1)\}$ and $s\in J$, such that

  $$\left|\right|Y\left|\right|={\iota }_1\left|\right|TY\left|\right|\le {\iota }_1\left[|Y_o|+{\displaystyle \frac{T^l}{\mathsf{\Gamma }(l+1)}}{\left[\omega \right|\left|Y\right||}^q+U]\right].$$

Finally, E is bounded. Therefore, the consider operator is relatively compact, so it is completely continuous. Based on the results obtained, we can infer that the operator T possesses a unique fixed point. □

6. Stability for Solution

Here, we discuss the stability criterion for solutions of the alcohol model $\left(2\right)$. In the following theorem, we prove the Hyers–Ulam stability, which obviously also improves the result of Hyers–Ulam–Rassias stability using the fractional derivative method.

Definition 3. 

([30]). Suppose that $ϵ>0$ and assume that Y be any solution of the inequality given by

$$\left|\right|Y-TY\left|\right|\le ϵ,\forall s\in J.$$

(12)

The solution of the Equation $\left(5\right)$ is Hyers–Ulam stable, if there exists a solution $\overline{Y}$ for the Equation $\left(5\right)$ with the constant $\mathbb{C}>0$ and satisfying

$$\left|\right|Y-\overline{Y}\left|\right|\le \mathbb{C}ϵ,\forall s\in J,$$

will also be generalized Hyers–Ulam stable, if for the function $\mathfrak{P}\in C(R,R)$ with $\mathfrak{P}\left(0\right)=0$, such that

$$\left|\right|\overline{Y}-Y\left|\right|\le \mathbb{C}\mathfrak{P}\left(s\right),\forall s\in J.$$

Remark 1. 

Notice that $Y\in B$ is termed as solution Equation $\left(12\right)$, if $\mathcal{O}\in C(J,R)$, we have

1. 

$\left|\mathcal{O}\right(s\left)\right|\le ϵ,$$\forall s\in J$.

2. 

${}^{C}D^{n}Y\left(s\right)$= $\nabla (s,Y(s\left)\right)+\mathcal{O}\left(s\right)$$\forall s\in J$.

Lemma 2. 

Suppose that $Y\left(s\right)$ be a solution of the equation

$$\left\{\begin{array}{c}{}^{C}D^{n}Y\left(s\right)=\nabla (s,Y\left(s\right))+\mathcal{O}\left(s\right)\hfill \\ Y\left(0\right)=Y_0.\hfill \end{array}\right.$$

(13)

Then the above relation will be satisfied.

$$|Y\left(s\right)-TY\left(s\right)|\le \aleph ϵ,where\aleph ={\displaystyle \frac{T^l}{\mathsf{\Gamma }(l+1)}}$$

(14)

Theorem 2. 

The solution of Equation $\left(5\right)$ is Hyers–Ulam stable and consequently generalized Hyers–Ulam stable with Lemma 2 if the condition holds ${\displaystyle \frac{T^{l}M}{\mathsf{\Gamma }(l+1)}}<1$.

Proof. 

Assume that Y and $\overline{Y}$ are a unique solution of Equation $\left(5\right)$, then taking

$$\begin{array}{cc}\hfill |Y\left(s\right)-\overline{Y}\left(s\right)|& =|Y\left(s\right)-T\overline{Y}\left(s\right)|\hfill \\ & \le |Y\left(s\right)-TY\left(s\right)|+|TY\left(s\right)+T\overline{Y}\left(s\right)|\hfill \\ & \le \aleph ϵ+{\displaystyle \frac{T^l\pounds }{\mathsf{\Gamma }(l+1)}}|Y\left(s\right)-\overline{Y}\left(s\right)\hfill \\ & \le {\displaystyle \frac{\aleph ϵ}{1-\frac{T_l\pounds }{\mathsf{\Gamma }(l+1)}}}.\hfill \end{array}$$

(15)

Then the fractional model $\left(5\right)$ is said to be Hyers–Ulam stable. □

Definition 4. 

Assume that $ϵ>0$ and let Y be any solution of the inequality given by function $u_a$

$${\left|\right|}^C{D}^{n}Y\left(s\right)=\nabla (s,Y\left(s\right))\left|\right|\le u_aϵ,\forall s\in J,$$

The solution of the Equation $\left(5\right)$ is Hyers–Ulam–Rassias stable, if there exists a solution $\overline{Y}\in B$ with the constant $\mathbb{C}>0$ and satisfy

$$\left|\right|\overline{Y}-Y\left|\right|\le \mathbb{C}{u}_aϵ,\forall s\in J,$$

Remark 2. 

If $Y\in B$ will satisfy Equation $\left(12\right)$, if $\mathcal{O}\in C(J,R)$, we have

1. 

$\left|\mathcal{O}\left(s\right)\right|\le u_aϵ,$$\forall s\in J$.

2. 

${}^{C}D^{n}Y\left(s\right)$= $\nabla (s,Y(s\left)\right)+\mathcal{O}\left(s\right)$$\forall s\in J$.

Lemma 3. 

Suppose that $Y\left(s\right)$ be a solution, so we can write it as follows

$$\left\{\begin{array}{c}{}^{C}D^{n}Y\left(s\right)=\nabla (s,Y\left(s\right))+\mathcal{O}\left(s\right)\hfill \\ Y\left(0\right)=Y_0.\hfill \end{array}\right.$$

Then the above expression, as follows Equation $\left(13\right)$ yields that

$$|Y\left(s\right)-TY\left(s\right)|\le \aleph u_aϵ,where\aleph ={\displaystyle \frac{T^l}{\mathsf{\Gamma }(l+1)}}$$

(16)

Theorem 3. 

The solution of (5) is Hyers–Ulam–Rassias stable and consequently generalized Hyers–Ulam–Rassias stable if the condition holds ${\displaystyle \frac{T^l\pounds }{\mathsf{\Gamma }(l+1)}}<1$ with the Lemma.

Proof. 

Assume that Y and $\overline{Y}$ be a unique solution of $\left(5\right)$, then taking

$$\begin{array}{cc}\hfill |Y\left(s\right)-\overline{Y}\left(s\right)|& =|Y\left(s\right)-T\overline{Y}\left(s\right)|\hfill \\ & \le |Y\left(s\right)-TY\left(s\right)|+|TY\left(s\right)+T\overline{Y}\left(s\right)|\hfill \\ & \le \aleph u_aϵ+{\displaystyle \frac{T^l\pounds }{\mathsf{\Gamma }(l+1)}}|Y\left(s\right)-\overline{Y}\left(s\right)\hfill \\ & \le {\displaystyle \frac{\aleph u_aϵ}{1-\frac{T^l\pounds }{\mathsf{\Gamma }(l+1)}}}.\hfill \end{array}$$

(17)

Then the fractional model $\left(5\right)$ is said to be the Hyers–Ulam–Rassias stable. □

7. Numerical Scheme

According to model $\left(2\right)$, the numerical scheme of the adopted fractional model using the Adams–Bashforth method demonstrates the technique. We assess the precision of this approach based on the exact results. In this section, we utilize graphs created through numerical simulation. We employed a fractional differential equation nonlinear system to mathematically analyze the alcohol model. Compared to ordinary derivatives, we discover a greater degree of flexibility in the fractional alcohol model. Through numerical simulation, we were able to obtain the compression for various values of n.

$$D^{n}h\left(s\right)=\omega (s,h\left(s\right)).$$

(18)

Furthermore, it is presumed that the Volterra integral of the above Equation $\left(18\right)$ is displayed as

$$h\left(s\right)=\sum _{i=0}^{\left(n\right)-1}{\displaystyle \frac{h_0^{\left(i\right)}{s}^i}{i!}}+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(n\right)}}{\int }_0^s{(s-\upsilon )}^{n-1}\omega (\upsilon ,h\left(\upsilon \right))d\upsilon .$$

Again integrating the above equation, we gain as

$$h\left(s_{y+1}\right)=\sum _{i=0}^{\left(n\right)-1}{\displaystyle \frac{h_0^{\left(i\right)}{s}^i}{i!}}+{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\omega (s_{y+1},h^{\upsilon }\left(s_{y+1}\right))+{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\sum _{x=0}^b{c}_{x,y+1}\omega (s_x,h\left(s_x\right)).$$

Observe that $q={\displaystyle \frac{T}{R}},s_y=yq,y=0,1,2,..R\in z^{+}$. By making use of the above scheme for the suggested novel model, we obtain the following solution

$$\begin{array}{cc}\hfill S\left(s_{r+1}\right)=& \mathbb{S}\left(s_0\right)+{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\{\Delta +\mathsf{\Omega }\mathsf{\Phi }R-\alpha \beta I_{a}S-S\rho \}\hfill \\ & +{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\sum _{p=0}^r{c}_{p,r+1}\{\Delta +\mathsf{\Omega }\mathsf{\Phi }R-\alpha \beta I_{a}S-S\rho \},\hfill \\ \hfill E\left(s_{r+1}\right)=& \mathbb{E}\left(s_0\right)+{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\{\alpha \beta I_{a}S-(\mathsf{\Psi }+\rho )E\}+{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\sum _{p=0}^r{c}_{p,r+1}\{\alpha \beta I_{a}S-(\mathsf{\Psi }+\rho )E\},\hfill \\ \hfill I_a\left(s_{r+1}\right)=& {\mathbb{J}}_1\left(s_0\right)+{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\{\gamma \mathsf{\Psi }E-(\rho +\lambda +\varrho )I_a\}\hfill \\ & +{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\sum _{p=0}^r{c}_{p,r+1}\{\gamma \mathsf{\Psi }E-(\rho +\lambda +\varrho )I_a\},\hfill \\ \hfill I_d\left(s_{r+1}\right)=& {\mathbb{J}}_2\left(s_0\right)+{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\{\varrho I_a+\lambda (1-\mathsf{\Phi })I_a-(\mu +\sigma +\rho )I_d\}\hfill \\ & +{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\sum _{p=0}^r{c}_{p,r+1}\{\varrho I_a+\lambda (1-\mathsf{\Phi })I_a-(\mu +\sigma +\rho )I_d\},\hfill \\ \hfill R\left(s_{r+1}\right)=& \mathbb{P}\left(s_0\right)+{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\{(1-\gamma )\mathsf{\Psi }E+\mu I_d+\lambda \mathsf{\Phi }{I}_a-(\rho +\tau )R\}\hfill \\ & +{\displaystyle \frac{q^n}{\mathsf{\Gamma }(n+2)}}\sum _{p=0}^r{c}_{p,r+1}\{(1-\gamma )\mathsf{\Psi }E+\mu I_d+\lambda \mathsf{\Phi }{I}_a-(\rho +\tau )R\}.\hfill \end{array}$$

(19)

Results and Discussion

Using MATLAB 2021a for graphical representation, we simulate the numerical schemes using Equation $\left(19\right)$. Using the initial data $S=10,E=5,I_a=3,I_d=2$ and $R=0$ with the parameters given in

Table 1. Parameters.

Parameters	Values	Source
$\Delta$	$0.7$	Supposed
$\mathsf{\Omega },\varphi$	$0.45,0.4$	[31]
$\beta ,\psi$	$0.25,0.8$	[32]
$\mathsf{\Psi }$	0.25	Assumed
$\lambda ,\eta$	$0.0027,0.4$	[33]
$\alpha$	0.3, 0.5	Supposed
$\mathsf{\Phi }$	0.05	[34]
$\mu ,\gamma$	$0.7$	[35]
$\sigma$	$0.01$	Supposed
$\tau$	$0.13978$	Supposed

, we provide this summary of the findings of a numerical simulation of addiction dynamics. The simulation results indicate that, at the peak of addiction, the proportions of exposed and addicted individuals stabilize at 0.06 and 0.10, respectively. The rapid increase in addiction levels initially causes a rise in the number of depressed individuals, which eventually stabilizes at 0.10. Meanwhile, the percentage of recovered individuals fluctuates over time as the number of exposed, addicted, and depressed individuals decreases. It finally settles at 0.2, with changes in the parameter $\rho$, as shown in Figure 2.

8. Conclusions

In this study, we used the Caputo fractional derivative to establish a connection between alcohol consumption and the impacts of alcohol addiction. This relationship was formulated as a Caputo fractional differential equation. To validate the authenticity of the proposed model, we verified the existence and uniqueness of the solution. Our analysis reveals that, under specific conditions, the model solution exists and is unique.

The alcohol-free equilibrium point highlights the importance of considering the population’s interplay with addiction and sadness to achieve a more realistic representation. Remarkably, the data generated by our model align closely with the actual observations, reinforcing the reliability of our projections. The inclusion of fractional derivatives has expanded the scope for analyzing the behavior of these categories. Specifically, we observed that fractional derivatives in the current model effectively represent the coexistence dynamics inherent in ecological phenomena.

Using fixed-point theory, we established the existence and uniqueness of solutions and demonstrated Hyers–Ulam stability. Numerical results were graphically represented using the fractional Adams–Bashforth method for iterative solutions, underscoring the significance of the fractional-order derivative. Our findings indicate that as the number of exposed individuals increases, the transmission of addiction also rises, leading to a corresponding increase in the number of depressed individuals.

From an epidemiological perspective, the study shows that reducing and maintaining the reproduction number below one can lead to the eradication of the disease. However, if the reproduction rate reaches one, the disease will persist and continue to spread throughout the population. Future studies can incorporate environmental factors such as seasonality, wealth, interaction rate, age structure, and optimal control.