Smoking as Epidemic : Modeling and Simulation Study

In this paper we propose smoking epidemic model which analyzes the spread of smoking in a population. The model consists of five compartments corresponding to five population classes, namely, potential-moderate-heavy-temporarily recoveredpermanently recovered class. The basic reproduction number R0 has been derived, and then the dynamical behaviors of both smoking free equilibrium and smoking persistent equilibrium are analyzed by the theory of differential equation, and Numerical simulation has been carried out and the results have confirmed the verification of analytical results. Sensitivity analysis of R0 identifies β1, the transmission coefficient from potential smokers to moderate smokers and β2, the transmission coefficient from potential smokers to heavy smokers, as the most useful parameters to target for the reduction of R0.


Introduction
Smoking is a practice in which a substance is burned and the resulting smoke breathed in to be tasted and absorbed into the bloodstream.Most commonly the substance is the dried leaves of the tobacco plant which have been rolled into a small square of rice paper to create a small, round cylinder called a "cigarette" [1].Epidemiology is concerned with the distribution of diseases in populations and the factors that influence the transmission of diseases [2].One of an interesting area in epidemiology to study is smoking subject.There are a lot of studies that has been done on the epidemics of smoking [3][4][5][6][7][8][9][10][11][12].In order to understand the dynamics of this disease, like many infectious diseases, mathematical models can be used.Most of these models are compartmental models, with the population being divided into compartments with the assumptions about the nature and time rate of transfer from one compartment to another.The basic concepts in these models are that all people in a community start as healthy.Healthy people may become infected by diseases.But infected people may become healthy again in community.
Castillo-Garsow et al. [3] presented a mathematical model for giving up smoking.In their model the total population is classified into three compartments namely; potential smokers, smokers and quitters (former smokers).Later on, Sharomi and Gumel [4] extended the basic model to account for variability in smoking frequency, by introducing two classes of smokers who temporarily and permanently quit smoking.Zaman [5] derived and analyzed a smoking model taking into account the occasional smokers compartment in the basic model.In [6], Z. Alkhudhari, S. Al-Sheikh and S. Al-Tuwairqi adopted the model developed and studied in [6] and considered the effect of peer pressure on temporary quitters.In 2014 [7], Z. Alkhudhari, S. Al-Sheikh and S. Al-Tuwairqi introduced a new model by dividing the smokers into two subclasses: occasional smokers and heavy smokers, and the impact of these two subclasses on the existence and stability of equilibrium points.They also studied the effect of occasional smokers on potential smokers, and later in 2015 [8] they studied the effect of heavy smokers on potential smokers.
In this paper, we develop a new model based on the idea given in [6, 7 and 8] by considering, basically, the effect of both moderate and heavy smokers on the potential smokers, and the effect of heavy smokers on temporarily quit smokers.Furthermore, it assesses the impact of this peer pressure on the existence and stability of steady state solutions, and the result is supported with numerical simulation.

Flow of People Among the Compartments
People will join the potential smoker's compartment ( ) at a constant recruitment rate of .This recruitment is due to the total natural birth rate .Some of these people will vacate this compartment at a constant death rate of , due to the total natural death rate .It is considered that the potential smokers start smoking due to peer influence of moderate and heavy smokers, will vacate the potential compartment with the rates and .Here, is the contact rate between potential and moderate smokers and is the contact rate between potential and heavy smokers.
The moderate smokers, ( ) are increased when potential smokers start to smoke, with rates and .Some other people will leave the compartment with the rates , and , in which, is the rate at which moderate smokers becomes heavy smokers without peer influence and is the contact rate between moderate and heavy smokers.People will join into heavy smoker's class ( ) with the rates , and .Some others will leave with the rates and .Here, and are the contact rate between heavy smokes and those who temporarily quit smoking and the rate of quitting smoking, respectively.
People will enter into temporary quitter's compartment with a rate (1 − ) and leave with the rates and , where, (1 − ) is the fraction of heavy smokers who temporarily quit smoking (at a rate ).
The permanent quitters, are increased when heavy smokers begin to smoke, with the rate and some people of this compartment will die with the rate , where, (0 < < 1) is the remaining fraction of heavy smokers who permanently quit smoking (at a rate ).

Model Assumptions
Here, for the above flow diagram of smoking endemic model, the following assumptions are made.
The total population is divided into two diseases (smoker) and three non diseases (non smoker) compartments.
The recruitment rate ( ) is different from the natural death rate ( ) so that the total population is not constant.
The potential smokers start to smoke under the peer influence of both moderate and heavy smokers.All the newly born people are assumed to join only potential smoker's compartment.
The people of all the compartments will die naturally with the same rate of .
No other influence to smoke other than smokers.
Temporarily quitted smokers could only be allowed to relapse into heavy smokers' class.
Legal age could be considered to start smoking, in the model.
No person is more popular than another, so that interactions between any two compartments are equally likely.

Model Equations
By adding the rates at which people enter the compartment and subtracting the rates at which people leave the compartment, we get an equation for the rate at which the population of each compartment changes over time.Thus, by considering the above assumptions, smoking epidemic model presented in figure 1 may be represented by the following system of first order nonlinear ordinary differential equations: With the initial densities By introducing = *, = +, = , = , and = , , equation (1) can be modified (scaled) as, $.
Provided * + + + + , + , = 1.The initial densities in equation (2) may also be modified as follows: Since the variable , of equation (3) does not appear in the first four equations, we shall consider the equations (3.4) − (3.#) and we shall find the feasible (solution) region of equations (3.4) − (3.#) so that the model makes biological sense, as follows: Therefore, as it has been shown in [14], the solution of equation We denote the boundary and the interior of H by IH and H * , respectively.

Analysis of the Model
In this section we are going to analyze the smoking endemic model in equations (3.4) − (3.K).

The Basic Reproduction Number (P M )
Like as it has been described in [9], analyzing the differential equations given in equation (3) can also give insight into how smoking spreads and how the spread can be limited.An important tool for analyzing a model of this type is its reproductive number.
The basic reproduction number, denoted E ' , is defined as the number of new infections produced by a typical infective individual in a susceptible population at a disease free equilibrium [10].In model (3) , taking the infected compartments to be moderate and heavy smokers, the basic reproduction number is then the sum of the transmission rate from potential smokers to moderate smokers divided by the sum of the natural death rate of the moderate smokers population and the proportion of individuals who enter into heavy smoker's compartment, i.e., However, the moderate smokers (+) in equation ( 6) can be computed from equation (3) as follows: By plugging equation (7) into equation (6) , the basic reproduction number at the disease free equilibrium E ' is given as, As it has been briefly discussed in [10], the dimensionless constant R ' tells us that if R ' < 1, then the DFE in ∂Ω, is locally asymptotically stable, and the disease cannot invade the population, but if R ' > 1, then the endemic equilibrium exists in Ω * , and is the DFE is unstable and invasion is always possible.

Sensitivity Analysis of P M
It is critical to take various actions to control the system parameters so that R ' is remarkably below one.This section is intended to propose some effective measures for achieving this goal.To examine the sensitivity of R ' to each of its parameters, following Arriola and Hyman [13], the normalized forward sensitivity index with respect to each of the parameters is calculated as follows: It can be seen that, among these six parameters, E ' is the most sensitive to the change in β , β and σ.An increase or decrease in σ will cause an increase or decrease in E ' with the same proportion.Physically, however, decreasing the natural birth rate σ is neither practical nor ethical.An increase (or decrease) in the value of β or β leads to a corresponding increase (or decrease) in E ' .As opposed to this, the other two parameters and have an inversely proportional relationship with E ' , an increase in and will bring about a decrease in E ' , with a proportionally smaller size of decrease.Recall that is the natural death rate of the population.It is clear that increase in is neither ethical nor practical.Additionally, the parameter ϵ may have directly or inversely proportional relationship with the reproduction number.Thus we choose to focus on one of three parameters: either , the transmission rate from potential smokers to moderate smokers or , the transmission rate from potential smokers to heavy smokers or , the rate of quitting smoking.Given E ' 's sensitivity to , and , it seems sensible to focus efforts on the reduction of and .In other words, this sensitivity analysis tells us that prevention is better than quitting smoking.

Local Stability of Smoking Free Equilibrium
The local stability of N ' = ( / , 0,0,0) is then determined from the signs of the eigenvalues of the Jacobian matrix.At the smoking free equilibrium N ' , the Jacobian matrix is given by: The characteristic equation of this matrix is given by det (st − l(N ' )) = 0 , where t is a square identity matrix of order 4. The equation thus becomes Where, The roots of the characteristic equation are the eigenvalues of the Jacobian matrix.It is clear that the one eigenvalue is determined from (s + ) = 0, and is s = − .All the other remaining eigenvalues are determined from the quadratic equation Now by using Routh-Hurwitz criterion [15], it can be stated that the smoking free equilibrium N ' is locally asymptotically stable if and only if the following inequalities satisfied: v > 0 and v > 0.
Theorem 1: If E ' < 1, then v and v are greater than zero.Proof: Since each parameter's value is greater than zero, it Hence, all roots of equation ( 11) have negative real parts if E ' < 1.So all eigenvalues of the Jacobian matrix are negative if E ' < 1, and hence, N ' is locally asymptotically stable.If E ' = 1 , then equation (10) may be modified as (s + ) (s + v s) = 0 which has the roots s = − , s = 0 and s = Q R A B − ( + + 2 ) < 0, and hence, N ' is locally stable.If E ' > 1 , then v < 0 and it is clear that the quadratic equation has some positive roots.This leads to conclude that the smoking free equilibrium becomes unstable.
For the smoking present equilibrium N * = (* * , + * , * , , * ), the Jacobian matrix is given as, Where, The characteristic polynomial for l(N * ) is given by det (st − l(N * )) = 0 , where t is a square identity matrix of order 4. The polynomial becomes Where, Algebraically, the epidemic equilibrium value is asymptotically stable by Routh-Hurwitz criteria [15], if v > 0, # > 0, K > 0 and v•# > # + v K.These conditions ensure that all of the four eigenvalues found from (13) have negative real parts.However, the stability of the endemic equilibrium N * will be illustrated using numerical simulations, by imposing the value of parameters and initial population of compartments.
Figure 2 show that every compartment approaches its smoking free equilibrium value N ' as time increases.The results show that the smoking free equilibrium value of the system is asymptotically stable, which is as expected from our theoretical analysis as E ' = A(Q R (B5W)5TQ S ) B(T5B)(B5W) = 0.5345 < 1 .Additionally, the stability of smoking free equilibrium value is not affected by the initial values of each compartment as it is presented in figure 2. On the other hand, we have considered another case in which E ' = 3.925 > 1 as it has been shown in figure 3. The existence and stability of smoking present equilibrium value(s) has been simulated in figure 3 even though it is messy to obtain analytically.Here, the stability of smoking present equilibrium value(s) is/are independent of the initial values.Furthermore, the parameters and have no significant on the stability of both smoking free and smoking present equilibrium values as long as their stability depend on the reproduction number E ' .

Conclusions
In this paper, we have proposed a mathematical model describing the dynamics of smoking.This model consists of five differential equations describing the rate of change of the subpopulation , , , and .The model has smoking free equilibrium and smoking present equilibrium.The threshold E ' is derived and it is shown that if E ' < 1, then the smoking will disappear, while if E ' > 1 , then the smoking persists with the population which means that after some period of time the smoking will become hazardous.Numerical simulation has been carried out and the results have confirmed the verification of analytical results.Sensitivity analysis of E ' identifies , the transmission coefficient from potential smokers to moderate smokers and , the transmission coefficient from potential smokers to heavy smokers, as the most useful parameters to target for the reduction of E ' .In future work, the possible extensions are to consider that Smoking related death will be introduced in the model and the temporary quitters decide to quit smoking permanently.
Here, we have given the flow diagram of the model.The compartments of the model are represented by rectangular colored boxes.The flow directions of the people among the compartments are represented by directed arrows.

Figure 1 .
Figure 1.Flow diagram of Smoking Epidemic Model.

Figure 2 .
Figure 2. Time series plots of Potential smokers, Moderate smokers, Heavy smokers, Temporary quitter smokers and Permanent quitter smokers with different initial densities for E ' = 0.5345 < 1.

Figure 3 .
Figure 3.Time series plots of Potential, Moderate, Heavy, Temporary and Permanent quitter smokers with different initial densities for E ' > 1.