A passive verses active exposure of mathematical smoking model: A role for optimal and dynamical control

Takasar Hussain; Aziz Ullah Awan; Kashif Ali Abro; Muhammad Ozair; Mehwish Manzoor; José Francisco Gómez-Aguilar; Ahmed M. Galal

doi:10.1515/nleng-2022-0214

Article Open Access

A passive verses active exposure of mathematical smoking model: A role for optimal and dynamical control

Takasar Hussain , Aziz Ullah Awan , Kashif Ali Abro , Muhammad Ozair , Mehwish Manzoor , José Francisco Gómez-Aguilar and Ahmed M. Galal

Published/Copyright: September 30, 2022

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From the journal Nonlinear Engineering Volume 11 Issue 1

Abstract

Smoking has become one of the major causes of health problems around the globe. It harms almost every organ of the body. It causes lung cancer and damage of different muscles. It also produces vascular deterioration, pulmonary disease, and ulcer. There is no advantage to smoking except the monetary one to the tobacco producers, manufacturers, and advertisers. Due to these facts, a passive verse active exposure of mathematical smoking model has been analyzed subject to the dynamical aspects for giving up smoking. In this context, mathematical modelling and qualitative analysis have been traced out for smoking model having five classes. Mathematical forms of smoke absent and smoke present points of equilibrium have been calculated for knowing optimal and dynamical control. By making use of the Lyapunov function theory, we have shown the global asymptotic behavior of smoke-free equilibrium for threshold parameter R 0 < 1 . The ability to observe theoretically and through graphs is invoked to study the general behavior of single smoke present point. To make effective, vigorous, authentic, and stable strategies to control the disease, we have performed the sensitivity examination of threshold parameter and disease, present apartments.

Keywords: smoking model; stability analysis; reproduction number; equilibria; optimal control

1 Introduction

Smoking is an exercise in which a material scorching and producing smoke is breathed for the taste. At the time of adolescence, most of our attitudes change in which one of them may be the wish of smoking. Smoking effects create huge problems in personal as well as occasionally in public matters. According to a strong medical documentation, there are many killing diseases with concealed cause of smoking [1]. Smokers experience the ill effects of lung cancer growth multiple times more than non-smokers, and smoking-related deaths also occurs [2]. Heart disease, emphysema, and chronic bronchitis have been diagnosed among 80% of smokers, and lung cancer occurs among 29% of smokers [3].

Cigarette smoking becomes a vogue among teenagers. World Health Organization (WHO) published an article on the worldwide tobacco epidemic, which identifies that significant number of people die or disabled in their most productive years as a result of smoking [4]. There are approximately 440,000 deaths in the United States and 105,000 deaths in the United Kingdom every year due to the smoking-related diseases. Despite of the fact that across the globe almost 4 millions casualties are occurring due to smoking-effected diseases, the quantity of smokers is increasing continuously [5]. If the current situation continues, then there is an apprehension that tobacco can kill or disable more than eight millions people every year by 2030 [6].

Mathematical modeling is a very effective way to represent any physical phenomenon. In this approach, we use the techniques of mathematical modeling to represent the problem in the mathematical form and correlate the solutions with the physical aspect of the problem. At that point how the infection spreads and the pattern of the model were concentrated by investigating the soundness of solutions. The investigation of smoking is one of the many intriguing regions with regards to the study of disease transmission. A great deal of work has been done on smoking scourges. To show signs of improvement understanding in the elements of this illness, individuals utilize scientific methods using mathematical techniques.

By considering the enlargement and influence of the ailments, caused by smoking, on the health of common people, Sharomi and Gumel [7] have designed a mathematical model consisting of nonlinear system of ODEs. Zaman qualitatively analyzed a mathematical model with a class of incidental smokers [8]. Alkhudhari et al. [9] investigated the impact of smoker on temporary quitters. Ullah et al. [10] theoretically showed that a mathematical model having absolute population size N ( t ) contains total five number os classes P ( t ) (people who are non-smokers yet they can smoke), occasional smokers ( O ( t ) ), smokers ( S ( t ) ), temporarily quitters ( Q t ( t ) ), and permanently quitters ( Q p ( t ) ) . This is the reality that a smoker who quits smoking temporarily must have desire to smoke again. This point had been discussed by Awan et al. [12] through a mathematical model. Mojeeb and Adu [13] thought that smokers can be treated through counseling, so they have discussed the epidemics of smoking by using this idea. Shah et al. [14] studied the tuberculosis caused by smoking. There exist many mathematical models, which explain the threats to life due to smoking. The one who has interest can read the points given in refs [15,16, 17,18,19, 20,21,22, 23,24,25, 26,27,28, 29,30]. Additionally, few dynamical models have been included here [31,32,33, 34,35,36, 37,38,39, 40,41,42, 43,44,45, 46,47].

Every year more than eight million people die due to smoking [48]. So it is reasonable to include the parameter in a smoking model, which represents the additional death rate of smokers due to smoking together with the natural mortality rate. In this article, we revisited the Ullah et al. model [10] by involving the demise rate because of smoking. The analysis of the present work include the following main characteristics, which were not present, to the best of our knowledge, in the models discussed previously:

Recognizable proof of delicate boundaries for reproduction number and endemic degrees of smoking classes.
Design the ideal control system based on affectability investigation.

The remaining part of this work is organized as follows:

Section 2 is devoted to the presentation of model framework, mathematical formulation, and region for the existence of solutions. Constant solutions and contact number are discussed in Section 3. It is proved in Section 4 that all non-constant solutions approach the equilibria on the basis of the contact number irrespective of initial conditions. The most important parameters that contribute significantly in the spread of smoking menace is identified in Section 5. The design of the optimal control problem is presented in Section 6, and it is analyzed mathematically and verified graphically. Concluding remarks are presented in the last section.

2 Model framework with flow diagram

Let us take that the whole population, denoted by N , is separated into five subclasses, which are P (potential smokers), O (occasional smokers), S smokers, Q t (smokers who quit smoking temporarily), and Q p (smokers quit smoking permanently), and then N = P + O + S + Q t + Q p . A schematic flow diagram for the aforementioned model is shown in Figure 1. We have the following governing equations, which represent our mathematical model of smoking:

(1) d P d t = Λ − P β S − P μ , d O d t = P β S − α 1 O − μ O , d S d t = α 1 O + α 2 S Q t − ( μ + γ + η ) S , d Q t d t = − α 2 S Q t + S ( 1 − δ ) γ − Q t μ , d Q p d t = δ γ S − Q p μ .

Table 1 presents complete description of all the parameters, with values and also sources from where they have taken.

Figure 1

Flow diagram of the model representing the transition of population among different compartments with various rates.

Table 1

Representation of values of parameters with sources

Variable	Definition	Value	Sources
Λ	Rate of recruitment of smokers given in class P	1	[10]
β	Effective contact rate between S and P	0.14	[10]
μ	Natural mortality rate	0.001	[10]
α 1	The rate at which smokers from class O move to class S	0.002	[10]
α 2	It represents the rate at which people move from class Q t to class S	0.0025	[10]
γ	The rate at which the smokers quit smoking	0.8	[10]
1 − δ	Part of smokers who briefly stop smoking	0.52	Assumed
δ	The proportion of smokers who leave the smoking permanently	0.48	Assumed
η	Disease-related death rate	0.00003	Assumed

Since each and every state variable is representing the human compartments, they are non-negative having non-negative initial values. For this, we prove the following theorem:

Theorem 2.1

All the solutions of the system (1) are non-negative with non-negative initial conditions.

Proof

Let t 1 = sup { t > 0 : P > 0 , O > 0 , S > 0 , Q t > 0 , Q p > 0 ∈ [ 0 , t ] } . Therefore t 1 > 0 . The first equation of the system (1) yields

d P d t + ( β S + μ ) P = Λ d d t P e μ t + ∫ 0 t β S ( u ) d u = Λ e ∫ 0 t β S ( u ) d u + μ t P ( t 1 ) e μ t 1 + ∫ 0 t 1 β S ( u ) d u − P ( 0 ) = ∫ 0 t 1 Λ e ∫ 0 y { β S ( u ) d u + μ y } d y

P ( t 1 ) = P ( 0 ) e − μ t 1 − ∫ 0 t 1 β S ( u ) d u + e − μ t 1 − ∫ 0 t 1 β S ( u ) d u ∫ 0 t 1 Λ e ∫ 0 y { β S ( u ) d u + μ y } d y > 0 .

Similarly, it can be shown that O > 0 , S > 0 , Q t > 0 , Q p > 0 , for all positive time.

We claim the subsequent result:

Lemma 2.2

The set ξ = { ( P , O , S , Q t , Q p ) ∈ R + 5 ∣ 0 ≤ P + O + S + Q t + Q p < Λ μ } is positively invariant.

Proof

The sum of total population results in the following equation:

(2) d N d t = Λ − N μ − η S .

Since d N d t ≤ Λ − N μ , we can see that d N d t ≤ 0 if N ≥ Λ μ . By the comparison theorem given in ref. [11], we can see that N ≤ N ( 0 ) e − μ t + Λ μ ( 1 − e − m u t ) . In particular, N ( t ) ≤ Λ μ if N ( 0 ) ≤ Λ μ . Thus, ξ is positively invariant.

3 Existence of constant solutions and analysis of contact number

By direct calculations, we may have the smoke absent or smoke-free equilibrium (SFE) point of system (1) as follows:

E 0 = Λ μ , 0 , 0 , 0 , 0 .

The premise reproduction number is utilized for the examination of illness elements. The premise multiplication number is characterized as the quantity of optional cases occurs through a contaminated individual of the powerless populace during the irresistible period. It will help us in obtaining the clear idea about the existence or removal of this irresistible disease in populace. By utilizing the method of next-generation strategy [49], the fundamental reproduction number is computed as follows:

R 0 = Λ μ β α 1 ( μ + γ + η ) ( α 1 + μ ) .

The expression of reproduction number, R 0 , is telling us that how many number of new infected people can be produced by one infected person in his/her whole infectious time. If the value of R 0 is less than the threshold value 1, it means that the disease is no more spreading, but if it is greater than one, then disease exists in the population and will go on increasing. When R 0 exceeds unity, the system ( 1 ) has a unique smoke present equilibrium (SPE) E ∗ = ( P ∗ , O ∗ , S ∗ , Q t ∗ , Q p ∗ ) , where

P ∗ = Λ ( β S ∗ + μ ) , O ∗ = β Λ S ∗ ( α 1 + μ ) , Q t ∗ = γ ( 1 − δ ) S ∗ ( α 2 S ∗ + μ ) , Q p ∗ = γ δ S ∗ μ ,

and S ∗ is uniquely obtained from

(3) a S ∗ 2 + b S ∗ + c = 0 ,

where

a = α 2 γ δ β ( α 1 + μ ) + μ ( α 1 + μ ) β α 2 + η ( α 1 + μ ) β α 2 > 0 , b = α 2 μ γ ( σ − 1 ) ( α 1 + μ ) − α 1 β Λ α 2 + ( μ + η + γ ) ( α 1 + μ ) ( β μ + μ α 2 ) , c = ( μ + γ + η ) ( α 1 + μ ) μ 2 − μ α 1 β Λ .

It is obvious from Eq. (3) that a remains positive, but c changes its sign when R 0 goes above or below unity. That is c , will be negative whenever R 0 > 1 and vice versa. So, it will be concluded as follows:

Whenever R 0 > 1 , the model (1) has one and only one equilibrium point, which is E ∗ = ( P ∗ , O ∗ , S ∗ , Q t ∗ , Q p ∗ ) .

4 Global stability of both smoke absent and smoke existing equilibrium points

4.1 Global stability of smoke-free equilibrium point

To analyze global behavior of SFE for the system (1), we construct the Lyapunov function with the help of the method given in ref. [50]:

Theorem 4.1

The smoke-free point of equilibrium of the model (1) is globally asymptotically stable inside the ξ when R 1 ≤ 1 , where R 1 = β Λ μ ( μ + η ) .

Proof

Assuming the following positive definite function,

L = O + Q p + Q t + S , d L d t = d O d t + d Q p d t + d Q t d t + d S d t = β P S − α 1 O − μ O + α 1 O + α 2 S Q t − ( μ + γ + η ) S + − α 2 S Q t − μ Q t + γ ( 1 − δ ) S − μ Q p + γ δ S = β P S − μ O − ( μ + η ) S − μ Q t − μ Q p ≤ β Λ μ S − ( μ + η ) S = β Λ μ ( μ + η ) − 1 ( μ + η ) S .

The aforementioned inequality is telling us that L ′ ≤ 0 if whenever R 1 ≤ 1 . Moreover, L ′ = 0 for O = S = Q t = Q p = 0 . Thus, we can explore that the largest compact unchangeable set for the system (1) is the only singleton set E ∘ . Lasalle’s principle of invariance [51] says that E ∘ is globally asymptotically stable in ξ .

Remark

From the aforementioned result, one can easily say that if we can make R 0 and R 1 less than unity, then the ailment will be disappeared from the community. It is easy to see that R 0 < R 1 , which means that R 1 < 1 gives a guarantee for the complete removal of smoking.

4.2 Global behavior of smoke present point of equilibrium

By making use of the graph theoretic approach [52,53], the global behavior of SPE will be investigated. To obtain complete insight of this approach, one can go through the following [50].

Theorem 4.2

The one and only one SPE admits GAS inside of ξ .

Proof

Suppose that

(4) D 1 = P − P ∗ − P ∗ ln P P ∗ , D 2 = O − O ∗ − O ∗ ln O O ∗ , D 3 = S − S ∗ − S ∗ ln S S ∗ , D 4 = Q t − Q t ∗ − Q t ∗ ln Q t Q t ∗ , D 5 = Q p − Q p ∗ − Q p ∗ ln Q p Q p ∗ .

Taking derivative and utilizing 1 − y + ln y ≤ 0 for y > 0 , we obtain

D 1 ′ = − P ′ P ∗ P − 1 D 1 ′ = − P ∗ P − 1 ( Λ − β P S − μ P ) D 1 ′ = − P ∗ P − 1 ( β P ∗ S ∗ + μ P ∗ − β P S − μ P ) D 1 ′ = − β P ∗ S ∗ P ∗ P − 1 1 − S P S ∗ P ∗ − μ P ∗ P P ∗ − 1 1 − P ∗ P D 1 ′ = − β P ∗ S ∗ P ∗ P + P S P ∗ S ∗ − S S ∗ k − 1 − μ ( P − P ∗ ) 2 P D 1 ′ ≤ β P ∗ S ∗ S S ∗ − S P S ∗ P ∗ + ln p P ∗ S S ∗ − ln S S ∗ ≔ a 13 G 13 ,

and similarly,

D 2 ′ ≤ β P ∗ S ∗ ln O ∗ O S P S ∗ P ∗ − ln S P S ∗ P ∗ − O ∗ S P O P ∗ S ∗ + S P S ∗ P ∗ ≔ a 21 G 21 , D 3 ′ ≤ α 1 O ∗ ln S ∗ O S O ∗ + O O ∗ − S ∗ O S O ∗ − l n O O ∗ + α 2 S Q t S ∗ S Q t Q t ∗ − ln S ∗ S Q t Q t ∗ − Q t Q t ∗ + ln Q t Q t ∗ ≔ a 34 G 34 + a 32 G 32 , D 4 ′ ≤ μ Q t ∗ S S ∗ + l n Q t ∗ S Q t S ∗ − Q t ∗ S Q t S ∗ − l n S S ∗ ≔ a 45 G 45 , D 5 ′ ≤ δ γ S ∗ S S ∗ + l n Q p ∗ S Q p S ∗ − Q p ∗ S Q p S ∗ − l n S S ∗ ≔ a 51 G 51 .

We develop a weighted digraph having five vertices and six arcs as shown in Figure 2.

It can be seen that there are two cycles and along each cycle (Eqs. (5), (6)).

(5) G 51 + G 45 + G 34 + G 13 = 0 ,

(6) G 21 + G 32 + G 13 = 0 .

Thus, by Lemma (4.5), given in ref. [50], there exists c 1 , c 2 , c 3 , c 4 , and c 5 such that D = c 1 D 1 + c 2 D 2 + c 3 D 3 + c 4 D 4 + c 5 D 5 . By using Lemma (4.4), given in [50], we have

c 3 a 32 = c 2 a 21 , c 1 a 13 = c 3 a 34 + c 3 a 32 , c 3 a 34 = c 4 k 8 a 45 , c 4 a 45 = c 5 a 51 .

Thus, we have

(7) D = α 1 O ∗ + α 2 S Q t β P ∗ S ∗ D 1 + α 2 S Q t β P ∗ S ∗ D 2 + D 3 + α 2 S Q t μ Q t ∗ D 4 + α 2 S Q t δ γ S ∗ D 5 .

Eq. (7) is the Lyapunov function for the model (2.1). By making use of this and LaSalle’s principle of invariance [51], we have global asymptotic stability of E ∗ inside ( ξ ) .

Figure 2

Weighted diagraph.

5 Sensitivity analysis

Our primary concern is to study the dissemination capacity of any infectious disease in a community. To obtain the those parameters that are the main sources of any contamination and its spread, we will go to the sensitivity analysis. The main objective is to remove the disease from the community completely, which is not possible practically, but with the help of sensitivity analysis, we can identify those parameters that give an idea that how we can reduce the spread of the ailment in the community.

To achieve this goal, we find the sensitivity index of each parameter by simply obtaining the ratio of difference of parameter values and change in the reproduction number. Our basic goal is to reduce fundamental reproduction number below one, and if we cannot do it, then our second purpose is to identify those parameters that increase or decrease the infectious substantially.

5.1 Sensitivity indices of R 0

We will establish those parameters that can be used to control the disease, and this will be done by calculating the sensitivity indices of R 0 .

Definition 5.1

The definition given in ref. [54], will be used to obtain the indices. If we want to calculate the sensitivity index of any variable, say T with respect to any variable Z , then it can be expressed as follows:

(8) Γ Z T = ∂ T ∂ Z × Z T .

This formula will be used to obtain the analytical values of the sensitivity indices of R 0 with respect to all parameters. For example, let us calculate it with respect to α 1 :

∂ R 0 ∂ α 1 = − β Λ α 1 ( α 1 + ( η + γ + μ ) μ ) 2 μ + β Λ ( α 1 + ( η + γ + μ ) μ ) μ Γ α 1 R 0 = ∂ R 0 ∂ α 1 × α 1 R 0 = ( α 1 + ( η + γ + μ ) μ ) μ − β Λ α 1 μ ( μ ( γ + η + μ ) + α 1 ) 2 + β Λ μ ( μ ( γ + η + μ ) + α 1 ) β Λ .

The earlier calculated expression is too much complicated to say any thing for the increasing or decreasing of R 0 with respect to α 1 , but it will give us the sensitivity index of R 0 0.166759 after using the values of parameters in Table 1. This numerical value is telling us that if one wants to obtain 1 % in R 0 , then vale of α 1 must be increased by 10 % . Figure 3 shows that the natural death rate, μ , is the most sensitive one (followed by β ) among all the parameters, with greatest numerical value ( − 1.16686 ). As to increase or decrease the natural death rate is not in our control, so we will focus on β , which is the second most sensitive parameter. Sensitivity index corresponding to β is 1, so 10% change in the value of β will change the value of R 0 by 10 % . This is suggesting to spread the awareness about smoking using the media transmission so that people can not become smokers.

$Figure 3 Sensitivity indices of R 0 {R}_{0} with respect to model parameters.$

Figure 3

Sensitivity indices of R 0 with respect to model parameters.

5.2 Devaluation in the magnitude of endemic random smokers and smokers

In Section 5.1, we determined those components that assume essential role in decreasing the fundamental proliferation number less than one.

We have determined the most compelling elements that assume an imperative role in decreasing the essential proliferation number below unity in Section 5.1. In any case, it is unthinkable for all intents and purposes to annihilate smoking totally. The expulsion of this disease from any community is a very difficult task. In any case, one can make a try to reduce the number of irregular and potential smokers. This can be done only by observing the change in the constant level of occasional and regular smokers comparative with the variation in parameter values. It can be understood as to normalize the every value with the smallest one, which will give us the main parameter that contributes to the decrease of both infected classes.

5.2.1 Impact of parameters on the endemic level of occasional smokers

Apparently we see, by observing Figures 4(a)–(h), that the parameters Λ (Figure 4(a)), μ (Figure 4(c)), α 1 (Figure 4(d)), η (Figure 4(e)), and γ (Figure 4(g)) are substantially effecting the endemic level of the smokers who smoke occasionally, and in any case, the figuring of the rate contrast of the estimations of O ∗ compared to rate distinction of the boundary esteems shows that the most touchy boundary is η , which is smoking-related passing rate; see Figure 4(e). The estimation of this rate distinction of endemic level is − 101092.7881 . It implies that the treatment of customary and incidental smokers can essentially decrease the quantity of infrequent smokers. Percent increase in the estimation of O ∗ comparative with all boundaries is presented in Table 2.

$Figure 4 Variation in the endemic level of O ∗ {O}^{\ast } (from a–h) with respect to model parameters.$

Figure 4

Variation in the endemic level of O ∗ (from a–h) with respect to model parameters.

Table 2

Sensitivity of O ∗ for all model parameters

Parameters	Initial	Final value	Difference	Percentage difference	Initial value of I ∗	Final value of I ∗	Difference	Percentage difference	C 5 ∕ 850	C 9 ∗ C 10
Λ	1	17	16	1,600	108.4	1,502	1393.6	1285.608856	1.882352941	2419.969611
β	0.001	0.08	0.079	7,900	108.4	128.5	20.1	18.54243542	9.294117647	172.3355763
μ	0.001	0.01	0.009	900	108.4	44.41	− 63.99	− 59.03136531	1.058823529	− 62.50379857
α 1	0.002	0.019	0.017	850	108.4	50.6	− 57.8	− 53.32103321	1	− 53.32103321
α 2	0.0025	0.1	0.0975	3,900	108.4	120.3	11.9	10.97785978	4.588235294	50.36900369
γ	0.0001	0.8	0.7999	799,900	130.8	108.4	− 22.4	− 17.12538226	941.0588235	− 16115.99208
η	0.00003	0.9	0.89997	2,999,900	108.4	77.35	− 31.05	− 28.64391144	3529.294118	− 101092.7881
δ	0.0001	0.9	0.8999	899,900	109.9	107.1	− 2.8	− 2.54777070	1058.705882	− 2697.339828

5.3 Impact of parameters on the endemic level of regular smokers

Figure 5(a)–(h) portray that the endemic stage of smokers can be decreased significantly by decreasing the values of the quantities Λ and γ . During our search for the percentage differences, we have again found η with highest value. It makes clear that the treatment of occasional and routine smokers is mandatory to decrease the enhancement of other diseases due to smoking and to decrease smoking-related deaths. Percent increase in the estimation of S* comparative with all boundaries is presented in Table 3.

$Figure 5 Variation in the endemic level of S ∗ {S}^{\ast } (from a–h) with respect to model parameters.$

Figure 5

Variation in the endemic level of S ∗ (from a–h) with respect to model parameters.

5.4 Variation of R 0

The key features to comprehend the dynamics of any epidemic, addiction, or any infectious disease is to observe the vital factors that significantly accelerate the dynamical process. The primary focus is to seek those methods that helps to decrease the basic reproduction number because by controlling this, we can overcome the addiction. We try to identify combined policies that play a major role in reducing the basic reproduction number less than unity. It is done by plotting the phase portrait of the basic reproduction number against combined effect of two parameters. If the basic reproduction number seems below unity with the variation of parameters, then control policies designed on these factors certainly help to curtail or overcome the addiction from the community. One can observe from Figure 3 that the basic reproduction number is highly influenced by three factors. i.e., (i) the effectual contact rate among the persons who are at a risk of obtaining smoking habit and smokers, (ii) the movement rate of smokers who smoke at particular event to regular smokers community, and (iii) the quitting rate of permanent smokers. The two parameters β and α 1 are directly related to the basic reproduction number, whereas the quitting rate of smokers has an inverse impact on the basic reproduction number, which means that greater the quitting rate of regular smokers, smaller will be the value of the basic reproduction number. We focus on reducing those parameters that leads to a decrease of the basic reproduction number less than unity. Now, we have made plot of R 0 as a function of β and α 1 ; see Figure 6, and then in Figure 7, we plotted R 0 as a function of β and γ . One can observe from Figure 6 that R 0 remains less than one when β approaches 0.00095 and α 1 is upto 0.004, but when both cross these limits, then R 0 is greater than unity. Similarly, Figure 7 shows that R 0 may be less than unity when γ is from 0.06 to 0.4 and β is less than 0.0001, but when γ is less than 0.06, then R 0 crosses unity.

$Figure 6 Variation of R 0 {R}_{0} with respect to β \beta and α 1 {\alpha }_{1} .$

Figure 6

Variation of R 0 with respect to β and α 1 .

$Figure 7 Variation of R 0 {R}_{0} with respect to β \beta and γ \gamma .$

Figure 7

Variation of R 0 with respect to β and γ .

6 Optimal control

Optimal control theory is very common in the study of infectious diseases. It tells us how a disease with different control measures can be controlled. The interested readers may go through refs [55,56,57, 58,59] for the deep understanding of optimal control problems. By keeping, in mind all the discussions presented in the previous section, it is very clear that we have to keep our attention on the parameters δ , γ , and β if we want to overcome this ailment. It makes us to change the system of Eq. (1) to obtain the influence of some control strategies to be certain counteractive action. The model having three controls is as follows:

(9) d P d t = − ( 1 − v 1 ) β P S − μ P + Λ , d O d t = ( 1 − v 1 ) β P S − α 1 O − μ O , d S d t = α 1 O + ( 1 − v 1 ) α 2 S Q t − ( μ + γ + η ) S − r 2 v 2 S − ( p + q ) v 3 S , d Q t d t = − ( 1 − v 1 ) α 2 S Q t − μ Q t + γ ( 1 − δ ) S + p v 3 S , d Q p d t = δ γ S − μ Q p + r 2 v 2 S + q v 3 S .

The control v 1 ( t ) is used to overcome the smokers of classes P and Q t so that they can not become smokers again, and this will be done to make the people aware about this disease with the help of media transmission, v 2 ( t ) represents anti smoking gum, and v 3 ( t ) depicts medications utilization against the nicotine. To investigate the perfect degree of endeavors to rein the illness, we build the useful functional J . One has

(10) J ( v 1 , v 2 , v 3 ) = ∫ 0 T A 1 S + v 1 2 B 1 2 + v 2 2 B 2 2 + v 3 2 B 3 2 d t ,

in Eq. (10) where A 1 , B 3 , B 2 , and B 1 constitute the non-negative weights. Our major purpose is to decrease smokers while making the price of controls v 3 ( t ) , v 2 ( t ) , and v 1 ( t ) lesser and lesser with the use of the aforementioned useful functional. The optimal control v 1 ∗ , v 2 ∗ , and v 3 ∗ may be seen as follows:

J ( v 1 ∗ , v 2 ∗ , v 3 ∗ ) = min { J ( v 1 , v 2 , v 3 ) , ( v 1 , v 2 , v 3 ) ∈ V } ,

where

V = { ( v 1 , v 2 , v 3 ) ∣ v j ( t ) is Lebsgue measurable on [ 0 , 1 ] , 0 ≤ v j ( t ) ≤ 1 , j = 1 , 2 , 3 }

is the control set. The solution of this problem and derivation of essential boundaries are acquired with the help of Pontryagin’s maximum principle [60].

6.1 Existence and analysis for optimal control

The occurrence and analysis of an optimal control can be verified through a prominent classical procedure. According to ref. [61], it is necessary to verify that the following hypotheses fulfilled:

( H 1 ) Controls set and set containing the values of state variables are nonempty.

( H 2 ) The acceptable control set U is closed and convex.

( H 3 ) RHS of the state framework is limited linearly in the state and control.

( H 4 ) The goal utilitarian J has raised integrand on U and limited beneath by c 1 ∑ i = 1 3 ∣ v i ∣ 2 τ 2 − c 2 ,

where c 1 , c 2 > 0 and τ > 1 .

The presence of arrangements of ODEs (9) is built up by utilizing the outcome given by Lukes ([62], Th 9.2.1, p 182). Along these lines, we affirm the aforementioned speculation. ( H 1 ) is fulfilled in light of the fact that the coefficients are limited. The boundedness of arrangements show that the control set verifies ( H 2 ) . The arrangement of conditions is bilinear in v 1 , v 2 , and v 3 , and arrangements are limited. Subsequently, RHS of (9) fulfills the measures H 3 . We know the convexity of all the control functions v 1 , v 2 , and v 3 and boundedness of smokers, thus one can pick the positive constants A 1 , B 1 , B 2 , B 3 , c 1 , c 2 , and τ > 1 in the way that inequality (11)

(11) A 1 S + 1 2 B 1 v 1 2 + 1 2 B 2 v 2 2 + 1 2 B 3 v 3 2 ≥ c 1 ∑ i = 1 3 ∣ v i ∣ 2 τ 2 − c 2 ,

is satisfied. Thus, the last condition is also verified. So, we have the following theorem:

Theorem 6.1

For the objective functional J ( v 1 , v 2 , v 3 ) = ∫ 0 T A 1 S + 1 2 ( B 1 v 1 2 + B 2 v 2 2 + B 3 v 3 2 ) d t , where U = { ( v 1 , v 2 , v 3 ) ∣ 0 ≤ v 1 , v 2 , v 3 ≤ 1 , t ∈ [ 0 , T ] } with respect to Eqs. (9) with initial conditions and ∃ is an optimal control v = ( v 1 ∗ , v 2 ∗ , v 3 ∗ ) so that J ( v 1 ∗ , v 2 ∗ , v 3 ∗ ) = min { J ( v 1 , v 2 , v 3 ) , ( v 1 , v 2 , v 3 ) ∈ U } .

The optimal solution is obtained by determining the Hamiltonian and Lagrangian for problem (9). Its Lagrangian is expressed as follows:

(12) L ( S , v 1 , v 2 , v 3 ) = A 1 S + 1 2 B 1 v 1 2 + 1 2 B 2 v 2 2 + 1 2 B 3 v 3 2 ≥ c 1 ( ∣ v 2 ∣ 2 + ∣ v 1 ∣ 2 ) .

For the minimal value of the Lagrangian, we define the Hamiltonian H , (13), as follows:

Let X = ( P , O , S , Q t , Q p ) , U = ( v 1 , v 2 , v 3 ) , and λ = ( λ 1 , λ 2 , λ 3 , λ 4 , λ 5 ) , then we have:

(13) H ( X , U , λ ) = A 1 S + B 1 2 v 1 2 + B 2 2 v 2 2 + B 3 2 v 3 2 + λ 1 ( Λ − ( 1 − u 1 ) β P S − μ P ) + λ 2 ( ( 1 − u 1 ) β P S − α 1 O − μ O ) + λ 3 ( α 1 O + ( 1 − u 1 ) α 2 S Q t − ( μ + γ + η ) S − r 2 u 2 S − ( p + q ) u 3 S ) + λ 4 ( − ( 1 − u 1 ) α 2 S Q t − μ Q t + γ ( 1 − δ ) S + p u 3 S ) + λ 5 ( δ γ S − μ Q p + r 2 u 2 S + u 2 Q t + q u 3 S ) .

6.2 The optimality system

The Pontryagin’s maximum principle as written in ref. [63] will be used to obtain compulsory optimal control conditions. It is proceeded in the following way.

For the optimal solution ( v 1 ∗ , v 2 ∗ , v 3 ∗ ) of problem (9), a non-trivial vector function λ ( t ) = ( λ 1 ( t ) , … , λ 5 ( t ) ) exists that completes the relevant conditions. The stated relation is expressed as follows:

d x d t = ∂ ∂ λ ( H ( v 1 ∗ , v 2 ∗ , v 3 ∗ , t , λ ( t ) ) ) ,

the optimality condition

0 = ∂ ∂ v ( H ( v 1 ∗ , v 2 ∗ , v 3 ∗ , t , λ ( t ) ) )

with the adjoint equation

d λ d t = − ∂ ∂ x ( H ( v 1 ∗ , v 2 ∗ , v 3 ∗ , t , λ ( t ) ) ) .

Now, necessary conditions are incorporated for the Hamiltonian H .

Theorem 6.2

For optimal controls v 3 ∗ , v 2 ∗ , and v 1 ∗ , and solutions O ∗ , P ∗ , S ∗ , Q p ∗ , and Q t ∗ of the relevant state systematic model (9), there are adjoint variables λ 1 , λ 2 , … , λ 5 fulfilling the following (14)

(14) d λ 1 d t = ( λ 2 − λ 1 ) ( 1 − v 1 ) β S − λ 1 μ , d λ 2 d t = ( λ 2 − λ 3 ) α 1 + λ 2 μ , d λ 3 d t = − A 1 + ( λ 1 − λ 2 ) ( 1 − v 1 ) β P + λ 3 μ + ( λ 3 − λ 4 ) γ + ( λ 4 − λ 5 ) δ γ + ( λ 3 − λ 4 ) p v 3 + ( λ 3 − λ 5 ) q v 3 + ( λ 4 − λ 3 ) ( 1 − v 1 ) α 2 Q t + ( λ 3 − λ 5 ) r 2 v 2 + λ 3 η , d λ 4 d t = ( λ 4 − λ 3 ) ( 1 − v 1 ) α 2 S + λ 4 μ , d λ 5 d t = λ 5 μ ,

with the associated conditions λ 1 ( T ) = λ 2 ( T ) = … = λ 5 ( T ) = 0 . In addition, v 3 ∗ , v 2 ∗ , and v 1 ∗ are written as follows:

v 1 ∗ = min max 0 , ( λ 2 − λ 1 ) β P S − ( λ 4 − λ 3 ) α 2 S Q t B 1 , 1 , v 2 ∗ = min max 0 , ( λ 4 − λ 5 ) Q t B 2 , 1 v 3 ∗ = min max ( λ 3 − λ 4 ) p S + ( λ 3 − λ 5 ) q S B 3 , 0 , 1 .

Proof

The transversality conditions and the adjoint equations are determined by utilizing the Hamilton H . Set P = P ∗ , O = O ∗ , S = S ∗ , Q t = Q t ∗ , Q p = Q p ∗ and obtain the derivative of Hamiltonian H with relative to variables P , O , S , Q t , Q p . Thus, we obtain Eq. (15):

(15) d λ 1 d t = ( λ 2 − λ 1 ) ( 1 − v 1 ) β S − λ 1 μ , d λ 2 d t = ( λ 2 − λ 3 ) α 1 + λ 2 μ , d λ 3 d t = − A 1 + ( λ 1 − λ 2 ) ( 1 − v 1 ) β P + λ 3 μ + ( λ 3 − λ 4 ) γ + ( λ 4 − λ 5 ) δ γ + ( λ 3 − λ 4 ) p v 3 + ( λ 3 − λ 5 ) q v 3 + ( λ 4 − λ 3 ) ( 1 − v 1 ) α 2 Q t + ( λ 3 − λ 5 ) r 2 v 2 + λ 3 η , d λ 4 d t = ( λ 4 − λ 3 ) ( 1 − v 1 ) α 2 S + λ 4 μ , d λ 5 d t = λ 5 μ ,

with associated conditions λ 5 ( T ) = λ 4 ( T ) , … , = λ 1 ( T ) = 0 . The following (16) consequences have been obtained from the optimality conditions and the property of the control space U as follows:

(16) v 1 ∗ = min max 0 , ( λ 2 − λ 1 ) β P S − ( λ 4 − λ 3 ) α 2 S Q t B 1 , 1 , v 2 ∗ = min max 0 , ( λ 4 − λ 5 ) Q t B 2 , 1 v 3 ∗ = min max ( λ 3 − λ 4 ) p S + ( λ 3 − λ 5 ) q S B 3 , 0 , 1 .

Now, the problem is solved numerically and observe the efficiency of applied controls. We assume that the optimal campaign continues for 4 months, values given in Table 1 have been used. All the postive weights are taken as A 1 = 1 , B 1 = 3 , B 2 = 7 , and B 3 = 4 . The initial conditions are ( 50 , 10 , 20 , 25 , 30 ) . We can easily see from Figure 8 that after the application of all the controls, we are reaching our objective, that is, deficiency in the smokers and enhancement in the potential smokers and temporary quitters. There is no considerable effect on the number of occasional smokers.

Figure 8

Controls applied to the population (from a–f).

Table 3

Sensitivity of S ∗ for all model parameters

Parameters	Initial	Final value	Difference	Percentage difference	Initial value of Y ∗	Final value of Y ∗	Difference	Percentage difference	C 5 ∕ 91.11111111	C 9 ∗ C 10
Λ	1	28	27	2,700	0.3035	9.275	8.9715	2956.01318	29.63414634	87598.92715
β	0.14	1	0.86	614.2857143	0.3035	0.3634	0.0599	19.73640857	6.742160279	133.0660299
μ	0.00001	0.01	0.00999	99,900	0.3374	0.1151	− 0.2223	− 65.8861885	1096.463415	− 72241.79522
α 1	0.002	0.9	0.898	44,900	0.3035	1.715	1.4115	465.0741351	492.8048781	229190.8024
α 2	0.05	0.001	− 0.049	− 98	0.3035	0.6142	0.3107	102.3723229	− 1.075609756	− 110.1126693
γ	0.8	0.01	− 0.79	− 98.75	0.3035	39.9	39.5965	13046.62273	− 1.083841463	− 14140.47068
η	0.00003	0.9	0.89997	2999900	0.3035	0.0945	− 0.209	− 68.86326194	32925.73171	− 2267373.287
δ	0.9	0.08	− 0.82	− 91.11111111	0.3035	0.2857	− 0.0178	− 5.86490939	− 1	5.864909391

7 Conclusion

The deterministic smoking model has been thoroughly investigated in this work. Worldwide conduct has been talked about through edge boundary. Worldwide conduct populace’s steady level with zero tainted classes has been demonstrated by the development of appropriate Lyapunov utilitarian. It has additionally been demonstrated that all populace having a place with various compartments approaches non-zero constant level irrespective of initial conditions. It has been proved through the application of graph theoretic methodology. It implies that this model can be applied to any network so as to figure the future patterns of smoking habit.

By thinking about the perfect circumstance, we played out the affectability investigation of the fundamental generation number. For the total destruction of smoking danger from the network, the technique for nearby affectability examination of boundaries has been received. The affectability lists of fundamental proliferation number as for model boundaries have been registered, and those boundary that gives the most noteworthy change is declared as the most delicate parameter.

In the real life, it is not possible to obtain rid of all the smokers from the population. We can only make the effort to reduce the constant level of infected compartments. By making the change in the parameters values, we can observe the corresponding variation in occasional smokers and regular smokers. The normalized percentage change of constant magnitude of individuals smoking occasionally and regular smokers helped us to decide the most influential parameter. The highest value of outcome demonstrates the most sensitive parameter.

To reduce the ailment of smoking, sensitivity examination guides us to design fruitful control strategies. All the results of this study are based on the data given in the published papers or by assuming the parameters values. But if one has the real data, then this approach can be very helpful in designing very effective programs to reduce smoking menace from the community. In the future work, we will include it.

Acknowledgments

This publication was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-12-09

Revised: 2022-05-31

Accepted: 2022-06-22

Published Online: 2022-09-30

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2022-0214

Keywords for this article

smoking model; stability analysis; reproduction number; equilibria; optimal control

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