# The Cost-Effectiveness Analysis and Optimal Strategy of the Tobacco Control

Authors: Liuyong Pang; Sanhong Liu; Xinan Zhang; Tianhai Tian
Journal: Computational and Mathematical Methods in Medicine
 (2019)

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## Abstract

This paper aims at investigating how the media coverage and smoking cessation treatment should be implemented, for a certain period, to reduce the numbers of smokers and patients caused by smoking while minimizing the total cost. To this end, we first propose a new mathematical model without any control strategies to investigate the dynamic behaviors of smoking. Furthermore, we calculate the basic reproduction numberR0 and discuss the global asymptotic stabilities of the equilibria. Then, from the estimated parameter values, we know that the basic reproduction number R0 is more than 1, which reveals that smoking is one of the enduring problems of the society. Hence, we introduce two control measures (media coverage and smoking cessation treatment) into the model. Finally, in order to investigate their effects in smoking control and provide an analytical method for the strategic decision-makers, we apply a concrete example to calculate the incremental cost-effectiveness ratios and analyze the cost-effectiveness of all possible combinations of the two control measures. The results indicate that the combination of media coverage and smoking cessation treatment is the most cost-effective strategy for tobacco control.


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## Body

## 1. Introduction

Tobacco use is the single greatest preventable cause of death in the world today. Currently, about 6 million people die from tobacco-related illnesses each year [1]. By 2030, this figure is expected to reach 10 million deaths [2]. If current patterns of smoking continue, about 500 million of the world’s population alive today will eventually be killed by smoking, half of them in productive middle age, losing 20 to 25 years of life [3]. Statistical data indicate that it will be very difficult to reduce tobacco-related deaths over the next 30–50 years, unless adult smokers are encouraged to quit [4]. Hence, smoking control and reducing smoking-related death are priority concerns that government organizations must face in the respective countries. Since tobacco contains nicotine which is addictive, it is very difficult to quit smoking [5]. Many different measures have been used to control smoking, including regulation of the packaging and labelling of tobacco products, higher taxes and prices of cigarettes, setting special smoking areas, mass media campaigns, and psychosocial and pharmacological treatment, all of which aim to enhance public consciousness and help tobacco users to give up smoking and avoid subsequent relapse [6].Many studies have been conducted to analyze the smoking phenomenon and investigate the effects of different control measures (Ham [7], Yen et al. [8], Ertürk et al. [9], Castillo et al. [10], Sharomi and Gumel [11], and Guerrero et al. [12]). Rowe et al., in 1992 [13], applied a dynamical model to investigate smoking behavior. Zeb et al., in 2013 [14], proposed a model with square-root incidence rate to describe smoking phenomenon. Lahrouz, et al., in 2011 [15], used deterministic and stochastic models to study the dynamic properties of smokers. Guerrero et al., in 2011 [16], used a mathematical model to successfully describe the characteristics of smoking habit in Spain. In 2002, the Canadian Cancer Society released a study which indicated that setting health warning on cigarette packages is very effective in discouraging smoking [17]. In 2015 [18], we proposed a mathematical model with saturated incidence rate to explore the effects of controlling smoking by setting special smoking areas and raising the price of cigarettes. Results indicate that setting special smoking areas and putting up the price of cigarettes are very effective in reducing the number of smokers.As a continuation of our previous work, we will further investigate the effects of media coverage and smoking cessation treatment in controlling smoking. We will use a concrete example to provide an analytical method for strategic decision-makers, so that we can find out which strategy is the most cost-effective for all possible combinations of the two tobacco control measures. The organization of this paper is as follows. In Section2, we will present a new mathematical model to describe the dynamic behavior of smokers. In Section 3, we will derive the concrete form of the basic reproduction number R0 and perform stability analysis of the model. In Section 4, we will introduce media coverage and smoking cessation treatment into the model to investigate the effects of two control measures as well as the combination of them. In Section 5, the cost-effectiveness analysis is carried out to gain insight to which strategy is most cost-effective in controlling smoking. Finally, the conclusions are summarized in Section 6.


## 2. Construction of the Mathematical Model

In order to facilitate discussion, we introduce new occasional smoker classLt and patient class Ct caused by smoking into our previous model [18]. Hence, we divide the total population into six subpopulations: potential smokers, occasional smokers, smokers, temporary quitters, permanent quitters, and patients caused by smoking, with sizes denoted by Pt, Lt, St, Qtt, Qpt, and Ct, respectively.The transitions among these subpopulations are shown graphically in Figure1, which shows that the number of potential smokers is increased at a constant recruitment rate Λ. In addition, potential smokers can become occasional smokers via effective “contact” with smokers. The incidence rate is bilinear (β is effective contact rate). The probability that an occasional smoker converts a smoker is assumed as ω. The rate of quitting smoking for smokers is γ. Smokers with the proportion γδ (δ∈0,1) are shifted into temporary quitters; nevertheless, smokers with the proportion γ1−δ become permanent quitters. The relapse rate of temporary quitters is α. The conversion ratios from occasional smokers, smokers, temporary quitters, and permanent quitters to patients caused by smoking are τ, τξ, ηθ (ξ,θ>1), and η, respectively. The natural death rates of all the subpopulations are μ, and the mortality rate due to the disease caused by smoking is d. Hence, we can establish the following model:(1)P˙=Λ−μP−βSP,L˙=βSP−μ+ω+τL,S˙=ωL+αQt−μ+γ+τξS,Q˙t=γδS−μ+α+ηθQt,Q˙P=γ1−δS−μ+ηQp,C˙=τL+τξS+ηθQt+ηQP−μ+dC.Figure 1
Flow chart of system (1).Thus, the total population size is given byNt=Pt+Lt+St+Qtt+QPt+Ct at time t. Adding all equations of system (1), we can get(2)N˙=Λ−μN−dC≤Λ−μN,which yields that(3)limt⟶∞Nt=Λμ.Therefore, the biologically feasible region(4)Ω=P,L,S,Qt,QP,C∈R+6:Pt+Lt+St+Qtt+QPt+Ct≤Λμ,is positively invariant.Since the first four equations in system (1) are independent of the variables Qp and C, it is sufficient to consider the following reduced system:(5)P˙=Λ−μP−βSP,L˙=βSP−μ+ω+τL,S˙=ωL+αQt−μ+γ+τξS,Q˙t=γδS−μ+α+ηθQt.


## 3. Basic Properties of the Model and Parameter Values

In this section, the basic reproductive numberR0 of model (5) will be calculated, and the stabilities of equilibria will be investigated. For convenience, we note a=μ+ω+τ,b=μ+γ+τξ,c=μ+α+ηθ.


### 3.1. The Basic Reproductive NumberR0

Apparently, model (5) always has a smoking-free equilibrium E0Λ/μ,0,0,0. Let X=L,S,Qt, from equation (5), we have(6)X˙=FX−VX,where(7)FX=βSP00,VX=aL−ωL−αQt+bScQt−γδS.By calculating, we obtain the Jacobina matrices ofFX and VX at the smoking-free equilibrium E0 as follows:(8)F=0βΛμ0000000,V=a00−ωb−α0−γδc.The inverse matrix ofV is given by(9)V−1=1Vbc−αγδ00ωccaαaωγδaγδab,where(10)V=abc−αγδ,bc−αγδ=μ+τξμ+α+ηθ+γμ+α+ηθ−αγδ=μ+τξμ+α+ηθ+γμ+ηθ+αγ1−δ.Clearly,bc−αγδ>0 when 0<δ<1. Then V=abc−αγδ>0.Hence, the basic reproductive numberR0 (i.e., the spectral radius of FV−1 [19]) is equal to(11)R0=βωcΛμabc−αγδ.Proposition 1.
IfR0>1, an unique positive equilibrium E∗P∗,L∗,S∗,Qt∗ exists in model (5), where P∗=Λ/μR0, L∗=Λ/aR0−1/R0, S∗=μR0−1/β, and Qt∗=μγδR0−1/βc.


### 3.2. The Stability Analysis of the Model

Theorem 1.
The smoking-free equilibriumE0 is globally asymptotically stable if R0<1 and unstable if R0>1.Proof.
The Jacobian matrix of model (5) at E0 is(12)JE0=−μ0−βΛμ00−aβΛμ00ω−bα00γδ−c,whose characteristic equation is given by(13)λ+μλ3+a1λ2+a2λ+a3=0.
Obviously,JE0 has an eigenvalue λ1=−μ, and the remaining eigenvalues satisfy(14)λ3+a1λ2+a2λ+a3=0,where(15)a1=a+b+c,a2=ab−βωΛμ+ac+bc−αγδ=abc−αγδc+αγδac−abc−αγδcR0+ac+bc−αγδ=abc−αγδ1−R0c+αγδac+ac+bc−αγδ,a3=abc−αγδ−βωΛcμ=abc−αγδ1−R0,a1a2−a3>cabc−αγδ1−R0c−abc−αγδ1−R0=0.
Given thatR0<1, we can obtain a2>0, a3>0, and a1a2−a3>0. Hence, by Routh–Hurwitz criterion, the smoking-free equilibrium E0 is locally asymptotically stable if R0<1. If R0>1, then a3<0, which implies that the smoking-free equilibrium E0 is unstable.
To discuss the global stability ofE0, we use a Lyapunov function(16)V1=ωcP−P∗−P∗lnPP∗+ωcL+acS+aαQt,where P∗=Λ/μ.
The derivative ofV1 along solutions of model (5) is calculated as follows:(17)dV1dt=ωcP−P∗PdPdt+ωcdLdt+acdSdt+aαdQtdt=ωcP−P∗PΛ−μP−βSP+ωcβSP−aL+acωL+αQt−bS+aαγδS−cQt=ωcP−P∗PμP∗−μP−βSP+ωcβSP−aL+acωL+αQt−bS+aαγδS−cQt=−μωcPP−P∗2+βωcΛμ−abc−αγδS=−μωcPP−P∗2+abc−αγδR0−1S.
ThendV1/dt≤0 if R0<1, and dV1/dt=0 only if P=P∗,S=0. Hence, P,L,S,QtdV1/dt=0=E0. Therefore, by the LaSalles Invariance Principle, every solution of model (5) approaches E0 as t⟶∞.Theorem 2.
The unique smoking-present equilibriumE∗ is globally asymptotically stable in Ω if R0>1.Proof.
The Jacobian matrix of system (5) at E∗ is(18)JE∗=−μ−βS∗0−βP∗0βS∗−aβP∗00ω−bα00γδ−c,and the characteristic equation is(19)λ4+b1λ3+b2λ2+b3λ+b4=0,where(20)b1=μ+βS∗+a+b+c,b2=μ+βS∗a+b+c+bc−αγδ+ab−ωβP∗+ac>ab−ωβP∗,b3=abc−βωP∗c−αγδa+μ+βS∗ab−ωβP∗+ac+bc−αγδ+βS∗ωβP∗,b4=μ+βS∗abc−βωP∗c−αγδa+βS∗βωP∗c,b1b2−b3=μ+βS∗2a+b+c+a+b+cab−ωβP∗+ac+bc−αγδ+μ+βS∗a+b+c2−βS∗βωP∗>μ+βS∗a+b+c2−βS∗βωP∗>βS∗ab−βS∗βωP∗=βS∗ab−βωP∗.
It is clear thatb1>0. Note that ab−ωβP∗=αγδa/c,abc−βωP∗c−αγδa=cab−ωβP∗−αγδa=0, then b2>0,b3=μ+βS∗ab−ωβP∗+ac+bc−αγδ+βS∗ωβP∗>0,b4=βS∗βωP∗c>0,b1b2−b3>0. We can also prove b1b2b3−b32−b12b4>0 (see Appendix A for details). According to Routh–Hurwitz criterion, the smoking-present equilibrium E∗ is locally asymptotically stable. Next, we will apply the novel approach based on the works [20–25] to explore the global stability of the smoking-present equilibrium E∗. From Theorem 1, we know that the smoking-free equilibrium E0 is unstable if R0>1. The instability of E0 and E0∈∂Ω indicates the uniform persistence, that is, there exist a constant const>0, such that(21)limt⟶∞infxt>const,x=P,L,S,Qt.
The uniform persistence, because of boundedness ofΩ, is equivalent to the existence of a compact set in the interior of Ω, which is absorbing for system (5).
Denotext=Pt,Lt,St,Qtt and mt=Xt,Yt,Zt,Wt, we assign the vector field generated by system (5) to fx. Then system (5) can be rewritten as(22)xt⋅=fx,mt⋅=∂f3∂xxm,where ∂f3/dx stands for the third additive compound matrix for system (5) (see Appendix B for details). It is given by(23)∂f3∂x=−3μ+βSE+Φ,where E is an identity matrix and(24)Φ=−ω+τ+γ+τξα00γδ−ω+τ+α+θηβPβP0ω−γ+τξ+α+ηθ000βS−ω+τ+γ+θη+τξ+α−βS.
Furthermore, the associated linear compound system is given by(25)X˙=−3μ+βS+ω+τ+γ+τξX+αY,Y˙=γδX−3μ+βS+ω+τ+α+θηY+βPZ+W,Z˙=ωY−3μ+βS+γ+τξ+α+ηθZ,W˙=βSZ−3μ+ω+γ+τ+τξ+α+θηW.
We construct a Lyapunov function given by(26)V2x,m=maxX+Y,LSZ+W.
Letm=Xt+Yt+Zt+Wt. Calculating the derivative of V2 along the positive solution of system (25) reduces to the following differential inequalities:(27)D+X+Y≤−3μ+βS+ω+τ+γ+τξ−γδX−3μ+βS+ω+τ+θηY+βPZ+W=−3μ+βS+ω+τ+τξ+γ1−δX−3μ+βS+ω+τ+θηY+βPZ+W≤−3μ+βS+ω+τX+Y+SLβPLSZ+W.
Similarly, we get(28)D+LSZ+W=LSL˙L−S˙SZ+W+LSD+Z+W≤L˙L−S˙S−3μ+γ+τξ+α+θηLSZ+W+ωLSY≤ωLSX+Y+L˙L−S˙S−3μ+γ+τξ+α+θηLSZ+W.
Combining (27) and (28) yields(29)D+V2t≤suph1t,h2tV2t,where(30)h1t=−3μ+βS+ω+τ+SLβP,h2t=ωLS+L˙L−S˙S−3μ+γ+τξ+α+θη.
Form system (5), we have(31)L˙L=βSPL−μ+ω+τ,S˙S=ωLS+αQtS−μ+γ+τξ.
Hence,(32)h1t=L˙L−2μ−βS≤L˙L−2μ,h1t=L˙L−2μ−αQtS−α−θη≤L˙L−2μ,which leads to(33)D+V2t≤L˙L−2μV2t.
Accordingly, from (33), we can obtain(34)V2t≤LtL0V20e−2μt≤ΛV20μL0e−2μt⟶0,ast⟶∞,which indicates that the associated linear compound system (25) is asymptotically stable. Hence, by results found in [20–23], the smoking-present equilibrium E∗ is globally asymptotically stable.


### 3.3. Parameter Values

In order to estimate the parameter values of the model (1), we make some reasonable hypothesis. Assume that the average age of people is 70 years old; then, we estimate the natural death rate μ=1/365∗70≈4×10−5 persons per day. We suppose to select a community with size about 1.5∗109 person as the object of our investigation. Thus, the recruitment rate of potential smokers is Λ≈1.5∗109∗μ=6000 persons per day. The convert rate of occasional smokers into smokers is estimated as ω=0.03 persons per day [14]. The average duration of smoking for a smoker is assumed as 10 years. Thus, the quit rate of a smoker is estimated as γ=1/365∗10≈2.74×10−4 persons per day. The ratio of quitters who temporarily quit smoking is assumed as δ=0.2. The average time-span for temporary quitters from the time quitting smoking to the time starting smoking again is assumed as 2 years, then α=1/2∗365=0.0014 persons per day. The average duration after which an occasional smoker will develop smoking-related illnesses is about 8 years. Thus, τ is estimated as 1/8∗365=3.42×10−4 persons per day. Based on that, a smoker have a higher probability of developing smoking-related illnesses than an occasional smoker; we assume that a smoker develops smoking-related illnesses at a rate τξ (where ξ=3). Similarly, the average duration after which a permanent quitter will develop smoking-related illnesses is assumed as 10 years. Hence, η is estimated as 1/10∗365=2.74×10−4 persons per day. Because that a temporary quitter have a higher probability of developing smoking-related illnesses than a permanent quitter, we assume that a temporary quitter develops smoking-related illnesses at a rate ηθ (where θ=6). It is assumed that an individual with smoking-related illnesses can averagely live for 20 years. Thus, the death rate due to illnesses is estimated as d=1/365×20=1.37×10−4. Goyal, in 2014, applied the data derived from Canada to deduce the effective contact rate between the potential smoker and the smoker as 8.2192×10−7 persons per year [26]. We take it as the effective contact rate β of this paper, i.e., β=8.2192×10−7/365=1.3177×10−11 persons per day. We list each parameter value of system (1) in Table 1 to provide a quick reference.Table 1
The biological meanings and estimated values of parameters.
Parameter
Description
Value
Source
μ
The natural death rate
4
×
10
−
5
[18]
Λ
The recruitment rate
6000
Assumed
β
The effective contact rate
1.3177
×
10
−
11
[26]
ω
The ratio of occasional smoker class converted to smoker class
0.03
[14]
γ
The quit ratio of smokers
2.74
×
10
−
4
[18]
δ
The ratio of quitters who quit smoking temporarily
0.200
[18]
α
The relapse rate of temporal quitters
0.0014
[18]
τ
The ratio of occasional smokers who develop smoking-related illnesses
3.42
×
10
−
4
Assumed
ξ
A positive constant
3.00
Assumed
η
The ratio of permanent quitters who develop smoking-related illnesses
2.74
×
10
−
4
Assumed
θ
A positive constant
6.00
Assumed
d
The death rate due to smoking-related illnesses
1.37
×
10
−
4
AssumedFrom the reasonably estimated parameter values in Table1, we can calculate the basic reproductive number R0=1.4840>1, which indicates that the smoking-present equilibrium E∗ is globally asymptotically stable, i.e., smoking is one of the enduring problems of society. Corresponding time series plots with different initial values of St are shown in Figure 2.Figure 2
Time series plots with different initial values ofSt when R0=1.4840>1.


## 3.1. The Basic Reproductive NumberR0

Apparently, model (5) always has a smoking-free equilibrium E0Λ/μ,0,0,0. Let X=L,S,Qt, from equation (5), we have(6)X˙=FX−VX,where(7)FX=βSP00,VX=aL−ωL−αQt+bScQt−γδS.By calculating, we obtain the Jacobina matrices ofFX and VX at the smoking-free equilibrium E0 as follows:(8)F=0βΛμ0000000,V=a00−ωb−α0−γδc.The inverse matrix ofV is given by(9)V−1=1Vbc−αγδ00ωccaαaωγδaγδab,where(10)V=abc−αγδ,bc−αγδ=μ+τξμ+α+ηθ+γμ+α+ηθ−αγδ=μ+τξμ+α+ηθ+γμ+ηθ+αγ1−δ.Clearly,bc−αγδ>0 when 0<δ<1. Then V=abc−αγδ>0.Hence, the basic reproductive numberR0 (i.e., the spectral radius of FV−1 [19]) is equal to(11)R0=βωcΛμabc−αγδ.Proposition 1.
IfR0>1, an unique positive equilibrium E∗P∗,L∗,S∗,Qt∗ exists in model (5), where P∗=Λ/μR0, L∗=Λ/aR0−1/R0, S∗=μR0−1/β, and Qt∗=μγδR0−1/βc.


## 3.2. The Stability Analysis of the Model

Theorem 1.
The smoking-free equilibriumE0 is globally asymptotically stable if R0<1 and unstable if R0>1.Proof.
The Jacobian matrix of model (5) at E0 is(12)JE0=−μ0−βΛμ00−aβΛμ00ω−bα00γδ−c,whose characteristic equation is given by(13)λ+μλ3+a1λ2+a2λ+a3=0.
Obviously,JE0 has an eigenvalue λ1=−μ, and the remaining eigenvalues satisfy(14)λ3+a1λ2+a2λ+a3=0,where(15)a1=a+b+c,a2=ab−βωΛμ+ac+bc−αγδ=abc−αγδc+αγδac−abc−αγδcR0+ac+bc−αγδ=abc−αγδ1−R0c+αγδac+ac+bc−αγδ,a3=abc−αγδ−βωΛcμ=abc−αγδ1−R0,a1a2−a3>cabc−αγδ1−R0c−abc−αγδ1−R0=0.
Given thatR0<1, we can obtain a2>0, a3>0, and a1a2−a3>0. Hence, by Routh–Hurwitz criterion, the smoking-free equilibrium E0 is locally asymptotically stable if R0<1. If R0>1, then a3<0, which implies that the smoking-free equilibrium E0 is unstable.
To discuss the global stability ofE0, we use a Lyapunov function(16)V1=ωcP−P∗−P∗lnPP∗+ωcL+acS+aαQt,where P∗=Λ/μ.
The derivative ofV1 along solutions of model (5) is calculated as follows:(17)dV1dt=ωcP−P∗PdPdt+ωcdLdt+acdSdt+aαdQtdt=ωcP−P∗PΛ−μP−βSP+ωcβSP−aL+acωL+αQt−bS+aαγδS−cQt=ωcP−P∗PμP∗−μP−βSP+ωcβSP−aL+acωL+αQt−bS+aαγδS−cQt=−μωcPP−P∗2+βωcΛμ−abc−αγδS=−μωcPP−P∗2+abc−αγδR0−1S.
ThendV1/dt≤0 if R0<1, and dV1/dt=0 only if P=P∗,S=0. Hence, P,L,S,QtdV1/dt=0=E0. Therefore, by the LaSalles Invariance Principle, every solution of model (5) approaches E0 as t⟶∞.Theorem 2.
The unique smoking-present equilibriumE∗ is globally asymptotically stable in Ω if R0>1.Proof.
The Jacobian matrix of system (5) at E∗ is(18)JE∗=−μ−βS∗0−βP∗0βS∗−aβP∗00ω−bα00γδ−c,and the characteristic equation is(19)λ4+b1λ3+b2λ2+b3λ+b4=0,where(20)b1=μ+βS∗+a+b+c,b2=μ+βS∗a+b+c+bc−αγδ+ab−ωβP∗+ac>ab−ωβP∗,b3=abc−βωP∗c−αγδa+μ+βS∗ab−ωβP∗+ac+bc−αγδ+βS∗ωβP∗,b4=μ+βS∗abc−βωP∗c−αγδa+βS∗βωP∗c,b1b2−b3=μ+βS∗2a+b+c+a+b+cab−ωβP∗+ac+bc−αγδ+μ+βS∗a+b+c2−βS∗βωP∗>μ+βS∗a+b+c2−βS∗βωP∗>βS∗ab−βS∗βωP∗=βS∗ab−βωP∗.
It is clear thatb1>0. Note that ab−ωβP∗=αγδa/c,abc−βωP∗c−αγδa=cab−ωβP∗−αγδa=0, then b2>0,b3=μ+βS∗ab−ωβP∗+ac+bc−αγδ+βS∗ωβP∗>0,b4=βS∗βωP∗c>0,b1b2−b3>0. We can also prove b1b2b3−b32−b12b4>0 (see Appendix A for details). According to Routh–Hurwitz criterion, the smoking-present equilibrium E∗ is locally asymptotically stable. Next, we will apply the novel approach based on the works [20–25] to explore the global stability of the smoking-present equilibrium E∗. From Theorem 1, we know that the smoking-free equilibrium E0 is unstable if R0>1. The instability of E0 and E0∈∂Ω indicates the uniform persistence, that is, there exist a constant const>0, such that(21)limt⟶∞infxt>const,x=P,L,S,Qt.
The uniform persistence, because of boundedness ofΩ, is equivalent to the existence of a compact set in the interior of Ω, which is absorbing for system (5).
Denotext=Pt,Lt,St,Qtt and mt=Xt,Yt,Zt,Wt, we assign the vector field generated by system (5) to fx. Then system (5) can be rewritten as(22)xt⋅=fx,mt⋅=∂f3∂xxm,where ∂f3/dx stands for the third additive compound matrix for system (5) (see Appendix B for details). It is given by(23)∂f3∂x=−3μ+βSE+Φ,where E is an identity matrix and(24)Φ=−ω+τ+γ+τξα00γδ−ω+τ+α+θηβPβP0ω−γ+τξ+α+ηθ000βS−ω+τ+γ+θη+τξ+α−βS.
Furthermore, the associated linear compound system is given by(25)X˙=−3μ+βS+ω+τ+γ+τξX+αY,Y˙=γδX−3μ+βS+ω+τ+α+θηY+βPZ+W,Z˙=ωY−3μ+βS+γ+τξ+α+ηθZ,W˙=βSZ−3μ+ω+γ+τ+τξ+α+θηW.
We construct a Lyapunov function given by(26)V2x,m=maxX+Y,LSZ+W.
Letm=Xt+Yt+Zt+Wt. Calculating the derivative of V2 along the positive solution of system (25) reduces to the following differential inequalities:(27)D+X+Y≤−3μ+βS+ω+τ+γ+τξ−γδX−3μ+βS+ω+τ+θηY+βPZ+W=−3μ+βS+ω+τ+τξ+γ1−δX−3μ+βS+ω+τ+θηY+βPZ+W≤−3μ+βS+ω+τX+Y+SLβPLSZ+W.
Similarly, we get(28)D+LSZ+W=LSL˙L−S˙SZ+W+LSD+Z+W≤L˙L−S˙S−3μ+γ+τξ+α+θηLSZ+W+ωLSY≤ωLSX+Y+L˙L−S˙S−3μ+γ+τξ+α+θηLSZ+W.
Combining (27) and (28) yields(29)D+V2t≤suph1t,h2tV2t,where(30)h1t=−3μ+βS+ω+τ+SLβP,h2t=ωLS+L˙L−S˙S−3μ+γ+τξ+α+θη.
Form system (5), we have(31)L˙L=βSPL−μ+ω+τ,S˙S=ωLS+αQtS−μ+γ+τξ.
Hence,(32)h1t=L˙L−2μ−βS≤L˙L−2μ,h1t=L˙L−2μ−αQtS−α−θη≤L˙L−2μ,which leads to(33)D+V2t≤L˙L−2μV2t.
Accordingly, from (33), we can obtain(34)V2t≤LtL0V20e−2μt≤ΛV20μL0e−2μt⟶0,ast⟶∞,which indicates that the associated linear compound system (25) is asymptotically stable. Hence, by results found in [20–23], the smoking-present equilibrium E∗ is globally asymptotically stable.


## 3.3. Parameter Values

In order to estimate the parameter values of the model (1), we make some reasonable hypothesis. Assume that the average age of people is 70 years old; then, we estimate the natural death rate μ=1/365∗70≈4×10−5 persons per day. We suppose to select a community with size about 1.5∗109 person as the object of our investigation. Thus, the recruitment rate of potential smokers is Λ≈1.5∗109∗μ=6000 persons per day. The convert rate of occasional smokers into smokers is estimated as ω=0.03 persons per day [14]. The average duration of smoking for a smoker is assumed as 10 years. Thus, the quit rate of a smoker is estimated as γ=1/365∗10≈2.74×10−4 persons per day. The ratio of quitters who temporarily quit smoking is assumed as δ=0.2. The average time-span for temporary quitters from the time quitting smoking to the time starting smoking again is assumed as 2 years, then α=1/2∗365=0.0014 persons per day. The average duration after which an occasional smoker will develop smoking-related illnesses is about 8 years. Thus, τ is estimated as 1/8∗365=3.42×10−4 persons per day. Based on that, a smoker have a higher probability of developing smoking-related illnesses than an occasional smoker; we assume that a smoker develops smoking-related illnesses at a rate τξ (where ξ=3). Similarly, the average duration after which a permanent quitter will develop smoking-related illnesses is assumed as 10 years. Hence, η is estimated as 1/10∗365=2.74×10−4 persons per day. Because that a temporary quitter have a higher probability of developing smoking-related illnesses than a permanent quitter, we assume that a temporary quitter develops smoking-related illnesses at a rate ηθ (where θ=6). It is assumed that an individual with smoking-related illnesses can averagely live for 20 years. Thus, the death rate due to illnesses is estimated as d=1/365×20=1.37×10−4. Goyal, in 2014, applied the data derived from Canada to deduce the effective contact rate between the potential smoker and the smoker as 8.2192×10−7 persons per year [26]. We take it as the effective contact rate β of this paper, i.e., β=8.2192×10−7/365=1.3177×10−11 persons per day. We list each parameter value of system (1) in Table 1 to provide a quick reference.Table 1
The biological meanings and estimated values of parameters.
Parameter
Description
Value
Source
μ
The natural death rate
4
×
10
−
5
[18]
Λ
The recruitment rate
6000
Assumed
β
The effective contact rate
1.3177
×
10
−
11
[26]
ω
The ratio of occasional smoker class converted to smoker class
0.03
[14]
γ
The quit ratio of smokers
2.74
×
10
−
4
[18]
δ
The ratio of quitters who quit smoking temporarily
0.200
[18]
α
The relapse rate of temporal quitters
0.0014
[18]
τ
The ratio of occasional smokers who develop smoking-related illnesses
3.42
×
10
−
4
Assumed
ξ
A positive constant
3.00
Assumed
η
The ratio of permanent quitters who develop smoking-related illnesses
2.74
×
10
−
4
Assumed
θ
A positive constant
6.00
Assumed
d
The death rate due to smoking-related illnesses
1.37
×
10
−
4
AssumedFrom the reasonably estimated parameter values in Table1, we can calculate the basic reproductive number R0=1.4840>1, which indicates that the smoking-present equilibrium E∗ is globally asymptotically stable, i.e., smoking is one of the enduring problems of society. Corresponding time series plots with different initial values of St are shown in Figure 2.Figure 2
Time series plots with different initial values ofSt when R0=1.4840>1.


## 4. Application of Optimal Control to the Tobacco Control Model

Based on the analysis results above mentioned, we know that smoking will become a huge social problem in the absence of any control measure. In order to combat this trouble, we will investigate the effects of media coverage and smoking cessation treatment in controlling smoking. (a) Effects of media coverage. The mass media campaigns which propagate that smoking is very harmful to health not only can cut down the relapse probability of an ex-smoker who returns to cigarettes and reduce the convert probability of an occasional smoker into a smoker, but also can decrease the probability of becoming an occasional smoker to a potential smoker due to the contact with a smoker. Based on that, the probability of a potential smoker becoming an occasional smoker is not only influenced by the media coverage, and we denote the relapse rate of an ex-smoker, the conversion rate of an occasional smoker into a smoker, and the probability of a potential smoker becoming an occasional smoker reduced by media coverage asμ1t, μ1t, and bμ1t, (where b1∈0,1), respectively. (b) Effects of smoking cessation treatment. Many treatment measures can be used for smoking cessation, including behavioral counseling and medications (such as nicotine replacement therapy and varenicline). These treatment measures do not only reduce withdrawal symptoms but also increase the success rate of quitting smoking. u2t represents the success rate of quitting smoking enhanced by smoking cessation treatment. Taking into account the extensions made above, system (1) is modified as the following system:(35)P˙=Λ−μP−β1−b1u1SP,L˙=β1−b1u1SP−μ+ω1−u1+τL,S˙=ω1−u1L+α1−u1Qt−μ+γ1+u2+τξS,Qt⋅=γ1+u2δS−μ+α1−u1+ηθQt,QP⋅=γ1+u21−δS−μ+ηQp,C˙=τL+τξS+ηθQt+ηQP−μ+dC.Our aim is to minimize the cost arising from the consumption of the social resources for smokers and the consumption of the medical and health resources for patients caused by smoking, as well as the costs incurred by media propaganda and smoking cessation treatment. For this end, the total cost functional is defined as(36)CTu=∫​0tfε0Lt+ε1St+ε2Ct+ε32u12t+ε42u22tdt,subject to the state system given by (35). We choose a linear functional for the costs arising from the occasional smokers, the smokers, and the patients. ε0 and ε1 represent the costs arising from the consumption of the social resources for every occasional smoker and smoker, respectively. ε2 denotes the cost produced by the consumption of the medical and health resources for every patient caused by smoking. However, we choose a quadratic functional to represent the costs incurred by media coverage and smoking cessation treatment; such a cost functional has been frequently used in [27–34]. ε3 and ε4 are cost weights associated with the controls u1 and u2, respectively. We seek to find an optimal control pair, u1∗ and u2∗, such that(37)CTu1∗,u2∗=minΦCTu1,u2,where the control set(38)Φ=u1,u2∈L∞0,tf20≤u1t≤u1max,0≤u2t≤u2max,t∈0,tf.Then the HamiltonianH associated with problems (35)–(37) reads(39)H=ε0Lt+ε1St+ε2Ct+ε32u12t+ε42u22t+λ1dPdt+λ2dLdt+λ3dSdt+λ4dQtdt+λ5dQPdt+λ6dCdt,where λi (i = 1, 2, …, 6) are the solutions of the following equalities:(40)λ1⋅=λ1μ+β1−b1u1S−λ2β1−bu1S,λ2⋅=−ε0+λ2μ+ω1−u1+τ−λ3ω1−u1−λ6τ,λ3⋅=−ε1+λ1−λ2β1−u1P+λ3μ+γ1+u2+τξ−λ4γ1+u2δ−λ5γ1+u21−δ−λ6τξ,λ4⋅=−λ3α1−u1+λ4μ+α1−u1+ηθ−λ6ηθ,λ5⋅=λ5μ+η−λ6η,λ6⋅=−ε2+λ6μ+d,satisfying the transversality condition(41)λitf=0,i=1,…,6.By Pontryagin’s Maximum Principle [35] and results obtained from Fleming and Rishel [36–38], we can get the following result:Theorem 3.
There exists an optimal strategyu∗=u1∗,u2∗∈Φ such that(42)CTu1∗,u2∗=minu1,u2∈ΦCTu1,u2,then the optimal controls u1∗,u2∗∈Φ are given by(43)u1∗t=minu1max,max0,λ2−λ1b1βS∗P∗+λ3−λ2ωL∗+λ3−λ4αQt∗ε3,u2∗t=minu2max,max0,λ3−λ4δ−λ51−δγS∗ε4,where λi (i = 1, 2,… 6) are the solutions of (40) and (41), and P∗, L∗, S∗, Qt∗ are optimal state solutions with associated optimal control variables u1∗,u2∗.Proof.
In order to prove the existence of optimal control strategyu∗ minimizing CTu, based on the method mentioned in [34, 36, 37], we need to verify whether the following hypotheses are met:(H1) The control set and state variables are nonempty(H2) The control setΦ is closed and convex(H3) The integrand of the objective functionalCT is convex in Φ and satisfies(44)∈0Lt+∈1St+∈2Ct+∈32u12t+∈42u22t≥ν2u12t+u22tν3/2−ν1,where ν1,ν2>0 and ν3>1.
(H4) The right hand side of system (35) is bounded by the sum of the bounded control variables and state variables and can be written as a linear equation of control variables with coefficient depending on time and state
FromPt+Lt+St+Qtt+QPt+Ct≤Λ/μ, we know that the solutions of state system are bounded. Furthermore, based on the result in [39], we can obtain the existence of the solution of system (35) with bounded coefficients. Hence, condition (H1) is satisfied. It is obvious that our control set Φ is closed and convex defined by Φ=u1,u2∈L∞0,tf20≤u1t≤u1max,0≤u2t≤u2max,t∈0,tf, which satisfies condition (H2). Since the integrand of the objective functional is positive and quadratic in the control variables, it is convex. Notice that(45)ϵ0Lt+ϵ1St+ϵ2Ct+ϵ32u12t+ϵ42u22t≥minϵ3,ϵ42u12t+u22t≥ν2u12t+u22tν32−ν1,with ν1>0, ν2=minε3,ε4/2>0, and ν3=2. Hence, the condition (H3) is satisfied. By definition, each right hand side of system (35) is continuous and can be written as a linear function of control u=u1,u2 with coefficients depending on time and states, which satisfies the condition (H4). Therefore, we conclude that there exists an optimal control.
Furthermore, by equating to zero the derivatives of the Hamiltonian with respect to the controls, we obtain(46)u1=λ2−λ1b1βS∗P∗+λ3−λ2ωL∗+λ3−λ4αQt∗ε3,u2=λ3−λ4δ−λ51−δγS∗ε4.
Using the property of the control space, we have(47)u1∗t=minu1max,max0,λ2−λ1b1βS∗P∗+λ3−λ2ωL∗+λ3−λ4αQt∗ε3,u2∗t=minu2max,max0,λ3−λ4δ−λ51−δγS∗ε4.
To find out the optimal control variables and state variables, we will numerically solve the above systems (35), (40), (41), and (43).


## 5. Numerical Results and Cost-Effectiveness Analysis

In this section, numerical simulations and cost-effectiveness analysis [40] are performed to illustrate the effects of control smoking by different control strategies. We apply the parameter values listed in Table 1 to obtain numerical results for the optimal system by using a forward-backward iterative method [41]. The cost-effectiveness of alternative combinations of the two control measures will be investigated.According to the method mentioned in [27, 29, 40] and our prior work [42], we will focus on comparing the following three control strategies. Strategy a: The combination of media coverage and smoking cessation treatment is implemented to combat smoking habit. For this case, u1 and u2 are defined as control variables. Strategy b: Single media coverage is performed. In this case, only u1 is taken as control variable. Strategy c: Single treatment measure is carried out. In this case, only u2 is seen as the control variable. Our purpose is to provide an analytical method for the strategic decision-makers. Due to the lack of the available literatures and data, as an example, we take cost coefficients εii=0,1,2,3,4 as ε0=1, ε1=1, ε2=1, ε3=36, ε4=36, and b1=0.8, respectively. The maximums of u1t and u2t are taken as 1 and 4, respectively. The smoking-present equilibrium E∗ is served as the initial point of system (35), and the control period is taken as 20 years, i.e., tf=20×365=7300 days whereE∗=1.011×108,6.4411×104,1.4693×106,2.6111×104,1.0258×106,1.0472×107.


### 5.1. Strategya: The Combination of Media Coverage and Smoking Cessation Treatment

For this strategy, corresponding optimal control variablesu1∗, u2∗ and optimal state variables S∗t, C∗t are depicted in Figures 3(a)–3(d), respectively.Figure 3
(a) Optimal control variableu1∗t for strategy a; (b) optimal control variable u2∗t for strategy a; (c) optimal state variable S∗t for strategy a; (d) optimal state variable C∗t for strategy a.
(a)
(b)
(c)
(d)Figures3(a) and 3(b) tell us that media coverage intensity and treatment intensity almost always take their maximum and then subsequently reduce to zero. From Figures 3(c) and 3(d), we know that adopting optimal combined control strategies u1∗t,u2∗t can significantly reduce the numbers of smokers and patients with smoking-related illnesses. In order to more clearly show the efficacy of strategy a, the efficacy function of smokers averted by strategy a is defined as(48)EaS=S0−Sa∗tS0,where S0 is the initial number of smokers and Sa∗t is the corresponding optimal state associated with optimal control strategy a. Efficacy function of smokers EaSt is depicted in Figure 4(a), which indicates that taking the optimal combined control strategy a can reduce the number of smokers be highest up to more than 98%. Similarly, the efficacy function of patients with smoking-related illnesses averted by strategy a is defined as(49)EaC=C0−Ca∗tC0,where C0 represents the initial number of patients with smoking-related illnesses and Ca∗t is the corresponding optimal state associated with optimal control strategy a. The corresponding plot is shown in Figure 4(b), from which we can know that taking the optimal control measures u1∗t,u2∗t can make decrement of patients be highest up to more than 60%. In a word, optimal strategy a is very effective in controlling smoking.Figure 4
(a) Efficacy functionEaSt for strategy a; (b) efficacy function EaCt for strategy a.
(a)
(b)To investigate the the cost-effectiveness of different control strategies, the total occasional smokers averted by the optimal strategya during the time period tf is firstly defined as(50)AaL=tfL0−∫​0tfLa∗tdt,where La∗t is optimal state associated with the optimal strategy a and L0 is the initial number of occasional smokers. Similarly, the total smokers and patients with smoking-related illness averted by the optimal strategy a during the time period tf are, respectively, defined as(51)AaS=tfS0−∫​0tfSa∗tdt,AaS=tfC0−∫​0tfCa∗tdt,where Sa∗t and Ca∗t are optimal states associated with the optimal control of strategy a, and S0 and C0 are the initial numbers of smokers and patients, respectively. We hope that the numbers of occasional smokers and smokers are as less as possible. Especially, we further expect that the number of patients with smoking-related illnesses approaches to 0. Hence, we use the weighted average of occasional smokers, smokers, and patients as the total cases averted by the optimal strategy a during the time period tf, i.e.,(52)TAa=ε0AaL+ε1AaS+ε2AaCε0+ε1+ε2.For strategya, we can, respectively, calculate the values of AaL, AaS, AaC, TAa, and CTa (see Table 2).Table 2
The cases averted and the total cost.
Strategyk
A
k
L
A
k
S
A
k
C
TA
k
C
T
k
a
3.0392
×
10
8
1.0016
×
10
10
2.5468
×
10
10
1.1929
×
10
10
5.1856
×
10
10
b
1.8916
×
10
4
9.3533
×
10
9
2.4397
×
10
10
1.1313
×
10
10
5.3703
×
10
10
c
4.0424
×
10
8
9.1958
×
10
9
2.0816
×
10
10
1.0139
×
10
10
5.7229
×
10
10


### 5.2. Strategiesb and c: Single Media Coverage and Single Smoking Cessation Treatment

For single media converge strategy, we takeu1 as the control variable to minimize the objective functional CT (36) and take u2=0. Similarly, for single smoking cessation treatment strategy, we take u2 as the control variable and set u1=0. The optimal control variables of u1∗t and u2∗t for strategies b and c are shown in Figures 5(a) and 5(b), respectively. Corresponding optimal state variables S∗t and P∗t are depicted in Figures 5(c) and 5(d), respectively.Figure 5
(a) Optimal controlu1∗t for strategy b; (b) optimal control u2∗t for strategy c; (c) optimal state variables S∗t for strategies b and c, respectively; (d) optimal state variables C∗t for strategies b and c, respectively.
(a)
(b)
(c)
(d)Similarly, the efficacy functions of smokers and patients averted for strategyk (k = b or c) are also, respectively, defined as(53)EkS=S0−Sk∗tS0,EkC=C0−Ck∗tC0,k=borc.The corresponding time series of efficacy functions are depicted in Figures6(a) and 6(b), respectively.Figure 6
(a) Efficacy functions ofEkSt (k = b and c) for strategies b and c; (b) efficacy functions of EkCt (k = b and c) for strategies b and c.
(a)
(b)The results of the strategyb are denoted by the blue dotted lines, while the results of the strategy c are represented by the red solid lines. From Figures 5(a), 5(c), 5(d), 6(a), and 6(b), we know that taking single high-intensity media coverage can effectively combat smoking behavior. Similarly, Figures 5(b)–5(d) tell us that single treatment measure can also obviously reduce the numbers of smokers and patients with smoking-related illnesses. In comparison, single optimal media coverage measure is more effective than single treatment strategy in controlling smoking and reducing the number of patients with smoking-related illnesses. Furthermore, for single media coverage strategy b and single smoking cessation strategy c, the values of AkL, AkS, AkC, TAk, and CTk (k = b or c) are, respectively, calculated and listed in Table 2.


### 5.3. Cost-Effectiveness Analysis

In this subsection, the cost-effectiveness of the three alternative strategies is investigated by the incremental cost-effectiveness ratio mentioned in [27–29, 40]. The differences between the costs and health outcomes of different control strategies are compared by this ratio, which is used to investigate the additional cost per additional health outcome. Based on the method mentioned in [27–29, 40], we rank the strategies in increasing order of effectiveness for the total cases (including occasional smokers, smokers, and patients) averted by strategy k (k = a, b, c) and list them in Table 3.Table 3
Incremental cost-effectiveness ratio in increasing order of total cases averted.
Strategyk
Total cases avertedTA
Total costCT
ICER
No strategy
0
0
−
Strategyc
1.0139
×
10
10
5.7229
×
10
10
5.6444
Strategyb
1.1313
×
10
10
5.3703
×
10
10
−3.0034
Strategya
1.1929
×
10
10
5.1856
×
10
10
−0.2315Next, we define and calculate the ICERs as follows:(54)ICERc=CTcTAc=5.7229×10101.0139×1010=5.6444,ICERb=CTb−CTcTAb−TAc=5.3703×1010−5.7229×10101.1313×1010−1.0139×1010=−3.0034.ComparingICERb with ICERc reveals a cost-saving of 3.0034 for strategy b over strategy c, which indicates that strategy b is more inexpensive and more effective than strategy c. Hence, the single smoking cessation strategy is ruled out from the set of alternatives strategies. We rule out strategy c and further compare strategy a and strategy b. Hence, we can get the following values of the ICER:(55)ICERb=CTbTAb=5.3703×10101.1313×1010=4.7470,ICERa=CTa−CTbTAa−TAb=5.1856×1010−5.3703×10101.1929×1010−1.1313×1010=−0.2315.From above calculations, we know that strategyb is more costly and less effective than strategy a. Therefore, strategy b is excluded. Consequently, strategy a, combining of mass media and smoking cessation treatment, is the most cost-effective among the three strategies considered.


## 5.1. Strategya: The Combination of Media Coverage and Smoking Cessation Treatment

For this strategy, corresponding optimal control variablesu1∗, u2∗ and optimal state variables S∗t, C∗t are depicted in Figures 3(a)–3(d), respectively.Figure 3
(a) Optimal control variableu1∗t for strategy a; (b) optimal control variable u2∗t for strategy a; (c) optimal state variable S∗t for strategy a; (d) optimal state variable C∗t for strategy a.
(a)
(b)
(c)
(d)Figures3(a) and 3(b) tell us that media coverage intensity and treatment intensity almost always take their maximum and then subsequently reduce to zero. From Figures 3(c) and 3(d), we know that adopting optimal combined control strategies u1∗t,u2∗t can significantly reduce the numbers of smokers and patients with smoking-related illnesses. In order to more clearly show the efficacy of strategy a, the efficacy function of smokers averted by strategy a is defined as(48)EaS=S0−Sa∗tS0,where S0 is the initial number of smokers and Sa∗t is the corresponding optimal state associated with optimal control strategy a. Efficacy function of smokers EaSt is depicted in Figure 4(a), which indicates that taking the optimal combined control strategy a can reduce the number of smokers be highest up to more than 98%. Similarly, the efficacy function of patients with smoking-related illnesses averted by strategy a is defined as(49)EaC=C0−Ca∗tC0,where C0 represents the initial number of patients with smoking-related illnesses and Ca∗t is the corresponding optimal state associated with optimal control strategy a. The corresponding plot is shown in Figure 4(b), from which we can know that taking the optimal control measures u1∗t,u2∗t can make decrement of patients be highest up to more than 60%. In a word, optimal strategy a is very effective in controlling smoking.Figure 4
(a) Efficacy functionEaSt for strategy a; (b) efficacy function EaCt for strategy a.
(a)
(b)To investigate the the cost-effectiveness of different control strategies, the total occasional smokers averted by the optimal strategya during the time period tf is firstly defined as(50)AaL=tfL0−∫​0tfLa∗tdt,where La∗t is optimal state associated with the optimal strategy a and L0 is the initial number of occasional smokers. Similarly, the total smokers and patients with smoking-related illness averted by the optimal strategy a during the time period tf are, respectively, defined as(51)AaS=tfS0−∫​0tfSa∗tdt,AaS=tfC0−∫​0tfCa∗tdt,where Sa∗t and Ca∗t are optimal states associated with the optimal control of strategy a, and S0 and C0 are the initial numbers of smokers and patients, respectively. We hope that the numbers of occasional smokers and smokers are as less as possible. Especially, we further expect that the number of patients with smoking-related illnesses approaches to 0. Hence, we use the weighted average of occasional smokers, smokers, and patients as the total cases averted by the optimal strategy a during the time period tf, i.e.,(52)TAa=ε0AaL+ε1AaS+ε2AaCε0+ε1+ε2.For strategya, we can, respectively, calculate the values of AaL, AaS, AaC, TAa, and CTa (see Table 2).Table 2
The cases averted and the total cost.
Strategyk
A
k
L
A
k
S
A
k
C
TA
k
C
T
k
a
3.0392
×
10
8
1.0016
×
10
10
2.5468
×
10
10
1.1929
×
10
10
5.1856
×
10
10
b
1.8916
×
10
4
9.3533
×
10
9
2.4397
×
10
10
1.1313
×
10
10
5.3703
×
10
10
c
4.0424
×
10
8
9.1958
×
10
9
2.0816
×
10
10
1.0139
×
10
10
5.7229
×
10
10


## 5.2. Strategiesb and c: Single Media Coverage and Single Smoking Cessation Treatment

For single media converge strategy, we takeu1 as the control variable to minimize the objective functional CT (36) and take u2=0. Similarly, for single smoking cessation treatment strategy, we take u2 as the control variable and set u1=0. The optimal control variables of u1∗t and u2∗t for strategies b and c are shown in Figures 5(a) and 5(b), respectively. Corresponding optimal state variables S∗t and P∗t are depicted in Figures 5(c) and 5(d), respectively.Figure 5
(a) Optimal controlu1∗t for strategy b; (b) optimal control u2∗t for strategy c; (c) optimal state variables S∗t for strategies b and c, respectively; (d) optimal state variables C∗t for strategies b and c, respectively.
(a)
(b)
(c)
(d)Similarly, the efficacy functions of smokers and patients averted for strategyk (k = b or c) are also, respectively, defined as(53)EkS=S0−Sk∗tS0,EkC=C0−Ck∗tC0,k=borc.The corresponding time series of efficacy functions are depicted in Figures6(a) and 6(b), respectively.Figure 6
(a) Efficacy functions ofEkSt (k = b and c) for strategies b and c; (b) efficacy functions of EkCt (k = b and c) for strategies b and c.
(a)
(b)The results of the strategyb are denoted by the blue dotted lines, while the results of the strategy c are represented by the red solid lines. From Figures 5(a), 5(c), 5(d), 6(a), and 6(b), we know that taking single high-intensity media coverage can effectively combat smoking behavior. Similarly, Figures 5(b)–5(d) tell us that single treatment measure can also obviously reduce the numbers of smokers and patients with smoking-related illnesses. In comparison, single optimal media coverage measure is more effective than single treatment strategy in controlling smoking and reducing the number of patients with smoking-related illnesses. Furthermore, for single media coverage strategy b and single smoking cessation strategy c, the values of AkL, AkS, AkC, TAk, and CTk (k = b or c) are, respectively, calculated and listed in Table 2.


## 5.3. Cost-Effectiveness Analysis

In this subsection, the cost-effectiveness of the three alternative strategies is investigated by the incremental cost-effectiveness ratio mentioned in [27–29, 40]. The differences between the costs and health outcomes of different control strategies are compared by this ratio, which is used to investigate the additional cost per additional health outcome. Based on the method mentioned in [27–29, 40], we rank the strategies in increasing order of effectiveness for the total cases (including occasional smokers, smokers, and patients) averted by strategy k (k = a, b, c) and list them in Table 3.Table 3
Incremental cost-effectiveness ratio in increasing order of total cases averted.
Strategyk
Total cases avertedTA
Total costCT
ICER
No strategy
0
0
−
Strategyc
1.0139
×
10
10
5.7229
×
10
10
5.6444
Strategyb
1.1313
×
10
10
5.3703
×
10
10
−3.0034
Strategya
1.1929
×
10
10
5.1856
×
10
10
−0.2315Next, we define and calculate the ICERs as follows:(54)ICERc=CTcTAc=5.7229×10101.0139×1010=5.6444,ICERb=CTb−CTcTAb−TAc=5.3703×1010−5.7229×10101.1313×1010−1.0139×1010=−3.0034.ComparingICERb with ICERc reveals a cost-saving of 3.0034 for strategy b over strategy c, which indicates that strategy b is more inexpensive and more effective than strategy c. Hence, the single smoking cessation strategy is ruled out from the set of alternatives strategies. We rule out strategy c and further compare strategy a and strategy b. Hence, we can get the following values of the ICER:(55)ICERb=CTbTAb=5.3703×10101.1313×1010=4.7470,ICERa=CTa−CTbTAa−TAb=5.1856×1010−5.3703×10101.1929×1010−1.1313×1010=−0.2315.From above calculations, we know that strategyb is more costly and less effective than strategy a. Therefore, strategy b is excluded. Consequently, strategy a, combining of mass media and smoking cessation treatment, is the most cost-effective among the three strategies considered.


## 6. Further Discussion and Conclusion

In this paper, we apply an example to investigate the effects of media coverage and smoking cessation treatment on controlling smoking. Firstly, we give the concrete form of the basic reproduction numberR0 and discuss the existence and stabilities of equilibria. Secondly, from the estimated parameter values, we obtain the basic reproduction number R0=1.4840>1, which indicates that the smoking is one of the enduring problems of society. Hence, we introduce two control measures (media coverage and smoking cessation treatment) into the previous model to find out which strategy is the most effective in combating smoking behavior. Finally, from the numerical results and cost-effectiveness analysis, we conclude that the combination of media coverage and smoking cessation treatment is the most cost-effective strategy. Although we have investigated the effects of media coverage and smoking cessation treatment in controlling smoking, we still do not consider the impact of second-hand smoke on nonsmokers. We will use the real data about tobacco in China to model the parameters and discuss the impact of second-hand smoke on individuals, society, and economy in our future work.



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*Source: 8189270-2019-02-04.xml*