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In this paper, the global stability of free smoking equilibrium point was evaluated and presented graphically. The linear stability of a developed mathematical model illustrates the effect on the population of chain, mild and passive smokers. MATLAB programming was used to simulate the solutions, the reproduction number
*R*
_{0} and the nature of the equilibria.

Castillo et al. [

E ˙ 1 = Δ − β N E 1 I − ( μ + v 1 ) E 1 , (1a)

I ˙ = β N E 1 I − ( μ + ϵ + θ ) I , (1b)

E ˙ 2 = θ I − β N E 2 I − ( μ + v 2 + ω ) I , (1c)

where E 1 = E 1 ( t ) , I = I ( t ) and E 2 = E 2 ( t ) . The parameters ( μ , v 1 , v 2 , ω , β , ϵ ) and θ represent respectively the rate of natural death, exit rate from E 1 to healthy population, exit rate form E 2 to healthy population, death rate of ex-smoker, infection rate of smoking, death rate of smokers and exit rate from I to E 2 . Δ is the average number of healthy people who are at risk of becoming active smokers.

Our paper is divided into: section (2), evaluating the global stability of the model in [

The equilibria of smoking-free equilibrium point in system (1a)-(1b), are given in [

P 1 = ( E 1 , 0 , I 0 , E 2 , 0 ) = ( Δ μ + v 2 , 0 , 0 ) , (2)

and the basic reproduction number R 0 is defined as R 0 = β μ + ϵ + θ .

Theorem 1.

If β ≤ δ 4 and R 0 ≤ 1 , then P 1 is globally asymptotically stable.

Proof.

The Lyapunov function is defined as

V 1 = E 1 , 0 ( E 1 E 1 , 0 − 1 − ln E 1 E 1 , 0 ) + E 2 + I ,

hence,

V ˙ 1 = ( 1 − E 1 , 0 E 1 ) + I ˙ + E ˙ 2 , (3)

By substituting (1a)-(1b) in (3),

V ˙ 1 = ( 1 − E 1 , 0 E 1 ) ( Δ − β N E 1 I − ( μ + v 1 ) E 1 ) + β N I ( E 1 + E 2 ) − ( μ + ϵ + θ ) I + θ I − β N E 2 I − ( μ + v 2 + ω ) E 2 ,

V ˙ 1 = ( 1 − E 1 , 0 E 1 ) ( Δ − ( μ + v 1 ) E 1 ) − β N E 1 I + β N E 0 , 1 I + β N I ( E 1 + E 2 ) − ( μ + ϵ + θ ) I + θ I − β N E 2 I − ( μ + v 2 + ω ) E 2 ,

Since E 1 , 0 = N , then we have

V ˙ 1 = ( 1 − E 1 , 0 E 1 ) ( Δ − δ 1 E 1 ) − δ 2 E 2 + ( β − ( μ + ϵ ) ) I .

where δ 1 = μ + v 1 and δ 2 = μ + v 2 + ω . Using (2) we get

V ˙ 1 = − δ 1 ( E 1 , 0 − E 1 ) 2 E 1 − δ 2 E 2 + ( β − ( μ + ϵ ) ) I .

By re-arranging the equation, we get

V ˙ 1 = − δ 1 ( E 1 , 0 − E 1 ) 2 E 1 − δ 2 R 0 E 2 + δ 2 δ 3 ( δ 3 ( R 0 − 1 ) ) E 2 + ( β − δ 4 ) I , (4)

where δ 3 = μ + ϵ + θ and δ 4 = μ + ϵ .

It is clear that V ˙ 1 ≤ 0 when β ≤ δ 4 and R 0 ≤ 1 for all E 1 , I , E 2 > 0 . Hence, the solutions of systems (1) is limited to Ω , the largest invariant subset of V ˙ 1 = 0 , where Ω is the region of solutions given in [

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In this section, system (1) was modified to include two classes of smokers, chain smokers S 1 and mild smoker S 2 (see

E ˙ 1 ( t ) = Δ − β N E 1 ( t ) ( η S 1 ( t ) + S 2 ( t ) ) − ( μ + v 1 ) E 1 ( t ) , (5a)

S ˙ 1 ( t ) = a β N ( η S 1 ( t ) + S 2 ( t ) ) ( E 1 ( t ) + E 2 ( t ) ) − ( μ + ϵ 1 + θ 1 ) S 1 ( t ) , (5b)

S ˙ 2 ( t ) = ( 1 − a ) β N ( η S 1 ( t ) + S 2 ( t ) ) ( E 1 ( t ) + E 2 ( t ) ) − ( μ + ϵ 2 + θ 2 ) S 2 ( t ) , (5c)

E ˙ 2 ( t ) = θ 1 S 1 ( t ) + θ 2 S 2 ( t ) − β N E 2 ( t ) ( η S 1 ( t ) + S 2 ( t ) ) − ( μ + v 2 + ω ) E 2 ( t ) , (5d)

where the parameters β , ω , v 1 , v 2 and μ are defined as in the system (1a)-(1c), ϵ 1 and ϵ 2 are death rate of chain smokers S 1 and mild smokers S 2 respectively, while parameters θ 1 and θ 2 are exit rates from chain smokers and mild smokers to the healthy population (outside population N). We assume that the exposed people become either a chain or mild smoker at probabilities (1 − a) and a with 0 < a < 1 ). The chain smokers have a higher probability of generating more new smokers (by a factor η ≥ 1 relative to the mild smokers). A simplified form of the model (5) can be written as

E ˙ 1 ( t ) = Δ − β N E 1 ( t ) ( η S 1 ( t ) + S 2 ( t ) ) − d 1 E 1 ( t ) , (6a)

S ˙ 1 ( t ) = a β N ( η S 1 ( t ) + S 2 ( t ) ) ( E 1 ( t ) + E 2 ( t ) ) − d 2 S 1 ( t ) , (6b)

S ˙ 2 ( t ) = ( 1 − a ) β N ( η S 1 ( t ) + S 2 ( t ) ) ( E 1 ( t ) + E 2 ( t ) ) − d 3 S 2 ( t ) , (6c)

E ˙ 2 ( t ) = θ 1 S 1 ( t ) + θ 2 S 2 ( t ) − β N E 2 ( t ) ( η S 1 ( t ) + S 2 ( t ) ) − d 4 E 2 ( t ) , (6d)

where d 1 = μ + v 1 , d 2 = ( μ + ϵ 1 + θ 1 ) , d 3 = ( μ + ϵ 2 + θ 2 ) and d 4 = ( μ + v 2 + ω ) . The initial conditions of system (6), are given by

E 1 ( α ) = φ 1 ( α ) , S 1 ( α ) = φ 2 ( α ) , S 2 ( α ) = φ 3 ( α ) , E 2 ( α ) = φ 4 (α)

where α ∈ R , ( φ 1 ( α ) , φ 2 ( α ) , φ 3 ( α ) , φ 4 ( α ) ) ∈ C R + 4 and C is the Banach space of continuous functions mapping the interval ( − ∞ , 0 ] into R + 4 . By the fundamental theory of functional differential equations [

Theorem 2. For system (6), there exist a positive number K such that the compact set Λ ,

Λ = { ( E 1 , S 1 , S 2 , E 2 ) ∈ R + 4 : 0 ≤ E 1 , S 1 , S 2 , E 2 ≤ K } .

is positively invariant.

Proof. We have

E ˙ 1 ( t ) | E 1 = 0 = Δ > 0 ,

S ˙ 1 ( t ) | S 1 = 0 = a β N S 2 ( E 1 + E 2 ) > 0 for S 2 , E 1 and E 2 > 0 ,

S ˙ 2 ( t ) | S 2 = 0 = ( 1 − a ) β N η S 1 ( E 1 + E 2 ) > 0 for S 1 , E 1 and E 2 > 0 ,

E ˙ 2 ( t ) | E 2 = 0 = θ 1 S 1 + θ 2 S 2 > 0 for S 1 and S 2 > 0 ,

so the solutions are non-negative. Now, to prove the boundedness of the solutions, we let N ( t ) = E 1 ( t ) + S 1 ( t ) + S 2 ( t ) + E 2 ( t ) , hence

N ˙ = Δ − β N E 1 ( η S 1 + S 2 ) − d 1 E 1 + a β N ( η S 1 + S 2 ) ( E 1 + E 2 ) − d 2 S 1 − d 3 S 2 + ( 1 − a ) β N ( η S 1 + S 2 ) ( E 1 + E 2 ) + θ 1 S 1 + θ 2 S 2 − β N E 2 ( η S 1 S 2 ) − d 4 E 2 ≤ Δ − d 1 E 1 − d 5 S 1 − d 6 S 2 − d 4 E 2 ≤ Δ − d ¯ N ( t ) ,

where d 5 = μ + ϵ 1 , d 6 = μ + ϵ 2 and d ¯ = min { d i } , i = 1 , 4 , 5 , 6 . That is, if N ( 0 ) ≤ 0 then N ( t ) < K where K = Δ / d ¯ . It follows that if E 1 ( 0 ) + S 1 ( 0 ) + S 2 ( 0 ) + E 2 ( 0 ) ≤ K then E 1 ( t ) , S 1 ( t ) , S 2 ( t ) , E 2 ( t ) ≤ K .

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Theorem 3. 1) If R 0 ≤ 1 , then there exists only one equilibrium point called smoking-free equilibrium P 0 .

2) If R 0 > 1 , there exist two equilibria that are smoking-free equilibrium P 0 and smoking present equilibrium P 1 .

Proof.

The system (6) has two equilibrium points which are: Smoking-free equilibrium point

P 0 = ( E 1 , 0 , S 1 , 0 , S 2 , 0 , E 2 , 0 ) = ( Δ d 1 , 0 , 0 , 0 ) , (7)

and smoking present equilibrium point

P 1 = ( E 1 * , S 1 * , S 2 * , E 2 * )

where

E 1 * = R 0 Δ ( R 0 − 1 ) β + R 0 ( μ + v 1 ) ,

S 1 * = a β ( R 0 − 1 ) R 0 d 2 ( x 1 + E 2 * ) ,

S 2 * = ( 1 − a ) β ( R 0 − 1 ) R 0 d 3 ( x 1 + E 2 * ) ,

E 2 * = Δ β ( R 0 − 1 ) [ ( 1 − a ) θ 2 d 2 + a θ 1 d 3 ] d 2 d 3 x 2 [ β ( R 0 − 1 ) + R 0 d 1 ] ,

and

x 1 = ( Δ R 0 R 0 d 1 + β ( R 0 − 1 ) ) ,

x 2 = d 4 + ( β ( R 0 − 1 ) R 0 d 2 d 3 ) ( d 2 d 6 ( 1 − a ) + a d 3 d 5 ) .

Using the next generation method [

F = ( 0 − η β − β 0 0 a η β a β 0 0 ( 1 − a ) η β ( 1 − a ) β 0 0 0 0 0 ) , V = ( d 1 0 0 0 0 d 2 0 0 0 0 d 3 0 0 0 0 d 4 ) ,

and hence evaluate the reproduction number R 0 which is the largest eigenvalue of F ( V ) − 1 ,

R 0 = β ( 1 − a ) d 2 + a η d 3 d 2 d 3 . (8)

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Theorem 4.

If R 0 < 1 , then P 0 is locally asymptotically stable.

Proof. The Jacobian matrix of system (6) at P 0 is

J ( P 0 ) = ( − d 1 − η β β 0 0 η a β − d 2 a β 0 0 ( 1 − a ) β η ( 1 − a ) β − d 3 0 0 θ 1 θ 2 − d 4 ) ,

with the eigenvalues λ 1 = − d 1 , λ 2 = − d 4 , and λ 3 , 4 = c ± b , where

c = ( 1 2 [ β ( 1 − a ) − d 3 ] + [ β a η − d 2 ] ) ,

and

b = 1 2 2 ( R 0 − 1 ) + ( d 2 − a β η ) 2 + ( d 3 − ( 1 − a ) β ) 2 + 2 a η β ( β ( 1 − a ) ) .

It is easy to show that c < 0 when R 0 < 1 . To study the nature of b, we start with the fact that,

β ( ( 1 − a ) d 2 + a η d 3 ) < d 2 d 3 ,

hence,

β ( 1 − a ) d 2 < d 2 d 3 , and β a η d 3 < d 2 d 3 ,

β ( 1 − a ) < d 3 , and β a η < d 2 .

Next, we use the numerical approach by assuming two small positive real numbers ξ 1 and ξ 2 such that β η a + ξ 1 = d 2 and β ( 1 − a ) + ξ 2 = d 3 hence

b = 1 2 2 ( R 0 − 1 ) + ξ 1 2 + ξ 2 2 + 2 ( d 2 − ξ 1 ) ( d 3 − ξ 2 ) = 2 2 ( R 0 − 1 ) + 1 2 ξ 1 2 + 1 2 ξ 2 2 + d 2 d 3 − ξ 2 d 2 − ξ 1 d 3 + ξ 1 ξ 2 = 2 2 ( R 0 − 1 ) + 1 2 ( ξ 1 + ξ 2 ) 2 + d 2 d 3 − ξ 2 d 2 − ξ 1 d 3 .

Since 1 2 ( ξ 1 + ξ 2 ) 2 ≅ 0 and d 2 d 3 < ξ 1 d 2 + ξ 2 d 3 then d 2 d 3 < d 2 + d 3 and

( R 0 − 1 ) + 1 2 ( ξ 1 + ξ 2 ) 2 + d 2 d 3 − ξ 2 d 2 − ξ 1 d 3 < 0.

Thus λ 3 , 4 are complex numbers. Finally, we see that all the real parts of λ i , i = 1 , 2 , 3 , 4 are negative so the point P 0 is locally asymptotically stable.

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Theorem 5. If β η ≤ d 5 , β ≤ d 6 and R 0 ≤ 1 , then P 0 is globally asymptotically stable.

Proof. As done in Theorem (1), we defined the Lyapunov function as

V 2 = E 1 , 0 ( E 1 E 1 , 0 − 1 − ln E 1 E 1 , 0 ) + S 1 + S 2 + E 2 ,

hence

V ˙ 2 = ( 1 − E 1 , 0 E 1 ) E ˙ 1 + S ˙ 1 + S ˙ 2 + E ˙ 2 . (9)

By substituting (6) in (9),

V ˙ 2 = ( 1 − E 1 , 0 E 1 ) ( Δ − β N E 1 ( η S 1 + S 2 ) − d 1 E 1 ) + a β N ( η S 1 + S 2 ) ( E 1 + E 2 ) − d 2 S 1 + ( 1 − a ) β N ( η S 1 + S 2 ) ( E 1 + E 2 ) − d 3 S 2 + θ 1 S 1 + θ 2 S 2 − β N E 2 ( η S 1 + S 2 ) − d 4 E 2 = ( 1 − E 1 , 0 E 1 ) ( Δ − d 1 E 1 ) + β N E 1 , 0 ( η S 1 + S 2 ) − d 2 S 1 − d 3 S 2 + θ 1 S 1 + θ 2 S 2 − d 4 E 2

By using the equilibrium conditions Δ = d 1 E 1 , 0 and E 1 , 0 = N ,

V ˙ 2 = − d 1 ( ( E 1 − E 1 , 0 ) 2 E 1 ) + ( β η − d 5 ) S 1 + ( β − d 6 ) S 2 − d 4 E 2 = − d 1 ( ( E 1 − E 1 , 0 ) 2 E 1 ) + ( β η − d 5 ) S 1 + ( β − d 6 ) S 2 + d 4 c 1 ( c 1 [ R 0 − 1 ] ) E 2 − R 0 d 4 E 2 , (10)

where c 1 = d 2 d 3 and c 2 = β ( 1 − a ) d 2 + a η d 3 . Clearly V ˙ 2 ≤ 0 , if β η ≤ d 5 , β ≤ d 6 and R 0 ≤ 1 , for all E 1 , S 1 , S 2 , E 2 > 0 . Hence, the solutions of system (6) are limited to ω , the largest invariant subset of V ˙ 2 = 0 . From (10) we see that V ˙ 2 = 0 if and only if E 1 = E 1 , 0 and S 1 = S 2 = E 2 = 0 . Using LaSalle’s invariance principle, we show that P 0 is globally asymptotically stable.

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Using MATLAB programming, the solution of system (6) is evaluated by assuming values for the parameters as Δ = 50 , ϵ 1 = 0.03 , ϵ 2 = 0.02 , θ 1 = 0.4 , θ 2 = 0.5 , μ = 0.05 , v 1 = 0.2 , v 2 = 0.3 , N = 200 , and ω = 0.001 with two sets of values for β , η and a.

Set (1): R 0 ≤ 1 , β = 0.05 , η = 1.5 , and a = 0.4 .

Set (2): R 0 > 1 , β = 0.5 , η = 2.5 , and a = 0.6 .

A three sets of values are applied and tested for the initial conditions:

IC1: E 1 ( 0 ) = 890 , S 1 ( 0 ) = 120 , S 2 ( 0 ) = 90 , E 2 ( 0 ) = 40 .

IC2: E 1 ( 0 ) = 480 , S 1 ( 0 ) = 60 , S 2 ( 0 ) = 40 , E 2 ( 0 ) = 30 .

IC3: E 1 ( 0 ) = 290 , S 1 ( 0 ) = 30 , S 2 ( 0 ) = 20 , E 2 ( 0 ) = 10 .

In Figures 2-5, the behavior of each smoking population is plotted as time increases.

state. Similar behavior was observed on the chain and mild populations for R 0 ≤ 1 ,

S 1 and S 2 . As the chain smokers’ number decreases, we notice an increase on the population of ex-smokers ( E 2 ) up to a maximum value at around one third of chain smokers’ initial values and around two-thirds of the mild smokers’ initial value,

As we observe from the Figures, we stress the importance of decreasing the number of chain smokers (either by using the media or educational direct seminars) on increasing the number of passive smokers (as the ex-smokers and mild are both joined) to create a healthy action.

Finally, we observe that when the contact between the two classes of smokers with the other populations is weak (small β) then the effect of the smoker’s classes (chain/mild) was not being enough to generate the new individuals to the smokers’ classes. So, the number of smoker’s groups decreased and approached to zero (because the death or stop smoking). On the other hand, if β increase, we found that the number of smoker’s groups increased also, since the rate of interaction between them and other groups (or the effect by smokers) was growing. However, we can see that, one of the solutions which is introduced to reduce the spread of smoking is prevent it in the larger places (such as educations) in order to minimize the rate of contact to be very small or zero.

The modified system/model shows interesting behavior between the different types of smoker’s populations. The main advantage was the more decrease in the size of heavy/mild smokers in comparison with earlier studies. This should encourage others (ex- or passive) smokers to interact better with chain/mild smokers to increase the number of quitters. By comparing our results with the results in [

The authors declare no conflicts of interest regarding the publication of this paper.

Alshareef, A.A. and Batarfi, H.A. (2020) Stability Analysis of Chain, Mild and Passive Smoking Model. American Journal of Computational Mathematics, 10, 31-42. https://doi.org/10.4236/ajcm.2020.101003