DYNAMIC MODEL OF SMOKERS AND ITS SENSITIVITY ANALYSIS

. We study a smoking dynamical model with two types of smokers: beginners and heavy smokers. The qualitative behavior of the model, such as the stability of the equilibrium points and the basic reproduction number, is investigated. We show some simulations to validate the analytical ﬁndings, such as solution dynamics at different time scales and phase portraits of solutions with varying initial conditions. We also present a normalized sensitivity analysis of the basic reproduction number to discover which parameter has the most impact on smoking transmission, and perform a time-dependent sensitivity analysis of parameters to examine their impact on population dynamics.


INTRODUCTION
Smoking tobacco increases the chance of several long-term health conditions, and it is the most common preventable disease, accounting for about 19% of adult deaths in the UK [1,2].
More than 16 million persons in the USA suffer illnesses as a result of tobacco use, and 527,736 of those fatalities are preventable, or 17.9% of all deaths each year [3]. The cost of treating illnesses brought on by smoking is believed to be $467 billion globally, with Europe and North America bearing the heaviest financial burden [4]. More than seven million people die each year as a direct result of smoking; six million more people die as a result of smoking; and approximately 900,000 nonsmokers are harmed by smokers, also known as passive smokers [5]. Researchers from a variety of fields are studying the dynamics of smoking, particularly with the help of mathematical models.
In order to characterize the dynamics of drug use among teenagers, specifically tobacco use, authors in [6] presented a general epidemiological model, then they constructed specific models by taking other factors into account that have been identified to have an impact on the rising trend of tobacco use. Authors in [7] offered a thorough mathematical analysis for evaluating the dynamics of smoking and its effects on community public health. Authors in [8] introduced a novel model for quitting smoking in which the interaction term is the square root of current and potential smokers. To investigate how media campaigns affect smoking cessation, authors in [9] examined a nonlinear mathematical model, with the focus of the analysis being on backward bifurcation. Authors in [10] suggested a mathematical model to investigate the dynamics of smoking habit under the influence of educational initiatives as well as human willpower to give up smoking. Authors in [11] examined the qualitative behavior of a smoking model in which the population is split into five classes: non-smokers, smokers, smokers who have temporarily given up smoking, smokers who have permanently given up smoking, and smokers who have a smoking-related ailment. In a delayed quitting smoking model with harmonic mean type incidence rate and relapse, authors in [12] examined the stability and Hopf bifurcation. Authors in [13] We investigated the existence of Hopf bifurcation and global stability in a delayed smoking model that included potential smokers, infrequent smokers, smokers, temporary quitters, permanent quitters, and smokers with some disease.
Because fractional order displays the past history and hereditary qualities in models, notably in the models of infectious diseases, fractional order mathematical models have been shown to be beneficial in mathematically displaying a wide range of phenomena than integer-order models [14], a fractional dynamic model of tobacco smoking is used by some scientists. Authors in [14] studied a Caputo fractional-order tobacco smoking model with snuffing class. Authors in [15] examined of the fractional order smoking model through computation. In [16], the fractional order smoking model is investigated and solved using the generalized Mittag-Leffler function method and the Sumudu transform method. Authors in [17] employed a numeric-analytic approach to approximate a fractional derivative-based model of quitting smoking. Authors in [18] considered the Atangana-Baleanu derivative to analyze the dynamics of the smoking model and its impact on public health.
Numerous scholars also investigated the smoking mathematical model in conjunction with the implementation of various prevention strategies. Authors in [19] investigated potential lightsmoker-quit smokers with two possible control variables in the form of education and therapy campaigns aimed at decreasing smoking attitudes. Authors in [20] studied the optimal control method for a discrete time smoking model with a fixed saturation incidence rate. Authors in [21] examined the optimal control scheme for a new model of quitting smoking that incorporates the class of chain smokers' continuous age-structure. Authors in [22] used four control variableseducational campaigns, anti-smoking gum, medications, and government bans on smoking in public spaces-along with a mathematical study of the harmonic mean type incidence rate of giving up smoking in order to reduce the use of smoking in the community. In a harmonic mean type dynamics of a delayed giving up smoking model, authors in [23] investigated the best legislative control strategy to reduce the number of smokers. Authors in [24] developed a control problem taking into account three control measures, namely; education campaign, anti-nicotine gum, and anti-nicotine medications, in order to manage the smoking behavior in the population of a giving up smoking model with relapse and harmonic mean type incidence rate. Recently, in a model of interactions between smokers in mixed populations of beginners and heavy smokers, authors in [25] examined a smoking cessation control, namely aducational campaign and nicotine therapy counselling.
The model in [25] considers the untreated and treated populations of smokers, since it incorporates a control strategy. In this research, we investigate the model without the controls, or, in other words, we investigate the reduced model by combining the untreated and treated beginners into a single population. The model's qualitative behavior, sensitivity of the basic reproduction number, and sensitivity of the model's parameters are then investigated. By using this strategy, we can identify the most sensitive parameters that have the greatest impact on smoking transmission and population dynamics.

MODEL
Herdiana et al. [25] proposed a mathematical model of the dynamics of active smokers in mix population incorporating with cessation controls. The model is as follows where P is non-smokers or potential smokers, B U is untreated beginner smokers, B T is treated beginner smokers, S U is untreated smokers, S T is treated smokers, and Q is smokers who quit smoking permanently. The cessation controls are ν 1 and ν 2 . The control ν 1 is educational campaign, while ν 2 is counseling with nicotine replacement.
When there is no control, that is ν 1 = ν 2 = 0, the model (1) becomes In this paper, we reduce the model (2) by grouping the untreated and treated beginners as one population, and also for the untreated and treated smokers, by defining B = B U + B T and S = S U + S T . We also neglect the smoking quit population, since in (2) it is a standalone equation. Thus, now we have a simpler model as follows where the description of the parameters and their value are given in Table 1. The population of potential smokers grows with constant rate Λ. The potential smokers interact with beginner or smokers with effective interaction rates α 1 and β 1 , respectively. This interaction makes the potential smokers becoming smokers. The beginner smokers may control theirselves to quit smoking with rate σ 1 . The beginner smokers can continue their behavior on smoking if they interact with smokers with effective rate δ 1 . All populations can die naturally with the rate µ.

STABILITY ANALYSIS
The equilibrium of system (3) is obtained by taking dP dt = dB dt = dS dt = 0. We get three equilibriums, namely Smoking-free equilibrium: Beginners equilibrium: where P * is the root of P(Z), and The Jacobian matrix of system (3) evaluated at any point E = (P, B, S) is as follows The local stability of smoking-free and beginners equilibriums are given in Theorem 1 and Theorem 2, respectively. Before that, the basic reproduction number is calculated. We use the next generation matrix method [27,28,29]. In model (3) Evaluating the Jacobian matrix of G and M at the smoking-free equilibrium E 0 yields Then, we have The basic reproduction number R 0 is the spectral radius of matrix GM −1 which is .

SIMULATION OF THE SOLUTION
We simulate system (3) using parameters value in Table 1 and initial conditions [19,25]: The phase portraits of system (3) with various initial conditions is presented in Figure 2.
Starting from any initial point, the trajectory (P(t), B(t), S(t)) declines in P-axis but at the same time it grows until reaching the highest level of B, and then it declines in B − axis but it grows until reaching the highest level of S, and then converging to the equilibrium point E S .

SENSITIVITY ANALYSIS
From previous analysis, we know that the basic reproduction number R 0 acts as initially smoking behavior transmission. The next step is to determine the normalized sensitivity index to see the the relative change of parameter (appeared in R 0 ) on the value of R 0 . This can be used to measure which parameter having most impact on R 0 . The normalized sensitivity index is defined as follows [29], where q ∈ {Λ, α 1 , σ 1 , µ}.
By the definition (6), we have I R 0 Λ = 1, I R 0 α 1 = 1, I R 0 σ 1 = − σ 1 σ 1 +µ , and I R 0 µ = − σ 1 +2µ σ 1 +µ . Hence, −1 < I R 0 σ 1 < 0 and I R 0 µ < −1. This means that parameters σ 1 and µ have negative impact on R 0 , while parameters Λ and α 1 have positive impact on R 0 . But, the natural death parameter µ is the most sensitive parameter, and thus reducing it will have highest proportional impact on the smoking behavior transmission, followed by Λ and α 1 . The comparison between these parameters by substituting their value from Table 1 is presented in Figure 3. Another interesting examination is to study the impact of changes of all parameters on the dynamics of populations. By this purpose, we perform a time-dependent sensitivity analysis.
This sensitivity analysis have been used in many papers, for example [30,31,32,33]. Let X = (P, B, S) be vector of populations, Q = (Λ, α 1 , β 1 , σ 1 , δ 1 , µ) be vector of parameters, and F = dX dt be the vector equations of the righ-side of (3). To see the effect of changes of parameters on populations, let us define a sensitivity function V = ∂ X ∂ Q . Now, by seeing V as a function of time t, we can make total derivation of V as follows The term ∂ F ∂ X is 3 × 3 Jacobian matrix J(X) as appeared in (4), V is a 3 × 6 matrix, and ∂ F ∂ Q is a 3 × 6 matrix which given as follows We solve the system of 18 differential equations (7) numerically with initial conditions all zeros, and then plot the solution. We write the sensitivity function as v x q = ∂ x ∂ q , where x ∈ {P, B, S} and q ∈ {Λ, α 1 , β 1 , σ 1 , δ 1 , µ}. The plot of v x q for each parameter is presented in Figure 4. Parameter Λ has positive impact on the populations. Parameters  We can observe that some parameters give positive and negative impact on populations as time goes by. To see which one of parameters that produces highest impact on all populations, we plot the sensitivity index of the sensitivity function after arriving at the equilibrium. We plot their comparison in Figure 5. We can see that parameter µ is the most sensitive parameter on population S, and it si followed by parameter δ 1 . On the other hand, parameter δ 1 is the most sensitive parameter on population B, and it is followed by µ. In the case of population P, parameter β 1 is the most sensitive parameter.

CONCLUSIONS
The model studied in this paper considers a population related to smoking behaviour of a system that consists of three compartments, namely potential or non-smokers, beginner smokers, and smokers. The stability of the system depends on the basic reproduction number. If the basic reproduction number is less than one, then the system converges to smoking-free equilibrium, if it is bigger than one, the system converges to smokers equilibrium. The normalized sensitivity analysis of the basic reproduction number reveals that the natural death rate parameter gives highest impact on the smoking behaviour transmission. But, in the time-dependent sensitivity analysis, this parameter gives highest impact only on the smokers population. Meanwhile the parameter of effecive contact rate between beginners and smokers gives highest impact on the beginners population, and the parameter of effective interaction rate between non-smokers and smokers has highest impact on the non-smokers population. Thus, to reduce the smoking behaviour, non-smokers population should avoid contact with smokers, and beginners should also avoid contact with smokers.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.