A novel technique to study the solutions of time fractional nonlinear smoking epidemic model

The primary goal of the current work is to use a novel technique known as the natural transform decomposition method to approximate an analytical solution for the fractional smoking epidemic model. In the proposed method, fractional derivatives are considered in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo senses. An epidemic model is proposed to explain the dynamics of drug use among adults. Smoking is a serious issue everywhere in the world. Notwithstanding the overwhelming evidence against smoking, it is nonetheless a harmful habit that is widespread and accepted in society. The considered nonlinear mathematical model has been successfully used to explain how smoking has changed among people and its effects on public health in a community. The two states of being endemic and disease-free, which are when the disease dies out or persists in a population, have been compared using sensitivity analysis. The proposed technique has been used to solve the model, which consists of five compartmental agents representing various smokers identified, such as potential smokers V, occasional smokers G, smokers T, temporarily quitters O, and permanently quitters W. The results of the suggested method are contrasted with those of existing numerical methods. Finally, some numerical findings that illustrate the tables and figures are shown. The outcomes show that the proposed method is efficient and effective.


Model description
By creating mathematical models and examining their dynamical behaviors, it is a crucial and effective technique to comprehend biological issues.In this study, the system consisting of five equations involving nonlinear differentials characterizing the smoking pandemic model is considered.Let N(τ ) denote the overall population, at time τ .We break down the population N(τ ) into five categories in order to better comprehend it: potential smokers V (τ ) , occasional smokers G(τ ) , smokers T(τ ) , temporarily quitting smokers O(τ ) , and permanently quitting smokers W(τ ) .The suggested smoking model 43 is given as a system of nonlinear differential equations with the following coefficients: The recruitment rate for potential smokers is represented by α , the effective contact rate between smokers and potential smokers is represented by ǫ , the natural death rate is represented by ϑ , the rate of quitting smoking is represented by ρ , the remaining percentage of smokers who permanently quit smoking is represented by σ , the rate at which occasional smokers become regular smokers are represented by ε 1 , and the contact rate between smokers and temporary quitters who return to smoking is represented by ε 2 .Table 1 provides information on the parameters employed in the system of Eq. (1).
The utilisation of fractional-order differential equations with time delay has been commonly employed in the field of biology in recent decades.Biological systems modelled using fractional-order differential equations produce more realistic and accurate outcomes in capturing the hereditary and memory characteristics of the system, as opposed to models based on integer-order differential equations.The majority of mathematical models of biological systems include a form of enduring historical memory 44 .Fractional order extension of the model (1) was initially investigated in 45,46 .Fractional differential equations are used to demonstrate the genuine biphasic decrease behaviour of disease infection, albeit at a slower rate.The fractional differential equation system is described by the following.
with the initial conditions The system is qualitatively examined in two different methods, namely endemic equilibrium and disease free equilibrium. (1) (2) Table 1.The parameters utilized in system (2), along with a description of their particular values.Interaction ratio between smokers and transient abstainers who resume smoking 0.0025

Equilibrium point and stability
Analysing the differential equations presented in Eq. ( 2) can provide valuable understanding of the propagation of smoking and the potential methods for restricting its spread.The reproductive number is a crucial tool for analysing a model of this nature.The basic reproduction number, represented as R 0 , and is defined as the number of new infections produced by a typical infective individual in a susceptible population at a disease free equilibrium.In the scenario of disease-free equilibrium, a value of R 0 < 1 indicates that the disease will ultimately disappear.Conversely, in the case of endemic equilibrium, a value of R 0 > 1 indicates that the disease will propagate throughout the population.
In system (2) we consider equilibrium point 47 , we take We achieved equilibria that were disease-free.
as well as the system being in endemic equilibrium where Theorem 1 15 The diseases free equilibrium E 0 is locally asymptotically stable for R 0 < 1 otherwise unstable.
Consider the Jacobian matrix as . Since the Jacobian matrix is J = F − Q then the matrix F and Q can be written as We know that B = FQ −1 and using the relation |B − I| = 0 for the eigen value , we get by substituting the values of each parameters, we get 0 < < 1 , which shows the reproductive number R 0 < 1 , so the constructed system is in diseases free state.Reproductive number in this system (2) is We see that all the eigen values are negative only for R 0 < 1 .Thus the disease free state is locally asymptotically stable for R 0 < 1 , otherwise unstable.Sensitivity analysis of R 0 : The sensitivity of R 0 to each of its parameters is It can be seen that R 0 is most sensitive to change in parameters here R 0 is increasing with ε 1 , ǫ, ϑ and decreasing with ρ .In other words it found that the sensitivity analysis shows that prevention is better than quit smoking. (4)

Preliminaries
In this section, we will present some fractional derivative definitions along with the NT.
Definition 4.1 48 The C derivative of g ∈ C q −1 , q ∈ N of order µ is as follows Definition 4.2 20 The CF derivative of the function g(τ ) with order µ is represented as Definition 4.3 49 The ABC derivative of the function g(τ ) with order µ is defined here E µ denotes the Mittag-Leffler function and B[µ] is a normalization function.
Definition 4.4 50 On employing the NT of the function g(τ ) is stated as Definition 4.5 51 On employing the NT of C derivative is defined as Definition 4.6 52 On employing the NT of CF derivative is given as Definition 4.7 51 On employing the NT of ABC derivative is defined as .

The methodology of NTDM
We study the given system of fractional nonlinear PDEs with the initial conditions utilizing singular and nonsingular kernel derivatives as given below to explain the fundamental concept of this approach.NTDM C : In view of Eq. ( 10) and initial conditions (3), we obtain Operating with the inverse NT on (13), we have The decomposition of the Adomian polynomials of nonlinear terms is as follow In the above Eq.( 15), the A k and B k are both Adomian polynomials, and by using the formula 53 , they are computed.

Existence and uniqueness
In this section we will present the existence and uniqueness results of the system (2), by considering the fractional derivative in the Caputo sense by making use of the approach given in 54 .Assume that With the use of Eqs. ( 2) and ( 30), we can write Applying fractional integral and using initial conditions, we have Let Using Eq. ( 33) in Eq. ( 32), we get Consider a Banach space = C[0, T] × C[0, T] , with a norm Let d : → be a mapping defined as Further, we impose some conditions on a nonlinear function as follows: Condition (i): There exist conditions Next, To show the system (31) has unique solution by using Banach-fixed point theorem.
Theorem 3 Suppose that condition (ii) holds.Then the system (31) has unique solution.
Proof Let X, X1 ∈ .Now Hence,d is the contraction.Using the Banach contraction theorem the system (31) has unique solution.
In similar lines, we can prove the uniqueness and existence for NTDM CF and NTDM ABC solutions of the system (2).

Illustrative example
In this section, the approximate solutions of nonlinear time fractional smoking model by applying three fractional derivatives are presented.
NTDM C : We get the following NTDM C solutions as, Vol Vol:.( 1234567890)

Numerical results and discussion
This study presents the approximate solutions of a non-linear time fractional smoking epidemic model.The NTDM is utilised to investigate this model by considering the fractional derivative in Caputo, CF and ABC sense.The proposed method presents the outcomes of the smoking model by utilising tables and figures to observe the effects of the parameters.Graphical representation of the five compartments provides an analysis of the behaviour exhibited by each class, offering explanations for their respective actions.These simulations illustrate that a modification in value influenced the dynamics of the epidemic.This aids our understanding of the evolution of smoking patterns over time.The non-integer order has a minor impact on the dynamics of the epidemic, as demonstrated in Figs. 1, 2, 3, 4 and 5. Furthermore, this demonstrates that the methodology employed for fractional differential equations yields more accurate approximations for various fractional models.We utilised integer and fractional order for potential smokers (non-smokers), and it is evident that the number of non-smokers increases at fractional values of µ .At the equilibrium point, one of the affected components (smokers) has a non-zero value, and its convergence can be observed for both integer and fractional values of µ .Meanwhile, the other affected component converges to zero.Furthermore, it is evident that the infection rate diminishes as the fractional values of µ decrease.It is evident from Figs. 1 and 3 the total number of potential smokers V (τ ) and smokers T(τ ) increases over time and increases as the integer order µ goes down.From Figs. 2 and 4 it is clear that the number of occasional smokers G(τ ) and temporarily quitters O(τ ) increases over time and decreases as the integer order µ goes down.Figure 5 depicts the quantity of permanently quitters W(τ ) increases over time and as the fractional parameter µ increases, but after a while the behavior reverses.Tables 2, 3, 4 and 5 present a comparison between the approximate solutions obtained using the present method and the existing methods in the literature, specifically LADM and q-HATM at various fractional order values µ .Tables 6 and 7 displays a comparison of the proposed method solutions and the two established techniques in the literature for the classical derivative.The proposed method solution demonstrates a significant degree of agreement with these methods.It is observed that the proposed method works well in producing approximations of solutions for the suggested mathematical model.We can infer from the results that the suggested approach is useful for comprehending behavior when using fractional derivatives.

Conclusion
Smoking increases the vulnerability of individuals to various perilous illnesses, such as oral, cervical, breast, and pancreatic cancers.The current framework utilises the natural transform decomposition approach for finding the approximate analytical solutions for the smoking epidemic model.The fractional derivatives included in the model under consideration are the Caputo, Caputo-Fabrizio, and Atangana-Baleanu-Caputo derivatives.In the context of the fractional-order smoking model, we analyse the concepts of reproduction number, endemic equilibrium point, and free disease equilibrium.The proposed method results demonstrate good agreement to limit smoking's negative effects over various time periods and to eliminate a leading cause of death worldwide.Upon comparing the results to those of the q-HATM and LADM, it is observed that the outcomes align with the proposed approach, where µ=1.Tables and graphs illustrate the characteristics of approximate solutions.The findings of this study will facilitate additional examination and the mitigation of diverse smoking-induced epidemics.In conclusion, we affirm that the proposed methodology is highly methodical and can be employed to analyse nonlinear fractional mathematical models that represent biological phenomena.Fractional calculus

Theorem 4
The NTDM C solution is convergent.Theorem 5 NTDM CF solution is convergent.Theorem 6 NTDM ABC solution is convergent.

Table 5 .
15proximate solution for permanently quit smokers W(τ ) at various fractional orders and τ.NTDM CF NTDM ABC q-HATM 55 LADM15NTDM C NTDM CF NTDM ABC q-HATM 55 LADM15 In the same manner, one arrives at these approximate solutions

Table 6 .
Approximate solution of Smoking epidemic model with fixed µ = 1 at various values of τ.