Smoking is closely related to health, and smoking ranks fourth among the top 10 risk factors for health, according to the World Health Organization report. Tobacco has been identified as a primary carcinogen around the world. Smokers are 10 to 30 times more likely to develop lung cancer than non-smokers. Smoking problem of people has become a significant public health concern. The behavior of smoking often causes a range of negative consequences. Long-term smoking produces negative changes in the heart, such as heart rate and blood pressure rise. Smoking damages almost all parts of the human body and contributes to a number of human diseases including lung cancer, respiratory disease, heart disease, alimentary canal effect and eventually death. Due to the increasing in the number of smokers, tobacco use is also as a disease to be treated.

In recent years, many types of epidemic models are discussed, such as virus dynamics models [1] [2] , tuberculosis models [3] [4] , and HIV models [5] [6] . One of the famous representative works in this area was done by Pastor-Satorras and Vespignani [7] . They presented a detailed analytical and numerical study on an SIS epidemic model in the highly heterogeneous networks (i.e., scale-free networks). The most striking result is that they found the absence of the epidemic threshold in these networks. That is, the threshold approaches zero in the limit of a large number of edges and nodes, and even a quite small infectious rate can produce a major epidemic outbreak. Liu XD et al. [8] presented an SIS (susceptible-infected-susceptible) epidemic model with infective medium and feedback mechanism on scale-free networks. They found that the epidemic is not only spread between individuals by direct contacts but also transmitted by medium, people’s initial response when epidemic disease outbreaks have been also considered. Moreno et al. [9] also found these similar conclusions for an SIR (susceptible-infected-removed) epidemic model on scale-free networks. The epidemic model is constantly evolving, such as SIRS (susceptible-infected-removed- susceptible), SEIR (susceptible-infected-exposed-removed), SIQRS (susceptible-infected-quarantined-recovered-susceptible) [10] [11] [12] [13] [14] .

Studying of smoking behavior has attracted the attention of many scholars and researchers recently. In order to explore the spread rule of smoking, some models are developed. Castillo-Garsow et al. [15] proposed a mathematical model for giving up smoking in the first time. In this model, a total constant population was divided into three states: potential smokers, that is, people who do not smoke yet but might become smokers in the future (P), smokers (S), and quit smokers (Q). Zaman [16] extended the work of Castillo-Garsow et al. [15] by adding the population of occasional smokers in the model, and presented qualitative behavior of the model. Zaman [17] presented the optimal campaigns in the smoking dynamics. They consider two possible control variables in the form of education and treatment campaigns oriented to decrease the attitude towards smoking and first showed the existence of an optimal control for the control problem.

However, in real life, some potential smokers may become light smoker since they contact with light smokers or persistent smokers. Some quit smokers may be only temporary quit smokers, so they will become potential smokers again. Enlightening by the previously mentioned cases, we present a PLSGP giving up smoking model on scale-free network. The paper is organized as follows: The model is formulated in Section 2. The basic reproduction number and existence of smoking equilibriums are calculated in Section 3. In Section 4, we analyze the stability of the equilibria. In Section 5, sensitivity analysis and numerical simulations are illustrated. In Section 6, we give some conclusions and discussions.

In this paper, we establish the giving up smoking model as Figure 1. From Figure 1, the total population is divided into four compartments, namely, the potential smokers compartment (P), light or occasion smokers compartment (L), persistent smokers compartment (S), and quit smokers group (G). The total recruitment number into this homogeneous social mixing community is b. Transmission coefficient from the potential smokers compartment to the light smokers compartment is ρ 1 , Transmission coefficient from the potential smokers compartment to the persistent smokers compartment is ρ 2 , Transmission coefficient from the light smokers compartment to the persistent smokers compartment is α , The permanent quit smoking rate is β , The relapse rate of which temporary quit people to become potential smokers is δ . Naturally death rate is μ . The total population size is N ( t ) . Let P k ( t ) , L k ( t ) , S k ( t ) , G k ( t ) be the relative densities of potential smokers, light smokers, persistent smokers and quit smokers nodes of degree k at time t, respectively.

With these assume, the dynamic mean-field equations of the P L S G P model can be written as follows:

{ d P k ( t ) d t = b + δ G k ( t ) − k ( ρ 1 Θ 1 + ρ 2 Θ 2 ) P k ( t ) − μ P k ( t ) d L k ( t ) d t = k ( ρ 1 Θ 1 + ρ 2 Θ 2 ) P k ( t ) − ( α + μ + γ ) L k ( t ) d S k ( t ) d t = α L k ( t ) − ( μ + β ) S k ( t ) d G k ( t ) d t = β S k ( t ) + γ L k ( t ) − ( μ + δ ) G k ( t ) (2.1)

where

Θ 1 ( t ) = ∑ i P ( i | k ) L i ( t ) = 〈 k 〉 − 1 ∑ i i P ( i ) L i ( t ) Θ 2 ( t ) = ∑ i P ( i | k ) S i ( t ) = 〈 k 〉 − 1 ∑ i i P ( i ) S i ( t ) (2.2)

where, P ( k ) > 0 is the probability that a node has degree k and thus ∑ k = 1 n P ( k ) = 1 , 〈 k 〉 = ∑ k = 1 n k P ( k ) denotes the average degree. ρ = ρ 1 Θ 1 + ρ 2 Θ 2 . Clearly, these variables obey the normalization condition:

P k ( t ) + L k ( t ) + S k ( t ) + G k ( t ) = 1 . (2.3)

The initial conditions for system can be given as follows S k ( 0 ) = 1 − P k ( 0 ) − L k ( 0 ) − G k ( 0 ) ≥ 0 , R k ( 0 ) ≥ 0 , S k ( 0 ) ≥ 0 , C k ( 0 ) ≥ 0 . In this model, we assumed μ equal to p.

{ d L k ( t ) d t = k ρ ( 1 − L k ( t ) − S k ( t ) − G k ( t ) ) P k ( t ) − ( α + μ + γ ) L k ( t ) d S k ( t ) d t = α L k ( t ) − ( μ + β ) S k ( t ) d G k ( t ) d t = β S k ( t ) + γ L k ( t ) − ( μ + δ ) G k ( t ) (2.4)

Theorem 1. Consider system (2.1). Define R 0 = 〈 k 2 〉 ( ρ 1 ( β + μ ) + ρ 2 α ) 〈 k 〉 ( β + μ ) ( α + μ + γ ) ) . There always exists the smoking-free equilibrium E 0 ( 1 , 0 , 0 , 0 ) . When R 0 > 1 , the system has an occasion smoking equilibrium E ∗ ( P k ∗ , L k ∗ , S k ∗ , G k ∗ ) .

Proof. To get the information-prevailing equilibrium solution E ∗ ( P k ∗ , L k ∗ , S k ∗ , G k ∗ ) , we need to make the right side of system equal to zero, it should satisfy

{ b + δ G k ∗ ( t ) − k ρ ∗ P k ∗ ( t ) − μ P k ∗ ( t ) = 0 k ρ ∗ P k ∗ ( t ) − ( α + μ + γ ) L k ∗ ( t ) = 0 α L k ∗ ( t ) − ( μ + β ) S k ∗ ( t ) = 0 β S k ∗ ( t ) + γ L k ∗ ( t ) − ( μ + δ ) G k ∗ ( t ) = 0 (3.1)

where ρ ∗ = ρ 1 Θ 1 ∗ + ρ 2 Θ 2 ∗ , we follow from (3.1) that

S k ∗ ( t ) = k ρ α ( μ + δ ) ( μ + δ ) ( μ + β ) ( μ + α + γ ) + k ρ [ ( μ + β ) ( 1 + γ ) + α ( β + ( μ + δ ) ) ] L k ∗ ( t ) = k ρ ( μ + δ ) ( μ + β ) ( μ + δ ) ( μ + β ) ( μ + α + γ ) + k ρ [ ( μ + β ) ( 1 + γ ) + α ( β + ( μ + δ ) ) ] (3.2)

Obviously, ρ ∗ = 0 satisfies (3.1). Hence, P k = 1 and L k = S k = R k = 0 is an equilibrium of (2.1), which is called the smoking-free equilibrium.

Substituting L k ∗ and S k ∗ of (3. 2) into ρ ∗

Let ρ ∗ ≅ f ( ρ ∗ )

Clearly, ρ ∗ = 0 is a solution of equation. To ensure the equation has a nontrivial solution, the following condition must should satisfied

d f ( ρ ∗ ) d ρ ∗ | ρ ∗ = ∂ f ( ρ ∗ ) ∂ L k ∗ + ∂ f ( ρ ∗ ) ∂ S k ∗ > 1 and f ( 1 ) ≤ 1 . (3.3)

We can obtain the reproductive number

R 0 = 〈 k 2 〉 ( ρ 1 ( β + μ ) + ρ 2 α ) 〈 k 〉 ( β + μ ) ( α + μ + γ ) .

Theorem 2. When R 0 < 1 , the smoking-free equilibrium of system (2.1) is globally asymptotically stable.

Proof. The Jacobian matrix of the smoking-free equilibrium of system (2.1), which is a 3 n × 3 n matrix, can be written as follows:

J = ( A 11 ⋯ A 1 n ⋮ ⋱ ⋮ A n 1 ⋯ A n n )

where

A 11 = ( − δ − μ − δ − ρ 1 P 1 − δ − ρ 2 P 1 0 ρ 1 P 1 − ( α + μ + γ ) ρ 2 P 1 0 α − μ − β )

A 1 n = ( 0 − ρ 1 P n − ρ 2 P n 0 ρ 1 P n ρ 2 P n 0 0 0 )

A 1 n = ( 0 − n ρ 1 P n − n ρ 2 P n 0 n ρ 1 P n n ρ 2 P n 0 0 0 )

A n n = ( − δ − μ − δ − n ρ 1 P n − δ − n ρ 2 P n 0 n ρ 1 P n − ( μ + α + γ ) n ρ 2 P n 0 α − μ − β )

A direct calculation leads to the characteristic polynomial of the smoking-free equilibrium in the following from:

( λ + ( μ + α + γ ) ) n − 1 ( λ + μ + β ) ( λ 2 + p λ + q ) = 0 ,

where p = ( μ + β ) + ( μ + α + γ ) − ρ 1 ∑ i = 1 n i P ( i ) and q = ( μ + β ) ( μ + α + γ + ρ 1 ) − α ρ 2 ∑ i = 1 n i P ( i ) .

Note that R 0 < 1 is equivalent to q > 0 and that R 0 < 1 also implies: ( μ + β ) + ( μ + α + γ ) > ρ 1 ∑ i = 1 n i P ( i ) , which means p > 0 . Note that R 0 < 1 is equivalent to q > 0 and that R 0 < 1 also implies and, which means p > 0 . Therefore, there exists a unique positive eigenvalue λ of J if and only if R 0 > 1 , otherwise, if R 0 < 1 , all real-valued eigenvalues of J are negative. By the Perron-Frobenius theorem, it implies that the maximal real part of all eigenvalues of J is positive if and only if R 0 > 1 . Then, a theorem of Lajmanovich and York [18] yields the results of this theorem. The proof is thus completed.

Next, the globally attractivity of positive endemic equilibrium is discussed. The main result is given in the following theorem.

Lemma 1. [19] if a > 0 , b > 0 and d x ( t ) d t ≥ b − a x , when t ≥ 0 and x ( 0 ) ≥ 0 , we have t ≥ 0 lim t → ∞ inf x ( t ) ≥ b a , if a > 0 , b > 0 and d x ( t ) d t ≤ b − a x , when t ≥ 0 and x ( 0 ) ≥ 0 , we have lim t → ∞ sup x ( t ) ≤ b a .

Theorems 3. Suppose that ( L k ( t ) , S k ( t ) , G k ( t ) ) is a solution of (2.4), with L k ( 0 ) > 0 , S k ( 0 ) > 0 , G k ( 0 ) > 0 and R 0 > 1 . If R 0 > 1 , then lim t → ∞ ( L k ( t ) , S k ( t ) , G k ( t ) ) = ( L k ∗ ( t ) , S k ∗ ( t ) , G k ∗ ( t ) ) , where ( L k ∗ ( t ) , S k ∗ ( t ) , G k ∗ ( t ) ) is the unique smoking equilibrium of (2.4) for k = 1 , 2 , ⋯ , n .

Proof: In the following, k is fixed to be any integer in ( 1 , 2 , ⋯ , n ) . There exists a sufficiently small constant ξ ( 0 < ξ < 1 ) and a larger enough constant T > 0 such that L k ( t ) ≥ ξ and S k ( t ) ≥ ξ for t > T , therefore ( ρ 1 + ρ 2 ) ξ < ρ ( t ) < ρ 1 + ρ 2 for t > T . Submit this into the first equation of (2.4) gives

L ′ k ( t ) ≤ k ( ρ 1 + ρ 2 ) ( 1 − L k ( t ) ) − ( α + μ + γ ) L k ( t ) ,     t > T

By Lemma 1, for any given constant 0 < ξ < μ + α + γ k ( ρ 1 + ρ 2 ) ( μ + α + γ ) , there exists a t 1 > T , such that L k ( t ) ≤ X k ( 1 ) + ξ 1 for t > t 1 , where

L k ( t ) ≤ X k ( 1 ) + ξ 1 = k ( ρ 1 + ρ 2 ) k ( ρ 1 + ρ 2 ) + ( α + μ + γ ) + ξ 1 < 1 ,     t > t 1 (4.1)

From the second equation of (2.4), it follows that

S ′ k ( t ) ≤ α ( 1 − S k ( t ) ) − ( μ + β ) S k ( t ) ,     t > t 1 (4.2)

Hence, for any given constant 0 < ξ 2 < min { 1 / 2 , ξ 1 , ( β + μ ) ( μ + α + β ) − 1 } , there exists a t 2 > t 1 , such that S k ( t ) ≤ Y k ( 1 ) − ξ 2 for t > t 2 , where

S k ( t ) ≤ Y k ( 1 ) + ξ 2 < α ( α + μ + β ) − 1 + ξ 2 < 1 ,     t > t 2 (4.3)

Then, it follows from the third equation of (2.4),

G ′ k ( t ) ≤ β ( 1 − G k ( t ) ) + γ ( 1 − G k ( t ) ) − ( μ + δ ) G k ( t ) ,     t > t 2 (4.4)

Similarly, for any given constant 0 < ξ 3 < min { 1 / 3 , ξ 2 , ( δ + μ ) ( μ + δ + β + γ ) − 1 + ξ 3 } , there exists a t 3 > t 2 , such that G k ( t ) ≤ Z k ( 1 ) + ξ 3 for t > t 3 , where

G k ( t ) ≤ Z k ( 1 ) + ξ 3 < ( β + γ ) ( μ + β + γ + δ ) − 1 + ξ 3 < 1 ,     t > t 3 (4.5)

Since ( ρ 1 + ρ 2 ) ξ < ρ ( t ) , we substitute this into the first equation of (2.4)

L ′ k ( t ) ≥ k ξ ( ρ 1 + ρ 2 ) ( 1 − L k ( t ) − Y k ( 1 ) − Z k ( 1 ) ) − ( α + μ + γ ) L k ( t ) ,     t > T (4.6)

So for any given enough small constant 0 < ξ 4 < min { 1 / 4 , ξ 3 , k ξ ( ρ 1 + ρ 2 ) ( 1 − L k ( t ) − Y k ( 1 ) − Z k ( 1 ) ) k ξ ( ρ 1 + ρ 2 ) + ( α + μ + γ ) } , there exists a t 4 > t 3 , such that L k ( t ) ≥ x k ( 1 ) − ξ 4 for t > t 4 , where

L k ( t ) ≥ x k ( 1 ) − ξ 4 = k ξ ( ρ 1 + ρ 2 ) ( 1 − L k ( t ) − Y k ( 1 ) − Z k ( 1 ) ) k ξ ( ρ 1 + ρ 2 ) + ( α + μ + γ ) − ξ 4 ,     t > t 4 (4.7)

It follows that

S ′ k ( t ) ≥ α x k ( 1 ) − ( μ + η ) S k ( t ) ,     t > t 4 (4.8)

So for any given enough small constant 0 < ξ 5 < min { 1 / 5 , ξ 4 , α x k ( 1 ) [ ( μ + β ) ] − 1 } , there exists a t 5 > t 4 , such that S ′ k ( t ) ≥ y k ( 1 ) − ξ 5 for t > t 5 , where

S ′ k ( t ) ≥ y k ( 1 ) − ξ 5 = α x k ( 1 ) ( μ + β ) − 1 − ξ 5 ,     t > t 5 (4.9)

From the third equation of (2.1) implies that

G ′ k ( t ) ≥ β y k ( 1 ) + γ x k ( 1 ) − ( μ + δ ) G k ( t ) ,     t > t 5 (4.10)

So for any given enough small constant 0 < ξ 6 < min { 1 / 6 , ξ 5 , [ β y k ( 1 ) + γ x k ( 1 ) ] ( μ + δ ) − 1 } , there exists a t 6 > t 5 , such that G k ( t ) ≥ z k ( 1 ) − ξ 6 for t > t 6 , where

G k ( t ) ≥ z k ( 1 ) − ξ 6 = ( β y k ( 1 ) + γ x k ( 1 ) ) ( μ + δ ) − 1 ,     t > t 6 (4.11)

Due to ξ is a small positive constant, we can derive that 0 < x k ( 1 ) ≤ X k ( 1 ) < 1 , 0 < y k ( 1 ) ≤ Y k ( 1 ) < 1 and 0 < z k ( 1 ) ≤ Z k ( 1 ) < 1 . Let

q ( j ) = 1 〈 k 〉 ∑ j = 1 n P ( i ) ( ρ 1 x i ( j ) + ρ 2 y i ( j ) )     , Q ( j ) = 1 〈 k 〉 ∑ j = 1 n P ( i ) ( ρ 1 X i ( j ) + ρ 2 Y i ( j ) )     ,     j = 1 , 2 , ⋯ (4.12)

We can easily get 0 < q ( j ) ≤ ρ ( t ) ≤ Q ( j ) < ρ 1 + ρ 2 , t > t 6 .

Again, from the first equation of (2.1), it has

L ′ k ( t ) ≤ k Q ( 1 ) ( 1 − L k ( t ) − y k ( 1 ) − z k ( 1 ) ) − ( μ + α + γ ) ,     t > t 6 (4.13)

Hence, for any given constant 0 < ξ 7 < min { 1 / 7 , ξ 6 } , there exists a, such that

Then, from the second equation of (2.1), we have

So, for any given constant, there exists a, such that

Consequently, from the third equation of (2.1), we have

Hence, for any given constant, there exists a, such that

Turning back, one has

So, for any given enough small constant, there exists a, such that for, where

So for any given enough small constant, there exists a, such that for, where

From the third equation of (2.1) implies that

So, for any given enough small constant, there exists a, such that for, where

Repeating the above analyses and calculation, we get six sequences,. Due to the first three are monotone decreasing sequences and the last three are monotone increasing, there exists a sufficiently large positive integer, such that:

, ,

, , (4.25)

,.

We can easy get that

Since the sequential limits of (4.26) exist, let, where and ,

Noting that, one has as. In the six sequences of (4.26), by taking, it follows from (4.26) that

, , ,

, ,. (4.27)

Substituting (4.27) and (4.28) into q and Q, respectively, one has

.

where

,

.

By subtracting the above two equations, it arrives at

.

It is obviously that, so, which sees that, for. From (4.26) and (4.27), it follows that

, ,.

Finally, substituting into (4.26), in view of (3.2) and (4.28), it obtains, , and. The proof is completed.

In this section, some sensitivity analyses are presented to illustrate the result of the smoking model (2.1). We consider the system (2.1) on a scale-free network with the degree distribution and. Consider system (2.1) with and.

Parameters used in the simulations list as follows: in Figure 2, we choose, , , , , , , , thus the threshold value. The figure show that when, and approach to zero, the smoking population will ultimately disappear, which means that smoking will disappear. In Figure 3, we choose, , , , , , , , thus the threshold value. The figure show that when, and maintain at a positive stationary level, which means that the smoking become endemic “disease”.

Figure 4 and Figure 5 show the dynamic behavior of light problem smoking and heavier problem smoking with different degree when. Figure 6 and Figure 7 show the dynamic behavior of light problem smoking and heavier problem smoking with different degree when. We find that the larger degree leads to larger value of the smoking level.

In this paper, we propose a PLSGP giving up smoking model on scale-free network.

We divide the smoker into two groups, light smoker and persistent smoker, considering individual’s birth and death rates. Through the mathematical calculation, we obtain the basic reproduction number and equilibriums. Using the comparison theorem and the iteration principle, we analyze the stability of the smoking free equilibrium, and also give the persistence and global attractivity of the smoking. If, the smoking-free equilibrium of the model is globally stable and smoking will disappear. If, the smoking-present equilibrium is global attractivity and maintaining a positive constant. Furthermore, the dynamic behavior has been analyzed in our model. The study may give us valuable guiding in effectively controlling the behavior of smoking.

This work is supported by the National Natural Science Foundation of China under Grants 61672112 and Project in Hubei province department of education under Grants B2016036.

Fei, Y.L. and Liu, X.D. (2018) Spreading Dynamic of a PLSGP Giving up Smoking Model on Scale-Free Network. Open Access Library Journal, 5: e4365. https://doi.org/10.4236/oalib.1104365