Non-standard finite difference scheme and analysis of smoking model with reversion class.

Smokers are at more risk to COVID-19 as the entertainment of smoking because their fingers are in touch with lips regularly during smoking that increases the probability of transmission of virus from hand to mouth. On other hand the smokers may have lung disease (or reduced lung capacity) which would greatly increase risk of serious illness especially COVID-19. For this esteem, in this research work, we first formulate a mathematical model contains the reversion class. Then, using different techniques for finding the local and global stability of the presented model related to equilibrium points that are free smoking and positive smoking equilibrium points. As the model consisting on the nonlinear equations, so we use the non-standard finite difference (NSFD) scheme, ODE45 and RK4 methods to find the numerical results. Finally, we show the graphs numerically through MATLAB.

Introduction

Smoking is that process, which not remains bounds to only the smokers but it affects the other society and economy of the country. Smoking sources such as water pipes often involve the sharing of mouth pieces and hoses, which could facilitate the transmission of COVID-19 in communal and social settings. Similarly, smokers are at more risk to COVID-19 as the entertainment of smoking because their fingers are in touch with lips that increases the probability of transmission of virus from hand to mouth. On other hand the smokers may have lung disease (or reduced lung capacity) which would greatly increase risk of serious illness specially COVID-19. It seems that the most of young smokers became regular smokers and then due to nicotine presence it is difficult to remove from this habit. Social habit of smoking was first discover by Christopher Columbus [1]. Jean Nicot was the first man who introduced the tobacco in 1560 and from his name nicotine is deduced. In 1920s, it is discovered that the smoking having a strong relation with cancers and from that time the first campaign against smoking take start. In start, people used the simple form of cigarettes as in the form of pipes while the modern cigarettes take start between 1960 and 1970 from Indonesia. Procedure of tobacco enlarged during the world battles and the cause was, delivering free cigarette to allied groups for their moral boosting exercise, however in 20th century as a result of knowledge about health smoking became less popular. In same sense, due to COVID-19 the peoples became jobless and fall in great tension due to which the smoking became enlarge. Mathematicians are working on infectious diseases to describe the dynamics mathematically for the world. Many mathematical models were presented by many authors for different types of diseases take start from the first mathematical model by Kermack and McKendrick [2], see therein [2], [3], [4], [5], [8], [7]. In recent days, smoking become a social habit, which is more dangerous for health because a cigarette contains 40 hundreds of chemicals consisting Benzene, Tar, Ammonia, Acetone etc., causes many cancers. Millions of smokers lost their lives due to the smoking. For these reasons, the mathematicians played an important role to present the smoking models for the reason that the people will aware about the dynamics of smoking. For the dynamics of smoking, the first model was presented by Castillo Garsow et al. [8] in 1997, in which they divided the whole population in three categories first one is potential smokers P, second one is smokers S and third one is permanently quit smokers Q. Then Sharomi and Gumel [9] extended the model by announcing the new class of temporarily quit smokers . Ham [10], described the different phases and processes of smoking. Zaman [11] made his contribution in the form of introducing the class of occasional smoker and presented dynamical interaction in an integer order. Then, square root dynamics of smoking model were presented by Zeb et al. [12] for the purpose that the model go to extinction in finite time. Many researchers tried on different aspects to developed smoking models and control on the epidemic models [8], [7], [10], [11], [12], [13], [14], [15], [16], [17]. In this work, first we frame a mathematical model contains the reversion class then study the equilibrium points that are free smoking and positive smoking equilibrium points. Then using different techniques for finding the local as well as the global stability of the presented model. As the model consisting on the nonlinear equations, so we use the non-standard finite difference scheme, ODE45 and RK4 methods to find the numerical results. Finally show the graphs sketch through MATLAB.

This paper is organized as follow: In Section “Model formulation”, the formulation of model is presented in which the special case occurs. The stability of the proposed model is presented in Section “Stability and equilibria of the proposed model” and numerical solution is presented in Section “Numerical method and results”. Finally, we give conclusion in Section “Conclusion”.

Model formulation

For the model formulation different classes are considered, which detail as follow:

The compartment represents the potential smokers (non-smokers) people who have not smoked yet but may be started in future. This compartment is increased at rate , which is the recruitment rate. The compartment stands for chain smokers, which is increased when potential smokers start to smoke with an incidence rate or contact rate between potential smokers and smokers and the rate of the relapse smokers who revert back to smoking. Some other people will leave or vacate the compartment with rates and . The compartment shows reversion (relapse) class are increased with rate . This compartment is decreased at rate and , where is relapse rate. On the similar way stands for permanent quitters.

The following five assumptions are assured for model formulation:

: The total population is divided into two habitual (smokers) and two non–habitual (non smokers) compartments.

: The natural death rate is approximately equal to recruitment rate .

: All the recently new born are assumed to join only potential smokers compartment.

:In all compartment’s the natural death rate is same .

: No person, is more popular than another, so that interaction between any two compartments are equally likely.

To describe an equation of difference for rate at which population of each compartment changes over discrete time, the entry of people to a compartment is represented by addition and taking off people from a compartment is shown by subtraction. Using assumptions , the Ice smoking mathematical model is described by the following differential equations: where the parameters used for this model is described in the following table. Now (see Table 1 )

Table 1

Parameters and variables of smoking model.

Parameter	Definition
λ	Birth or migration rate to the host population
β	The incidence rate for susceptible population to regular smokers class
ρ_1	Rate by which the regular smokers go to quit smokers
ρ_2	Represents the rate at which the reversion individual joins quit individual
μ	Natural death rate for all classes
γ	Rate at which regular smokers go to reversion class
α	Rate at which the relapse class people go to regular class

Suppose the total population is ; then Eq. (2) leads the following solution

Thus, population of the feasible solution for model (1) is restricted to the region;

The region is positively invariant because all the solutions of system (1) are bounded and enter the . That shows, every solution of model (1), with initial conditions in , remains there for all .

Stability and equilibria of the proposed model

Without loss of generality first three equations of model (1) are independent of the variable , so we omit the forth equation for simplicity and discuss the dynamics of system consists on first three equations. So the free smoking equilibrium point of model (1) is a steady state solution, for the entire population is non-smokers. Which is

Use the next generation matrix method [18] for smoking generation number as follow. Let ; then the reduced form of model (1) can be written assuch that

Now the Jacobian of Eq. (4) isat ,simplywhere

Similarly, the jacobian of Eq. (5) isat ,this can be written in form ofwhere and

So the required reproductive number is

System (1) has free smoking equilibrium point and for a positive smoking equilibrium point , one has three cases:

If and , one has no positive equilibrium point.

If , one has one positive equilibrium point .

If , one has no positive equilibrium point.

To find the positive smoking equilibrium point for system (1) by setting , as

So Eqs. (10) and (12) implies that and from Eq. (11), we have

The Eq. (15) has solutions of the formwhere and are

From here, we conclude that and for . Hence, we have only one positive solution:

While and there is no-positive solutions for . Further, if and then implies that there are no-positive solutions. □

If , then point of model (1) is locally asymptotically stable, and if , then is unstable.

Evaluating the Jacobian matrix of model (1) at by linearizing the model providesso

For finding the eigen values solve the equation which follows thatand

Hence, the eigenvalues of are and . If , then . So, for all eigenvalues are negative and hence is locally asymptotically stable and for , which follows that there exists a positive eigenvalues, so is unstable. □

Further, we explore the local stability of smoking positive equilibrium by using the following lemma.

(see [21]). Let M be a real matrix. If and are all negative, then all of the eigenvalues of M have negative real part.

in the previous lemma is well-known as the second additive compound matrix with the following definition.

(see [21](second additive compound matrix)) Suppose be an real matrix. The second additive compound of A is the matrix defined as follows;

The positive smoking equilibrium of model (1) is locally asymptotically stable if .

Linearizing model (1) at the equilibrium provideshere the termnow from Eq. (12), it follows that

So becomes

The second additive compound matrix of isand is

Now the is and in similar way the , if . Therefore, through Lemma 3, is locally asymptotically stable if .  □

If , then system (1) is globally stable.

For proof, first we construct the Lyapunov function Las;after differentiating and algebraic simplification along , we have

Therefore, it is clear that if , then the system (1) is globally stable at , if then the system (1) at is unstable. □

For global stability at consider the following theorem.

([21]) Let be a vector field which is piecewise smooth on and which satisfies the conditions in the interior of , where is the normal vector to and is the Lipschitz continuous field in the interior of and

Then the differential equations and has no periodic solutions, homoclinic loops, and oriented phase polygons inside .

Let . is positively invariant, and . Thus, it leads the following theorem.

Model (1) has no periodic solutions, homoclinic loops, and oriented phase polygons inside the invariant region .

Let and denote the right hand side of equations in model (1) respectively. Using the equation to rewrite and in the equivalent forms, as

Suppose be a vector field, where:

Then

Clearly on . As

Using the normal vector to , we have

Hence, model (1) has no periodic solutions, homoclinic loops, and oriented phase polygons inside the invariant region . This complete the proof. □

Consequently, we have the following result.

If , then the positive smoking equilibrium point of system (1) is globally asymptotically stable.

We know that, if in , then is unstable. Also is a positively invariant subset of and the limit set of each solution of system (1) is a single point in since there is no periodic solutions, homoclinic loops, and oriented phase polygon inside . Therefore model (1) at is globally asymptotically stable. □

Numerical method and results

NSFD method is used for numerical solution of proposed model (1). Basically NSFD scheme [19], [20], [21], [22] is an iterative method in which we get near to solution through iteration. Let non-standard ODEs is given belowwhere , then by NFSD method

Here, using NSFD method for numerical solution of system (1) will leadssimilarly

In Fig. 1, Fig. 2, Fig. 3, Fig. 4 , the approximate solution of smoking model of potential smokers, regular (chain) smokers, relapse smokers and quit smokers (quitters) are represented respectively. From figures, it is clear that results of NSFD and RK4 are matching very well for different values of parameters. The parameters values used for the numerical simulations are , with initial values .

Fig. 1

The population of potential smokers.

Fig. 2

The regular smokers.

Fig. 3

Relapse class.

Fig. 4

The quitters.

Conclusion

In this paper, a smoking mathematical model is presented in the light of some assumptions. First, we formulated the smoking model then discussed the dynamics of proposed model. It is observed that the population of the feasible solution for model (1) is restricted to the region;

The following results are obtained.

If , the free smoking equilibrium point of model (1) is locally asymptotically stable, and if , then is unstable.

The positive smoking equilibrium of model (1) is locally asymptotically stable if .

If , then the system (1) is globally stable.

Finally, a discrete-time, finite difference scheme was constructed using the nonstandard finite difference (NSFD) method. Numerical results of NSFD were compared with RK4 and ODE45, which showed a good agrement.

Availability of data and materials

The authors confirm that the data supporting the findings of this study are available within the article cited there in.

Authors Contribution

Authors are equally contributed in preparing this manuscript.

Funding

This article is supported by the Deanship of Scientific Research (DSR), , Jeddah, under Grant No. D 1439-158-130.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.