^{1}

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A new PLSGP (potential smokers-light smokers-persistent smokers-giving up smokers-potential smokers) model with birth and death rates on complex heterogeneous networks is presented. Using the mean-field theory, we obtain the basic reproduction number
*R*
_{0}
and find that basic reproduction number for constant contact is independent of the topology of the underlying networks. When
*R*
_{0}
＜1, the smoking-free equilibrium is globally asymptotically stable, then the smoking will disappear. When
*R*
_{0}
＞1, the smoking-present equilibrium is global attractivity, then the number of smoker will remain stable and smoking will become endemic. Numerical simulations illustrated theoretical results. Our result shows that the model is very important to control the spread of the smoking.

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 [

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. [

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

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 [

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

Lemma 1. [

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

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

So, for any given constant

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

Hence, for any given constant

Turning back, one has

So, for any given enough small constant

So for any given enough small constant

From the third equation of (2.1) implies that

So, for any given enough small constant

Repeating the above analyses and calculation, we get six sequences

We can easy get that

Since the sequential limits of (4.26) exist, let

Noting that

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

Finally, substituting

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

Parameters used in the simulations list as follows: in

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

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