Global dynamics and computational modeling approach for analyzing and controlling of alcohol addiction using a novel fractional and fractal–fractional modeling approach

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 behavior 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 fractal–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–Lindelöf 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.

The study is performed in the following sections: After a brief introduction in the first section, we provide basic definitions in ""Preliminaries" section.The model formulation in the classical case is presented in "Model construction in classical sense" section.In "Fractional extension of the alcohol addiction model" section, we extend the alcohol addiction model to the fractional case and conduct a comprehensive qualitative analysis."Numerical scheme and simulation discussion" Section includes the numerical solutions, dynamic visuals, and discussions.Additionally, in "Alcohol addiction model in fractal-fractional perspective" section, we introduce the construction of the fractal-fractional alcohol addiction model with a concise theoretical and numerical analysis.The concluding remarks of our work are presented in "Conclusion" section.

Preliminaries
We recall some necessary definitions from the literature 18,29 .
Definition 1 Consider a function V(t) in the space C m , where m is a natural number and m − 1 < ϑ ≤ m rep- resents the fractional order (FO), the Caputo derivative is defined as follows: C D ϑ t (V(t)) approaches to integer case when ϑ → 1.
Definition 2 For the system if f (t, ν * ) = 0 , then the point ν * is called an equilibrium point.Assume that the function V(t) exhibits continuous fractal differentiability over the interval (c 1 , c 2 ) .We con- sider the following fractal-fractional derivative and integral in the Caputo case, as referenced in 29 : Definition 3 The fractal-fractional operator based on the power-law is stated as follows: where, m − 1 < ϑ, ζ ≤ m ∈ N are the fractional and fractal dimensions respectively, and dV (s) ds ζ = lim t→s V(t)−V(s)

Definition 4
The fractal-fractional integral operator of V(t) with order ϑ is given through the following formula:

Model construction in classical sense
The construction process for the alcohol behavior model is briefly described in this section.The total population is categorized into seven groups according to their epidemiological status.At the time instance t, the potential drinkers are denoted by P(t) ; occasional or moderate alcohol drinkers are M(t) ; Heavy alcohol drinkers are H(t) ; under treatment drinkers are T (t) ; violence creating alcohol drinkers are V(t) ; heavy alcohol drinkers who cause traffic accident are A(t) and the recovered or quitter drinkers are placed in Q(t) class.The class P(t) includes individuals aged over adolescence and adulthood and has the potential to become drinkers.The population in P(t) are recruited at the rate and reduces due to contacts with the individuals in M(t) and H(t) compartments at the rates β 1 an β 2 respectively.The potential drinkers are further decreased because of natural death at a rate ν.
The class of moderate or occasional drinkers includes those individuals who can manage their alcohol consumption during some social gatherings, or whose consumption is hidden from their social circle.These drinkers do not experience any social issues or negative effects from alcohol use.These drinkers do not regularly consider drinking or frequently feel the desire to drink, which is one of their traits.They do not frequently fight, lose their cool, or act violently.The potential drinkers move to the occasional drinkers at a rate β 1 resulting in an increase in the population in this compartment.Furthermore, the interaction of P(t) with H(t) at a rate β 2 results in an increase in this compartment.It is diminished by the natural death rate ν and due to the transition to heavy drinkers at a rate ψ.
The class of heavy drinkers includes individuals who exhibit severe alcohol addiction and can potentially pass on the addiction to individuals of the class P(t) when they interact with heavy drinkers 33,34 .A person with alcoholism has trouble controlling or restricting their harmful use of alcohol.The moderate drinkers become heavy drinkers and join H class by the rate ψ .The class of heavy drinkers decreased because of the transition of individuals from H(t) to A(t) , V(t) , Q(t) and T (t) groups at the rates θ 2 , θ 1 , θ 3 and φ respectively.In addition, this class is reduced by the alcohol-induced death rate δ 1 and the natural death rate ν.
Treatment of an alcohol addictive person is a complex and multifaceted process that often requires a combination of medical, psychological, and social support.Keeping the importance of treatment interventions, in this study, we extend the model 23 by adding a treated class for the heavy drinkers under the assumption that some of the heavy drinkers become quitter/recovered passing through proper medications.Common medications ds m ds. (1) include naltrexone, acamprosate, and disulfiram 2 .Additionally, unlike the previous work, we assume that the heavy drinkers who exhibit severe alcohol addiction can potentially pass on the addiction to individuals in the class of potential drinkers P(t) .The class of treated individuals formed as a result of treating heavy drinkers at rate φ and are reducing at the rate δ 2 (death rate of treated individuals), γ 1 (the rate at which individuals recover and quit drinking) and ν of natural mortality rate.
The individuals in violence drinkers class V(t) engage in lengthy and excessive alcohol consumption and carry out numerous violent activities.The violent heavy drinker's population class is enhanced by the rate θ 1 and reduced at a rate γ 2 (a rate of V(t) who become healthier and become quitters), and δ 3 (the alcohol-induced in V(t) class).Moreover, this class is declined due to natural mortality at a rate ν.
Heavy drinkers who caused road accidents join the class A(t) by the rate θ 2 .Moreover, this class is decreased by the quitting rates γ 3 , the death rate δ 4 induced by A(t) and the natural motility rate ν.The recovered or quitter drinkers refer to those who have either temporarily or permanently stopped drinking.This class is increased by the recovery of individuals in various groups at the rates γ 1 , γ 2 , γ 3 and θ 3 .The recovered or quitter individuals are decreased due to natural death at a rate ν.
The system of differential equations listed below demonstrates the model's behavior and the transmission among different compartments is given in Fig. 1.

Fractional extension of the alcohol addiction model
In recent years, mathematical modeling using fractional differential equations has been recognized as an effective technique.Based on various features (as mentioned in the introduction), such differential equations are widely used in engineering and sciences such as mechanics, physics, chemistry, economics, and computational biology.The major reason is that fractional differential operators are a valuable tool to observe the influence of memory effects, which exists in most biological systems 13,16 .We reformulate the classical integer order model (4) using the system of fractional differential equations in the Caputo sense.Thus, in the system a time correlation function or memory kernel appears, providing a more accurate representation of the dynamics of such problem 15,16 .The system (4) in the fractional case can be reformulated as follows: where C D ϑ t is the Caputo derivative having order ϑ.

Basic analysis
A detailed analysis of the aforementioned alcohol addiction model ( 5) is provided in this part.

Positively invariant region
Through some mathematical results, we can easily demonstrate that the fractional model in Caputo sense has � ν as an upper bound of N (t) i.e., N (t) ≤ � ν , if N (0) ≤ � ν .Thus, the feasible region is structured as follows: The positively invariantness of region can be shown using Laplace transom.

Basic reproductive number and equilibrium states
The alcohol model (5) have two equilibria namely the alcohol-free equilibrium (AFE) D 0 and the endemic equilibrium (EE).The AFE can be given as: Utilizing the well-established method from 35 , we compute the threshold number R 0 as follows,

Uniqueness and existence of the fractional model
We will investigate the uniqueness and existence of the suggested alcohol model in the Caputo sense.The model can be restructured in the subsequent problem such that u(t) = (P, M, H, T , V, A, Q) , presents the subclasses and F is given below By using the integral operator we obtained After transforming the initial valued problem (IVP) (9) via the Picard iteration, it can be expressed in the following form Lemma 1 The vector function F (t, u(t)) defined in Eq. (9) satisfies the Lipschitz condition with respect to u on the set [0, T] × R 7 + , and this condition holds with the following Lipschitz constant (6) ( * + ν)q 1 q 2 q 5 = ψθ 2 * q 1 q 2 q 5 P * = c 5 * P * , Vol.:(0123456789)

Lemma 2
The IVP (5) has a unique solution u(t) ∈ C(J) , if the conditions in (12) holds.
Proof To attain the desired result, solution of the ( 5) is considered as + ) , defines the Picard operator given as: Moreover, it gives If ηW Ŵ(ϑ+1) < 1, then W gives a contraction and thereby ensuring that there exits unique solution of the prob- lem.

Stability analysis
Mathematical stability theory explores the resilience of differential equation solutions, including both fractional and non-fractional forms, and the robustness of dynamical system trajectories when faced with minor changes to their initial states.Ulam-Hyers (UH) and Ulam-Hyers-Rassias (UHR) stability analysis is one of the widely used techniques for evaluating stability in the context of fractional derivatives.UH stability was first discussed in 36 , but in a later publication by Rassias 37 , it was given a more comprehensive sense.When finding precise solutions is difficult, UH and UHR stability offers useful tools for controlling the behavior of a proposed model.Following the UH and UHR stability criteria, we will investigate analogous ideas from [38][39][40] while looking at the stable solutions of the Caputo fractional alcohol addiction behavior model (5).
Theorem 1 The addiction model (5) , where i and ϒ are given as If the assumptions (D1) given by M N ≤ 1 , H N ≤ 2 are true.
Proof Let δ 1 > 0 and P * * * ∈ Y such that Following remark 1, there exists a function p 1 (t) such that and p 1 (t) ≤ δ 1 .Therefore, From Lemma (2), we assume P ∈ Y as the unique solution of the system in fractional case, then P(t) is expressed as below ( 16) Proof For each δ 1 > 0 and for all P * * * ∈ Y such that there exists p 1 (t) such that Hence, Based on the information in Lemma (2), the assumption of the existence of unique solution for (5) with P ∈ Y is imposed.Therefore, P(t) is We get r, P * * * (r), M * * * (r), H * * * (r), T * * * (r), V * * * (r), A * * * (r), Q * * * (r) dr.
In similar way, we have where Hence, we conclude that the fractional model ( 5) is UHR stable.

Numerical scheme
This subsection presents an efficient numerical approach for obtaining an approximate solution to the Caputo alcohol addiction behavior model.To achieve this, the Adams-Bashforth-Moulton scheme is utilized.This technique provides a convergent and stable solution of the proposed fractional model.We rewrite the system (5) as follows: where, y = (P, M, H, V, T , A, Q) ∈ R 7 + and F (t, y(t)) shows a real-valued and continuous function.Utilizing the Caputo integral, Eq. (19) gives To apply the procedure outlined in 41 , we consider a uniform grid on [0, T] with a step size of h = T N , where N ∈ N .This results in a discrete-time points t u = uh for u = 0, 1, 2, ..., N .In a consequence, we formulate the iterative scheme for the proposed model (5) as follows: (

Results and discussion
The alcohol addiction fractional model ( 5) is simulated in this part using the Adams-Bashforth moulton method with the Caputo operator.1 leads the value of R 0 less than unity in the Figs. 2 and 3.It can be seen that for all fractional values ϑ the solution paths are convergent towards the AFE state.• Case 2. Simulation in the second scenario is performed by enhancing the the parameter value β 1 to its baseline value so that the value of R 0 is greater than 1.The population level dynamics are illustrated in Figs. 4 and 5.
It can be seen that the solution paths are convergent towards the EE state.It is important to note that in both cases for greater values of ϑ , the model solution converges more rapidly to steady states.
Impact of contact rates β 1 , β 2 and FO ϑ This subsection explains the impact of the contact rates ( β 1 , β 2 ) on moderate, heavy, violent heavy drinkers and individuals who cause accidents due to heavy drinking.Initially, the graphical results are produced using the parameter baseline values.After that, the parameter values are decreased by 25%, 50% , and 75% to the given tabulated values.Furthermore, the simulation results are performed using two different fractional values.The graphical results for the moderate, heavy, violent heavy drinkers and individuals who cause accidents due to heavy drinking are depicted in Figs. 6 and 7. Additionally, it is clear from the Fig. 6 that fractional value ϑ = 0.88 gives biologically more significant insights of the dynamics of the disease.Figure 8 depict the impact of β 1 on cumulative classes using different fractional values while keeping value of β 2 fixed.It is noted that the population significantly declines as the effective contact rates decrease.Based on the various numerical simulations presented, it is evident that the entire classes in the model eventually attain stability after a certain duration, and they all converge to a fixed population point.Furthermore, the variations in the transmission coefficient with different rates, a similar stable solution can be observed.From the biological perspective, such behaviors of the model revealed that alcohol addiction can be eradicated (or minimized) by implementing effective contorting interventions such as isolating, and proper medical, psychological, and social interventions of the addictive individuals.

Alcohol addiction model in fractal-fractional perspective
Many real-life phenomena exhibit self-similar patterns.In epidemics, such behavior refers to the presence of repeated and similar structures at different time intervals within the spread of diseases.Understanding selfsimilar patterns in epidemics is essential for modeling and predicting future dynamics accurately.Moreover, it can help public health officials and researchers develop more effective preventive strategies for disease control.In addition, recognizing these patterns can aid in the early detection of emerging outbreaks and the optimization of healthcare resources.To address the scientific challenges posed by phenomena exhibiting self-replicating structures, novel differential and integral operators called fractal-fractional operators were introduced.We extend the fractional alcohol addiction model using the Caputo fractal-fractional operator defined in (2).The fractal-fractional compartmental model comprising the alcohol addiction behavior can be illustrated as follows www.nature.com/scientificreports/under the assumption that the state variables are all positive, and with appropriate initial conditions.In the model (20), FF D ϑ,ζ 0,t (.) is fractal-fractional Caputo operator where the fractional and fractal parameters are denoted as ϑ and ζ respectively.In the subsequent section, we will commence the fundamental analysis of this model.

Application of fixed point theory to the alcohol addiction model
The existence and uniqueness of solutions for complex mathematical models can often be confirmed using fixed-point theory.This approach involves finding fixed points for certain mapping functions associated with model equations.When a unique fixed point is evaluated, it indicates the existence of a unique solution to the corresponding model.In this part of the study, we focused into establishing the existence and uniqueness of the solution for the proposed alcohol addiction model.This is accomplished by using the well-established Picard-Lindelof theorem in conjunction with the fixed-point approach.To proceeds, we can represent the Caputo fractal-fractional system (20) as a generalized Cauchy problem, which can be expressed as follows: where g(t) is composed of state variables and J represents a continuous vector function defined in the right sides of the model (20).The corresponding initial values are specified by g 0 .The problem (21) yields the following outcome upon the utilization of the fractional integral: After substituting the right side with the Caputo operator and subsequently integrating, the resulting expression is as follows 43 : Moreover, using the Picard Lindelö f approach, we give where Defining the following operator where, (23)   Our primary objective is to validate that the operator given in (24) maps a complete normed metric space into itself.Furthermore, it is essential to verify that this operator also satisfies the conditions of a contraction mapping.In our initial attempt, we aim to establish that: We proceed by considering the norm defined as follows: where, and the norm is defined by ( 24)   www.nature.com/scientificreports/Additionally, by letting r = ty , the preceding expression is transformed into: When considering φ 1 and φ 2 as continuous functions in the interval [I n (t n ), A b (t n )] , we have derived the fol- lowing outcome: Hence, we conclude that the contraction criteria is achieved upon the fulfillment of the following condition.

Numerical scheme for fractal-fractional alcohol addiction model
We provide a concise overview of a novel numerical scheme designed for the proposed transmission model in fractal-fractional perspective (20).The purpose of this scheme is to visualize the effects of fractional and fractal orders graphically in a simultaneous pattern.We adopt the procedure outlined in 43 .To begin, the system established in ( 20) is initially rewritten into Volterra integral since the fractional integral is differentiable.Subsequently, the fractal-fractional alcohol addition model in Riemann-Liouville sense is express as below The initial value problem stated in (21) can be converted as follows: where www.nature.com/scientificreports/and Furthermore, we replace the derivative in Riemann-Liouville sense by the Caputo case to facilitate the utilization of integer-order initial values.The subsequent results are then obtained by applying the Riemann-Liouville fractional integral to both sides:  12a-g.Since, the basic reproductive number R 0 > 1 in this case therefore, we can see that the solution curves converge to the EE E * for all values of fractal and fractional operators.
Case 2: In the second case, the parameters are set in a way that R 0 > 1 , and the model is simulated when the fractional dimension ϑ is considered to be the integer value and the fractional dimension ζ is varied such  (20).that ζ = 0.85, 0.90, 0.95, 1 .The graphical behavior of model's solution under this case is depicted in Fig. 13a-g revealing the convergence of the EE E * .
Case 3: Finally, in this case, the addiction fractional-fractal is simulated when both the fractional ϑ and fractal dimension ζ are varied simultaneously i.e., ϑ = 1, 0.95, 0.90, 0.85 and ζ = 1, 0.95, 0.90, 0.85 .The visual dynamics of model's compartments are analyzed in Fig. 14a-g.Similar to the previous cases, the solution curves converge to the EE E * for all fractional and fractal dimensions.www.nature.com/scientificreports/Similar to the scenario when R 0 < 1 , the simulation of this case demonstrates that the solution graphs in population groups converge quickly to a stable equilibrium at higher fractal-fractional orders, whereas the convergence is slower at lower fractal-fractional orders.Furthermore, at higher fractal-fractional orders, there is an increase in the density of potential drinkers, while the density of the remaining population groups decreases.the existence and uniqueness criterions for the solution of the fractal-fractional model using well-known fixed point theorems.The necessary components of the models including possible equilibrium points and the threshold parameter are evaluated.In addition, the stability analysis was investigated using the Ulam-Hyers as well as the Ulam-Hyers-Rassias stability approaches.The visual global dynamics of the fractional and fractal-fractional addiction model are analyzed whenever the basic reproduction is less than and greater than unity.We concluded that for R 0 < 1 , the model solutions converge rapidly to alcohol-free equilibrium at higher fractal-fractional orders, while the convergence is slower at lower fractal-fractional orders.Thus, the alcohol addiction can be control in the community.On the other hand, for R 0 > 1 , the model solutions converge to endemic steady states for all values of fractal and fractional parameters and the addiction will persist.In particular, we observed biological more feasible results of the model at higher fractional and fractal-fractional orders as all population classes in the model reach to a stability after a certain time period and all converge to a fixed point of the population.Moreover, the impact of memory index and variation in some controlling parameters are shown

Figure 1 .
Figure 1.Flow diagram of the proposed alcohol model.

Figure 2 .
Figure 2. Dynamics of the (a) potential drinkers (b) moderate/occasional drinkers (c) heavy drinkers, (d) under treatment drinkers when R 0 = 0.870517 < 1 and taking four values of fractional order ϑ .These plots show the stability of AFE of the fractional alcohol addiction model (5).

Figure 3 .
Figure 3. Dynamics of the (a) heavy drinkers creating violence (b) drinkers causing road accidents (c) Quitters/recoverd, when R 0 = 0.870517 < 1 and taking four values of fractional order ϑ .These plots show the stability of AFE of the fractional alcohol addiction model (5).

Figure 4 .
Figure 4. Simulation of the (a) potential drinkers (b) moderate/occasional drinkers (c) heavy drinkers, (d) under treatment drinkers when R 0 = 1.30577 > 1 and taking four values of fractional order ϑ .These plots demonstrate the stability of EE of the fractional alcohol addiction model (5).

Figure 5 .
Figure 5. Simulation of the (a) heavy drinkers drinkers creating violence (b) drinkers causing road accidents (c) Quitters/recoverd, when R 0 = 1.30577 > 1 and taking four values of fractional order ϑ .These plots depict the stability of EE of the fractional alcohol addiction model (5).

Figure 8 .
Figure 8. Impact of variation in contact rate β 1 on cumulative individuals in M , H , V and A groups for different values of ϑ..

Table 1 .
Description of the alcohol model's parameters.