EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6507 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Fractional Smoking Transmission Model Utilizing the Caputo-Fabrizio Operator: A Comprehensive Analysis and Simulation Supriya Bhosale1,, Chhaya Lande2,∗, Hossein Jafari3,, D. K. Raut4, 1 Symbiosis International Deemed University lavale, Pune, India 2 Symbiosis International Deemed University lavale Pune, Symbiosis Institute of Technology lavale Pune, India 3 Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa 4 Shivaji Mahavidyalaya, Renapur, Latur, India Abstract. Over the past few years, several novel formulations of fractional derivatives have emerged, facilitating the construction of mathematical models that can address a broad spec- trum of real-world applications. This study presents and investigates a mathematical model for smoking behavior incorporating the Caputo–Fabrizio fractional differential operator, which is par- ticularly valued for its non-singular kernel and ability to capture memory effects in dynamic systems without the complexities of singularities. Here, the population is divided into six distinct compart- ments: mainly susceptible, snuffing, irregular, habitual, regular, and quit. To explore the model’s dynamics, the Laplace Adomian Decomposition Method (LADM) and the Aboodh Adomian De- composition Method (AADM) are employed. A comparative analysis of LADM and AADM is carried out using MATLAB software, with numerical simulations conducted for various fractional orders to assess the accuracy and efficiency of both techniques. Both methodologies demonstrated a high level of accuracy and consistency in their results, highlighting their reliability for modeling complex systems. 2020 Mathematics Subject Classifications: 26A33, 34A08, 65L05 Key Words and Phrases: Caputo-Fabrizio fractional derivative, Smoking Model, Adomain decomposition method, Laplace Transform, Aboodh Transform, Numerical simulations 1. Introduction Mathematical modeling is an effective tool in this era to analyze dynamic processes, simulate real-world situations, identify important variables, test hypotheses, increase effi- ciency, evaluate risks, create cutting-edge technologies and an advance knowledge across ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6507 Email addresses: supriya.bhosale@sitpune.edu.in (S. Bhosale), chhaya.lande@sitpune.edu.in (C. Lande), jafari.usern@gmail.com (H. Jafari), dkraut1983@gmail.com (D. K. Raut) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 2 of 21 disciplines. Generally, mathematical modeling is based on differential equations, but in recent years, the introduction of fractional-order derivatives has enhanced these models by capturing memory effects and nonlocal behaviors. This advancement has led to more accurate and realistic representations of complex systems where traditional models often fall short in depicting long-term dependencies and heterogeneous dynamics [1]. Mathe- matical models supported by numerical simulations are increasingly used by researchers to investigate the transmission mechanisms and control strategies of communicable dis- eases [2]. Almost every branch of science and engineering has been interested in fractional order models [3]. In biomedical engineering, infectious diseases are one of the main area that have caught the attention of researchers in recent years, especially after Covid 19 pandemic. Fractional order analysis has garnered increasing attention due to its effective- ness in the study of biological models [4]. The dynamics of infectious diseases are more accurately and precisely represented with the use of Fractional Calculus [5] [6]. Fractional differential equations (FDEs) are differential equations that have derivatives of fractional order. They give a more extensive and precise mathematical tool for simulating intricate physical processes with memory effects, like biological systems, viscoelastic materials, and anomalous diffusion [3] [4]. The smoking model serves as a vital tool for analyzing one of the most widespread and troublesome social issues faced globally. These models (theoretical or mathematical) are used to explain tobacco usage dynamics, smoking behaviour, and its impacts on humans population. They have the ability to analyze and forecast pattern of smoking initiation, cessation, and the health effects of tobacco use in a variety of disciplines, including epi- demiology, public health, economics, and behavioural science [7]. A smoker has a 70 percent higher risk of having a heart attack than a nonsmoker. Similar to this, smokers have an incidence rate of lung cancer that is 10 percent higher than nonsmokers.[8] Ac- cording to the World Health Organization, the effects of smoking on various body organs cause around 5 million deaths worldwide, and by 2030, that number might rise to up to 8 million annually [9]. To safeguard human health, numerous researchers, mathemati- cians, and medical professionals are working to curb smoking. As a result, many studies have focused on analyzing smoking behaviour through a variety of mathematical modeling techniques. The pioneering work in the theory of epidemics is presented by Kermack and Mck- endrick (1927). They created the compartmental mathematical model, that explains the evolution of infectious diseases [10]. The n-order model possesses the capacity to capture the transmission dynamics of contagious illnesses with greater depth and flexibility. By incorporating memory effects and nonlocal behavior—especially through fractional deriva- tives like Caputo-Fabrizio—it enables more realistic and accurate predictions of disease spread over time. In Mathematical modeling, researchers typically divide the population into three distinct compartments: Susceptible (S), Infected (I), and Removed (R), which collectively capture the dynamics of disease transmission. S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 3 of 21 The basic compartment smoking mathematical model was developed in 1997 by O Sharomi et. al. [11] which gives the qualitative analysis of the dynamics of smoking and its impact on public health. The model was then extended to the variability of smok- ing habits and related illnesses. In addition, in 2007, Ham conducted a survey in Korea that documented the different phases and smoking habits of the students cited [12]. In the same era integer-order model with a dynamic interaction is described in the literature that explains the behaviour of the smokers who smoke infrequently. Several additional researchers have contributed to the literature by developing smoking models formulated in both fractional and integer orders, enriching the understanding of behavioral dynam- ics and intervention strategies.[13] [14]. Numerical techniques are used by researchers to develope a smoking model [13]. Many researchers studied smoking model using various Numerical techniques like LADM (Laplace-Adomian Decomposition method) [15], HPM (Homotopy Perturbation technique) [9], Reduced Diferential Transform technique [16], Atangana–Toufik Method [17], and Fractional Differential Transformation technique [18]. With these successful attempts, there is a scope of research in this area. Some parameters like “Habitual” has not been given much attention by the researchers. In this paper, an expanded smoking model is presented with added compartments, in- cluding susceptible, snuff, irregular, regular, and quit to represent more enduring behaviour and outside factors. The special inclusion of a ’Habitual’ section enables us to differenti- ate between habitual (deep-rooted) smokers and occasional or light smokers based on the frequency and intensity of their smoking behaviour. This refinement captures the effects of long-term dependence and behavioural reinforcement by incorporating fractional-order derivatives, enabling a more precise simulation of memory effects and gradual transitions. The creation of more successful stage-specific intervention tactics is supported by these extensions, which offer a deeper understanding of the course of tobacco use. The smoking population is generally categorized as a single group in many theoretical frameworks. But in reality, over time, some smokers gradually become more devoted to the habit and more consistent with it. A compartment ”Habitual” is included in this model to appropriately capture this distinction. This particular category of smokers developed a strong, long- lasting habit that is difficult to break. Habitual smokers have developed a physical and psychological need for nicotine. The inclusion of this compartment enhances the ability of the presented model to better understand long-term smoking behaviour and is helpful in designing more effective interventions to support individuals in their efforts to quit. 2. Preliminaries 2.1. Fundamental concepts Before delving into the main result, first recall some essential definitions and theoretical preliminaries. Definition 1. Let r(u) ∈ H1(0, α), α > 0, and let q ∈ (0, 1) S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 4 of 21 then, Caputo-Fabrizio fractional derivative of function r(u) is given by,[19] [20] CFDq u[r(u)] = M(q) 1− q ∫ u 0 r′(y) exp [ −q (u− y) 1− q ] dy (1) where M(q) is the normalized constants satisfying M(0)=M(1)=1. The Sobolev space H1(0, α) consists of the functions on (0, α) that are square integrable and have square integrable first (weak) derivatives. Definition 2. Caputo-Fabrizio fractional integral with q ∈ (0, 1) for function r(u) is defined by,[9] CFIqu[r(u)] = 1− q N(q) r(u)] + q N(q) ∫ u 0 r(y)dy, u ≥ 0. (2) Definition 3. The Caputo-Fabrizio fractional derivative in general formula for Laplace Transform is as follows,[21] L {CFDq ur(u) } (s) = sL[r(u)]− r(0) s+ q(1− s) , s ≥ 0. (3) Definition 4. The Caputo-Fabrizio fractional derivative in general formula for Aboodh Transform is given by,[22][23] A {CFDq ur(u) } = sA[l(s)]− l(0) s s+ q(1− s) (4) Definition 5. The Aboodh Transform of the function is defined as,[24][22][23] A[r(u)] = A[l(s)] = 1 s ∫ ∞ 0 r(u)e−su du, u ≥ 0, q1 ≤ s ≤ q2, q1, q2 > 0. (5) 3. Model Formulation In this section, A a comprehensive framework is provided, which is required for the execution of the proposed methodology in the smoking model and in its fractional rep- resentation. The application of mathematical modeling can serve as an effective strategy to reduce the proliferation of tobacco consumption, as it has been recognized as a critical instrument for understanding pandemics in recent years. The flow chart given in Figure 1 is designed to depict the structure and flow of the model, which gives a clear interpretation of the functional behavior. The node points are the compartments used in the model, and the parameters are represented by lines. A generic model is the susceptible-exposed-infected-recovered(SEIR) so, in order to achieve a more profound comprehension of the qualitative assessment and the numerical S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 5 of 21 Figure 1: Flow diagram of the smoking model iterative methodologies employed for its resolution, the revised iteration of smoking model will be examined as follows, dA(u) du = ρ− ϵ1A(u)B(u) + σF (u)− µA(u), dB(u) du = ϵ1A(u)B(u)− ϵ2B(u)C(u)− ρ1B(u)− µB(u), dC(u) du = ϵ2B(u)C(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C(u), dE(u) du = (κ+ ψ)C(u)− (µ+ ϕ+ ρ3)E(u), dF (u) du = (κ1 + κ2)C(u)− (η + µ+ σ)F (u), dG(u) du = ηF (u) + ϕE(u)− µG(u), (6) The population is segregated into six classes as a description of model compartments, which is given in the TABLE I. TABLE II, shows the description of model parameters with different age groups like 18-24, 25-44, and 45 and above. S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 6 of 21 Table 1: Description of model compartments variable Description A(u) Susceptible smokers B(u) Snuffing class C(u) Irregular smokers E(u) Habitual smokers F (u) Regular smokers G(u) Quit smokers Table 2: Description of model parameters Parameter Description ρ Recruitment frequency (birth or migration) ρ1 Snuffing class death rate because of smoking ρ2 Death rate due to smoking ρ3 Death rate due to Habitual smoking ϵ1 vulnerable population transition rate for age group 18-24 to Snuffing class ϵ2 Snuffing rate becomes Irregular smokers for age group 18-24 κ Irregular smokers rate become Habitual smokers for age group 18-24 κ1 Irregular smokers rate turning to Regular smokers for age group 18-24 κ2 Irregular smokers rate become Regular smokers for age group 25-44 η Departing rate of active smokers in 45 and above age group µ natural death rate σ Recovery rate in 25-44 age group ϕ Habitual smokers rate to quit smokers with support system for 45 and above age group ψ Irregular smokers rate becomes Habitual smokers for 25-44 age group 4. Model Solution In the context of Caputo-Fabrizio fractional derivative, the updated smoking model is examined and explained as follows, CFDq u(A)(u) = ρ− ϵ1A(u)B(u) + σF (u)− µA(u), CFDq u(B)(u) = ϵ1A(u)B(u)− ϵ2B(u)C(u)− ρ1B(u)− µB(u), CFDq u(C)(u) = ϵ2B(u)C(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C(u), CFDq u(E)(u) = (κ+ ψ)C(u)− (µ+ ϕ+ ρ3)E(u), CFDq u(F )(u) = (κ1 + κ2)C(u)− (η + µ+ σ)F (u), CFDq u(G)(u) = ηF (u) + ϕE(u)− µG(u), (7) with initial Conditions A(0) = n1, B(0) = n2, C(0) = n3, E(0) = n4, F (0) = n5, G(0) = n6. (8) S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 7 of 21 In the following subsection, we found the solution for the smoking model represented by equation (7) with the initial conditions. The techniques used to solve these equation by Laplace Adomain Decomposition Method and Aboodh Adomain Decomposition Method. 4.1. Laplace Adomain Decomposition method Now, applying Laplace transform on equation (7) with the initial condition and the normalized constant M(ρ) = 1 so, The obtained equation will be L [ CFDq u(A)(u) ] = L [ ρ− ϵ1A(u)B(u) + σF (u)− µA(u) ] , L [ CFDq u(B)(u) ] = L [ ϵ1A(u)B(u)− ϵ2B(u)C(u)− ρ1B(u)− µB(u) ] , L [ CFDq u(C)(u) ] = L [ ϵ2B(u)C(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C(u) ] , L [ CFDq u(E)(u) ] = L [ (κ+ ψ)C(u)− (µ+ ϕ+ ρ3)E(u) ] , L [ CFDq u(F )(u) ] = L [ (κ1 + κ2)C(u)− (η + µ+ σ)F (u) ] , L [ CFDq u(G)(u) ] = L [ ηF (t) + ϕE(u)− µG(u) ] , (9) with the initial conditions, L ( A(u) ) = A(0) s + s+ q(1− s) s L ( ρ− ϵ1A(u)B(u) + σF (u)− µA(u) ) , L ( B(u) ) = B(0) s + s+ q(1− s) s L ( ϵ1A(u)B(u)− ϵ2B(u)C(u)− ρ1B(u)− µB(u) ) , L ( C(u) ) = C(0) s + s+ q(1− s) s L ( ϵ2B(u)C(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C(u) ) , L ( E(u) ) = E(0) s + s+ q(1− s) s L ( (κ+ ψ)C(u)− (µ+ ϕ+ ρ3)E(u) ) , L ( F (u) ) = F (0) s + s+ q(1− s) s L ( (κ1 + κ2)C(u)− (η + µ+ σ)F (u) ) , L ( G(u) ) = G(0) s + s+ q(1− s) s L ( ηF (u) + ϕE(u)− µG(u) ) , (10) Consider the solution for the compartments A(u), B(u), C(u), E(u), F(u), G(u) in the form of infinite series, A(u) = ∞∑ z=0 Az(u), B(u) = ∞∑ z=0 Bz(u), C(u) = ∞∑ z=0 Cz(u), E(u) = ∞∑ z=0 Ez(u), F (u) = ∞∑ z=0 Fz(u), S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 8 of 21 G(u) = ∞∑ z=0 Gz(u), A(u)B(u) = ∞∑ z=0 Tz(u), B(u)C(u) = ∞∑ z=0 Xz(u). (11) Nonlinear terms are decomposed using Adomian polynomials and is defined as, [25] Tz(u) = 1 n! dn dρn [ n∑ z=0 ρzAz(u) n∑ z=0 ρzBz(u) ] ρ=0 , Xz(u) = 1 n! dn dρn [ n∑ z=0 ρzBz(u) n∑ z=0 ρzCz(u) ] ρ=0 . Now applying the equation (11) in equation (10), The obtained equations are, L [ ∞∑ z=0 Az(u) ] = A(0) s + s+ q(1− s) s L [ ρ− ϵ1 ∞∑ z=0 Tz(u) + σ ∞∑ z=0 Fz(u)− µ ∞∑ z=0 Az(u) ] , L [ ∞∑ z=0 Bz(u) ] = B(0) s + s+ q(1− s) s L [ ϵ1 ∞∑ z=0 Tz(u)− ϵ2 ∞∑ z=0 Xz(u)− ρ1 ∞∑ z=0 Bz(u)− µ ∞∑ z=0 Bz(u) ] , L [ ∞∑ z=0 Cz(u) ] = C(0) s + s+ q(1− s) s L [ ϵ2 ∞∑ z=0 Xz(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ) ∞∑ z=0 Cz(u) ] , L [ ∞∑ z=0 Ez(u) ] = E(0) s + s+ q(1− s) s L [ (κ+ ψ) ∞∑ z=0 Cz(u)− (µ+ ϕ+ ρ3) ∞∑ z=0 Ez(u) ] , L [ ∞∑ z=0 Fz(u) ] = F (0) s + s+ q(1− s) s L [ (κ1 + κ2) ∞∑ z=0 Cz(u)− (η + µ+ σ) ∞∑ z=0 Fz(u) ] , L [ ∞∑ z=0 Gz(u) ] = G(0) s + s+ q(1− s) s L [ η ∞∑ z=0 Fz(u) + ϕ ∞∑ z=0 Ez(u)− µ ∞∑ z=0 Gz(u) ] . (12) Equating both sides of the above equations, the follwing is obtained. L[A0(u)] = n1 s , L[B0(u)] = n2 s , L[C0(u)] = n3 s , L[E0(u)] = n4 s , L[F0(u)] = n5 s , L[G0(u)] = n6 s , L ( A1(u) ) = s+ q(1− s) s L ( ρ− ϵ1T0(u) + σF0(u)− µA0(u) ) , L ( B1(u) ) = s+ q(1− s) s L ( ϵ1T0(u)− ϵ2X0(u)− ρ1B0(u)− µB0(u) ) , L ( C1(u) ) = s+ q(1− s) s L ( ϵ2X0(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C0(u) ) , L ( E1(u) ) = s+ q(1− s) s L ( (κ+ ψ)C0(u)− (µ+ ϕ+ ρ3)E0(u) ) , L ( F1(u) ) = s+ q(1− s) s L ( (κ1 + κ2)C0(u)− (η + µ+ σ)F0(u) ) , L ( G1(u) ) = s+ q(1− s) s L ( ηF0(u) + ϕE0(u)− µG0(u) ) , S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 9 of 21 L ( A2(u) ) = s+ q(1− s) s L ( ρ− ϵ1T1(u) + σF1(u)− µA1(u) ) , L ( B2(u) ) = s+ q(1− s) s L ( ϵ1T1(u)− ϵ2X1(u)− ρ1B1(u)− µB1(u) ) , L ( C2(u) ) = s+ q(1− s) s L ( ϵ2X1(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C1(u) ) , L ( E2(u) ) = s+ q(1− s) s L ( (κ+ ψ)C1(u)− (µ+ ϕ+ ρ3)E1(u) ) , L ( F2(u) ) = s+ q(1− s) s L [ (κ1 + κ2)C1(u)− (η + µ+ σ)F1(u) ) , L ( G2(u) ) = s+ q(1− s) s L ( ηF1(u) + ϕE1(u)− µG1(u) ) , (13) and so on .... L ( Az+1(u) ) = s+ q(1− s) s L ( ρ− ϵ1Tz(u) + σFz(u)− µAz(u) ) , L ( Bz+1(u) ) = s+ q(1− s) s L ( ϵ1Tz(u)− ϵ2Xz(u)− ρ1Bz(u)− µBz(u) ) , L ( Cz+1(u) ) = s+ q(1− s) s L ( ϵ2Xz(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)Cz(u) ) , L ( Ez+1(u) ) = s+ q(1− s) s L ( (κ+ ψ)Cz(u)− (µ+ ϕ+ ρ3)Ez(u) ) , L ( Fz+1(u) ) = s+ q(1− s) s L ( (κ1 + κ2)Cz(u)− (η + µ+ σ)Fz(u) ) , L ( Gz+1(u) ) = s+ q(1− s) s L ( ηFz(u) + ϕEz(u)− µGz(u) ) . (14) Now taking inverse Laplace transform on both sides,the obtained equations are, A0 = n1, B0 = n2, C0 = n3, E0 = n4, F0 = n5, G0 = n6, A1(u) = [ρ− ϵ1n1n2 + σn5 − µn1] [1 + q(u− 1)], B1(u) = [ϵ1n1n2 − ϵ2n2n3 − ρ1n2 − µn2] [1 + q(u− 1)], C1(u) = [ϵ2n2n3 − (κ+ ρ2 + µ+ κ1 + κ2 + ψ)n3] [1 + q(u− 1)], E1(u) = [(κ+ ψ)n3 − (µ+ ϕ+ ρ3)n4] [1 + q(u− 1)], F1(u) = [(κ1 + κ2)n3 − (η + µ+ σ)n5] [1 + q(u− 1)], G1(u) = [ηn5 + ϕn4 − µn6] [1 + q(u− 1)], A2(u) = [ρ][1 + q(u− 1)]− [ϵ1(a11n2 + n1b11) + σf11 − µa11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , B2(u) = [ϵ1(a11n2 + n1b22)− ϵ2(c11n2 + n3b11) + ρ1b11 + µb11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 10 of 21 C2(u) = [ϵ2(c11n2 + n3b11)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)c11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , E2(u) = [(κ+ ψ)c11 − (µ+ ϕ+ ρ3)e11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , F2(u) = [(κ1 + κ2)c11 − (η + µ+ σ)f11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , G2(u) = [ηf11 + ϕe11 − µg11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] . (15) Moreover, the remaining variables may be derived in an analogous manner correspondingly. Thus the indeterminate quantities in the preceding equations are delineated as follows: a11(u) = [ρ− ϵ1n1n2 + σn5 − µn1] [1 + q(u− 1)], b11(u) = [ϵ1n1n2 − ϵ2n2n3 − ρ1n2 − µn2] [1 + q(u− 1)], c11(u) = [ϵ2n2n3 − (κ+ ρ2 + µ+ κ1 + κ2 + ψ)n3] [1 + q(u− 1)], e11(u) = [(κ+ ψ)n3 − (µ+ ϕ+ ρ3)n4] [1 + q(u− 1)], f11(u) = [(κ1 + κ2)n3 − (η + µ+ σ)n5] [1 + q(u− 1)], g11(u) = [ηn5 + ϕn4 − µn6] [1 + q(u− 1)]. (16) 4.2. Aboodh Adomain Decomposition Method Now, applying the Aboodh Transform in Equation (7) with the initial conditions and the normalized constant, M(ρ) = 1 The obtained equation will be , A [ CFDq u(A)(u) ] = A [ ρ− ϵ1A(u)B(u) + σF (u)− µA(u) ] , A [ CFDq u(B)(u) ] = A [ ϵ1A(u)B(u)− ϵ2B(u)C(u)− ρ1B(u)− µB(u) ] , A [ CFDq u(C)(u) ] = A [ ϵ2B(u)C(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C(u) ] , A [ CFDq u(F )(u) ] = A [ (κ+ ψ)C(u)− (µ+ ϕ+ ρ3)E(u) ] , A [ CFDq u(E)(u) ] = A [ (κ1 + κ2)C(u)− (η + µ+ σ)F (u) ] , A [ CFDq u(G)(u) ] = A [ ηF (u) + ϕE(u)− µG(u) ] , (17) S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 11 of 21 with the initial conditions, A ( A(u) ) = A(0) s2 + s+ q(1− s) s A ( ρ− ϵ1A(u)B(u) + σF (u)− µA(u) ) , A ( B(u) ) = B(0) s2 + s+ q(1− s) s A ( ϵ1A(u)B(u)− ϵ2B(u)C(u)− ρ1B(u)− µB(u) ) , A ( C(u) ) = C(0) s2 + s+ q(1− s) s A ( ϵ2B(u)C(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C(u) ) , A ( E(u) ) = E(0) s2 + s+ q(1− s) s A ( (κ+ ψ)C(u)− (µ+ ϕ+ ρ3)E(u) ) , A ( F (u) ) = F (0) s2 + s+ q(1− s) s A ( (κ1 + κ2)C(u)− (η + µ+ σ)F (u) ) , A ( G(u) ) = G(0) s2 + s+ q(1− s) s A ( ηF (u) + ϕE(u)− µG(u) ) , (18) A(u) = ∞∑ z=0 Az(u), B(u) = ∞∑ z=0 Bz(u), C(u) = ∞∑ z=0 Cz(u), E(u) = ∞∑ z=0 Ez(u), F (u) = ∞∑ z=0 Fz(u), G(u) = ∞∑ z=0 Gz(u), A(u)B(u) = ∞∑ z=0 Tz(u), B(u)C(u) = ∞∑ z=0 Xz(u). (19) Nonlinear terms are decomposed using Adomian polynomials and is defined as, Tz(u) = 1 n! dn dρn [ n∑ z=0 ρzAz(u) n∑ z=0 ρzBz(u) ] ρ=0 , Xz(u) = 1 n! dn dρn [ n∑ z=0 ρzBz(u) n∑ z=0 ρzCz(u) ] ρ=0 . Now applying the equation(19) in equation (18), the follwing is obtained. A [ ∞∑ z=0 Az(u) ] = A(0) s2 + s+ q(1− s) s A [ ρ− ϵ1 ∞∑ z=0 Tz(u) + σ ∞∑ z=0 Fz(u)− µ ∞∑ z=0 Az(u) ] , A [ ∞∑ z=0 Bz(u) ] = B(0) s2 + s+ q(1− s) s A [ ϵ1 ∞∑ z=0 Tz(u)− ϵ2 ∞∑ z=0 Xz(u)− ρ1 ∞∑ z=0 Bz(u)− µ ∞∑ z=0 Bz(u) ] , A [ ∞∑ z=0 Cz(u) ] = C(0) s2 + s+ q(1− s) s A [ ϵ2 ∞∑ z=0 Xz(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ) ∞∑ z=0 Cz(u) ] , A [ ∞∑ z=0 Ez(u) ] = E(0) s2 + s+ q(1− s) s L [ (κ+ ψ) ∞∑ z=0 Cz(u)− (µ+ ϕ+ ρ3) ∞∑ z=0 Ez(u) ] , A [ ∞∑ z=0 Fz(u) ] = F (0) s2 + s+ q(1− s) s L [ (κ1 + κ2) ∞∑ z=0 Cz(u)− (η + µ+ σ) ∞∑ z=0 Fz(u) ] , A [ ∞∑ z=0 Gz(u) ] = G(0) s2 + s+ q(1− s) s A [ η ∞∑ z=0 Fz(u) + ϕ ∞∑ z=0 Ez(u)− µ ∞∑ z=0 Gz(u) ] . (20) S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 12 of 21 Equating both sides of the above equations, obtained equation will be, A[A0(u)] = n1 s2 , A[B0(u)] = n2 s2 , A[C0(u)] = n3 s2 , A[E0(u)] = n4 s2 , A[F0(u)] = n5 s2 , A[G0(u)] = n6 s2 , A [ A1(u) ] = s+ q(1− s) s A [ ρ− ϵ1T0(u) + σF0(u)− µA0(u) ] , A [ B1(u) ] = s+ q(1− s) s A [ ϵ1T0(u)− ϵ2X0(u)− ρ1B0(u)− µB0(u) ] , A [ C1(u) ] = s+ q(1− s) s A [ ϵ2X0(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C0(u) ] , A [ E1(u) ] = s+ q(1− s) s A [ (κ+ ψ)C0(u)− (µ+ ϕ+ ρ3)E0(u) ] , A [ F1(u) ] = s+ q(1− s) s A [ (κ1 + κ2)C0(u)− (η + µ+ σ)F0(u) ] , A [ G1(u) ] = s+ q(1− s) s A [ ηF0(u) + ϕE0(u)− µG0(u) ] , A [ A2(u) ] = s+ q(1− s) s A [ ρ− ϵ1T1(u) + σF1(u)− µA1(u) ] , A [ B2(u) ] = s+ q(1− s) s A [ ϵ1T1(u)− ϵ2X1(u)− ρ1B1(u)− µB1(u) ] , A [ C2(u) ] = s+ q(1− s) s A [ ϵ2X1(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)C1(u) ] , A [ E2(u) ] = s+ q(1− s) s A [ (κ+ ψ)C1(u)− (µ+ ϕ+ ρ3)E1(u) ] , A [ F2(u) ] = s+ q(1− s) s A [ (κ1 + κ2)C1(u)− (η + µ+ σ)F1(u) ] , A [ G2(u) ] = s+ q(1− s) s A [ ηF1(u) + ϕE1(u)− µG1(u) ] , (21) and so on .... A [ Az+1(u) ] = s+ q(1− s) s A [ ρ− ϵ1Tz(u) + σFz(u)− µAz(u) ] , A [ Bz+1(u) ] = s+ q(1− s) s A [ ϵ1Tz(u)− ϵ2Xz(u)− ρ1Bz(u)− µBz(u) ] , A [ Cz+1(u) ] = s+ q(1− s) s A [ ϵ2Xz(u)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)Cz(u) ] , A [ Ez+1(u) ] = s+ q(1− s) s A [ (κ+ ψ)Cz(u)− (µ+ ϕ+ ρ3)Ez(u) ] , A [ Fz+1(u) ] = s+ q(1− s) s A [ (κ1 + κ2)Cz(u)− (η + µ+ σ)Fz(u) ] , A [ Gz+1(u) ] = s+ q(1− s) s A [ ηFz(u) + ϕEz(u)− µGz(u) ] . (22) S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 13 of 21 Now applying the Inverse Aboodh Transform on both sides, The obtained equation will be, A0 = n1, B0 = n2, C0 = n3, E0 = n4, F0 = n5, G0 = n6, A1(u) = [ρ− ϵ1n1n2 + σn5 − µn1] [1 + q(u− 1)], B1(u) = [ϵ1n1n2 − ϵ2n2n3 − ρ1n2 − µn2] [1 + q(u− 1)], C1(u) = [ϵ2n2n3 − (κ+ ρ2 + µ+ κ1 + κ2 + ψ)n3] [1 + q(u− 1)], E1(u) = [(κ+ ψ)n3 − (µ+ ϕ+ ρ3)n4] [1 + q(u− 1)], F1(u) = [(κ1 + κ2)n3 − (η + µ+ σ)n5] [1 + q(u− 1)], G1(u) = [ηn5 + ϕn4 − µn6] [1 + q(u− 1)], A2(u) = [ρ][1 + q(u− 1)]− [ϵ1(a11n2 + n1b11) + σf11 − µa11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , B2(u) = [ϵ1(a11n2 + n1b22)− ϵ2(c11n2 + n3b11) + ρ1b11 + µb11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , C2(u) = [ϵ2(c11n2 + n3b11)− (κ+ ρ2 + µ+ κ1 + κ2 + ψ)c11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , E2(u) = [(κ+ ψ)c11 − (µ+ ϕ+ ρ3)e11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , F2(u) = [(κ1 + κ2)c11 − (η + µ+ σ)f11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , G2(u) = [ηf11 + ϕe11 − µg11] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] . (23) The remaining variables can be obtained in a similar way. The indeterminate quantities in the earlier equations are described as follows: a11(u) = [ρ− ϵ1n1n2 + σn5 − µn1] [1 + q(u− 1)], b11(u) = [ϵ1n1n2 − ϵ2n2n3 − ρ1n2 − µn2] [1 + q(u− 1)], c11(u) = [ϵ2n2n3 − (κ+ ρ2 + µ+ κ1 + κ2 + ψ)n3] [1 + q(u− 1)], e11(u) = [(κ+ ψ)n3 − (µ+ ϕ+ ρ3)n4] [1 + q(u− 1)], f11(u) = [(κ1 + κ2)n3 − (η + µ+ σ)n5] [1 + q(u− 1)], g11(u) = [ηn5 + ϕn4 − µn6] [1 + q(u− 1)]. (24) It is observed that both LADM and AADM yield identical terms, demonstrating their consistency in solving nonlinear fractional-order differential equations (FODEs). This alignment confirms the accuracy and reliability of each method in handling such com- plex systems. The agreement in results underscores the effectiveness of both approaches S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 14 of 21 and supports the validity of the proposed fractional smoking model. Furthermore, by preserving the fundamental dynamics of the system, these consistent results highlight the robustness of LADM and AADM as validation tools for fractional models. The use of frac- tional calculus further strengthens the applicability of these techniques, thus increasing confidence in both analytical and numerical solutions to real-world problems. 4.3. Discussion of Numerical Simulation In this section, hypothetical values are allocated to the specific parameter and these values are used in the model (7) by n1 = 70, n2 = 45, n3 = 35, n4 = 24, n5 = 10, n6 = 8, ρ = 0.1, ρ1 = 0.002, ρ2 = 0.0025, ρ3 = 0.003, σ = 0.003, µ = 0.0015, κ = 0.04, κ1 = 0.003, κ2 = 0.002, ϕ = 0.005, ψ = 0.001, η = 0.03, ϵ1 = 0.003, ϵ2 = 0.002. Using parametric values in proposed model (7), the following is obtained, A0 = 70, B0 = 45, C0 = 35, E0 = 24, F0 = 10, G0 = 8, A1(u) = [−9.425] [1 + q(u− 1)], B1(u) = [6.1425] [1 + q(u− 1)], C1(u) = [1.4] [1 + q(u− 1)], E1(u) = [1.207] [1 + q(u− 1)], F1(u) = [−0.17] [1 + q(u− 1)], G1(u) = [0.408] [1 + q(u− 1)], A2(u) = [0.1][1 + q(u− 1)]− [0.0039] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , B2(u) = [−0.517] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , C2(u) = [0.486] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , E2(u) = [0.0459] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , F2(u) = [0.0129] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , G2(u) = [0.0003] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] . S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 15 of 21 Further, the solutions for first few terms can be found as, A(u) = 70− [9.325] [1 + q(u− 1)]− [0.0039] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , B(u) = 45 + [6.1425] [1 + q(u− 1)]− [0.517] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , C(u) = 35 + [1.4] [1 + q(u− 1)] + [0.486] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , E(u) = 24 + [1.207] [1 + q(u− 1)] + [0.0459] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , F(u) = 10− [0.17] [1 + q(u− 1)] + [0.0129] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] , G(u) = 8 + [0.408] [1 + q(u− 1)]− [0.0003] [q2u2 2! − 2q2u+ 2qu+ (q − 1)2 ] . 5. Analysis of Model The simulation is done using MATLAB and graphs are plotted. It gives the idea about the interrelation between the parameters. Figure 2: Susceptible smokers(A) vs time(u) by Laplace Trans. Figure 3: Susceptible smokers(A) vs time(u) by Aboodh Trans. S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 16 of 21 Figure 4: Snuffing smokers(B) vs time(u) by Laplace Trans. Figure 5: Snuffing smokers(B) vs time(u) by Aboodh Trans. Figure 6: Irregular smokers(C) vs time(u) by Laplace Trans. Figure 7: Irregular smokers(C) vs time(u) by Aboodh Trans. The graphs show how different groups of people move through smoking behaviors over time using different fractional orders in the model. Figures 2–13 present solutions versus various fractional-order q. As shown clearly in figures 2–13, A(u),B(u),C(u),E(u), F (u),G(u), increases fast for lower fractional values of q = 0.70 and increases slowly for higher fractional values of q = 0.95. The Caputo-Fabrizio operator-based fractional-order smoking model illustrates the temporal evolution of each compartment with increasing trends across the system. The progression begins with the Susceptible group, representing individuals at risk of tobacco exposure. Over time, the transition to the Snuffing stage reflects early experimentation, followed by an upward shift in the Habitual compartment, indicating developing patterns of smoking behavior. S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 17 of 21 Figure 8: Habitual smokers(E) vs time(u) by Laplace Trans. Figure 9: Habitual smokers(E) vs time(u) by Aboodh Trans. Figure 10: Regular smokers(F) vs time(u) by Laplace Trans. Figure 11: Regular smokers(F)vs time(u) by Aboodh Trans. The Irregular and Regular smoker populations also exhibit rising curves, showing further reinforcement of the habit. Eventually, the model depicts a steady increase in the Quit compartment, as individuals gradually succeed in cessation. The growth patterns depict how individuals move through increasingly dependent smoking stages before attempting cessation. The smooth flow between compartments underscores the memory effect embed- ded in the Caputo-Fabrizio fractional derivative, allowing a more realistic representation of smoking dynamics and behavioral relapse. S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 18 of 21 Figure 12: Quit smokers(G) vs time(u) by Laplace Trans. Figure 13: Quit smokers(G) vs time(u) by Aboodh Trans. 6. Conclusion This research introduces an innovative methodology for the representation of smoking behavior through the implementation of a fractional-order mathematical framework uti- lizing the Caputo-Fabrizio derivative. In comparison with the traditional integer-order models, the application of fractional calculus facilitates a more comprehensive. Especially when the parameters like memory and hereditary influences are involved, which affect crucially in the dynamics of human behaviour particularly when it comes to addiction and relapse. According to the concept, smoking behavior develops gradually, starting with those who are susceptible and moving through snuffing, frequent use, intermittent smoking, and finally regular smoking. The cohort of regular smokers remains dominant throughout the experiment, even if the number of people quitting smoking keeps rising. This phenomenon demonstrates how difficult it is to break smoking habits once they have been formed. The increasing number of people quitting smoking is a sign that intervention efforts are working, but it is not enough to reverse the overall trend. This discordance is largely caused by strong behavioral habits and a tendency to relapse. To strengthen the impact of smoking cessation programs, efforts should emphasize prompt engagement and continuous assistance for individuals in advanced stages of tobacco dependency. In this study, the interconnection between the parameters involved in the smoking models is presented. Numerical methodologies utilizing two specific transforms, LADM- Laplace Adomian Decomposition Method and AADM-Aboodh Adomian Decomposition Method, are used to solve the model. Both methodologies exhibited a high degree of consistency and precision in their outcomes. This strong agreement shows that these methods can be trusted or relied upon when used to model complex systems, such as nonlinear fractional-order differential equations. Numerical simulation is performed using S. Bhosale et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6507 19 of 21 MATLAB for the various compartments (such as susceptible smokers, snuffing class, ir- regular smokers, habitual smokers, regular smokers & quit smokers) and the graphs are plotted. The model highlights a deeper understanding of the model development and possible control using fractional-order methods. In summary, the study explains the significant advantages of fractional-order models for behavioural epidemiology research, particularly those that contain the Caputo-Fabrizio operator. Future studies are also encouraged by the work to investigate different kinds of fractional operators or broaden the model by including more social or psychological parameters. Overall, the study shows that memory (represented by the fractional or- der) greatly affects smoking behaviour, especially in helping or delaying the move toward quitting. Strong public health efforts are needed, especially for Regular, Irregular, and Habitual smokers, to overcome the influence of long-term habits. References [1] Laécio Carvalho de Barros, Michele Martins Lopes, Francielle Santo Pedro, Estevão Esmi, José Paulo Carvalho dos Santos, and Daniel Eduardo Sánchez. The memory effect on fractional calculus: an application in the spread of covid-19. Computational and Applied Mathematics, 40(3):72, 2021. 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