Detectable sensation of a stochastic smoking model

Abdullah Alzahrani; Anwar Zeb

doi:10.1515/math-2020-0068

Article Open Access

Detectable sensation of a stochastic smoking model

Abdullah Alzahrani and Anwar Zeb

Published/Copyright: September 26, 2020

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

Abstract

This paper is related to the stochastic smoking model for the purpose of creating the effects of smoking that are not observed in deterministic form. First, formulation of the stochastic model is presented. Then the sufficient conditions for extinction and persistence are determined. Furthermore, the threshold of the proposed stochastic model is discussed, when noises are small or large. Finally, the numerical simulations are shown graphically with the software MATLAB.

Keywords: stochastic smoking model; Itô’s formula; extinction; persistence; numerical analysis

MSC 2020: 37H30; 26A18; 28Dxx; 34Cxx

1 Introduction

Epidemiology is used for understanding the spreading nature of infectious disease in a society. The epidemic models are mostly based on the similarity of the models but not necessary that all practical models possess all probable properties but somewhat include the mechanisms in the simplest probable mode so as to contain major constituents that influence disease spread. The prediction of real sensations in epidemic models must be carried out very carefully [1]. The first epidemic model was introduced by Kermack and McKendrick [2], in which they divided the whole population into three classes based on different aspects of spreading of disease. The work presented in [1,2] is the groundwork of epidemic models. First, two epidemic models were studied: one is SIS (susceptible; infected; susceptible) model and the other is SIR (susceptible; infected; recovered) model [3,4]. After that, many authors studied different epidemic models based on different types of diseases by including linear, bilinear, saturated incidence rates and similarly using delay differential equations, fractional differential equations and partial differential equations [1,2,3,4,5,6]. A major health concern is a social habit of smoking, which is common in people. Both preventable and premature type of deaths occur in the US and also worldwide due to smoking. According to a previous report, in a year over 4,40,000 deaths occur due to smoking-related disease in the US annually and 1,05,000 at UK [7]. The risk of heart attack is 70% more for smokers rather than nonsmokers. There is 78% chance of lung cancer and 24% of heart disease for smokers. Similarly, throat, mouth, stomach, cervix, breast and pancreas cancers are somehow related to smoking because one cigarette contains more than 4,000 chemicals. According to the third edition of cancer atlas jointly released by International Agency for Research on Cancer, American Cancer Society and Union for International Cancer Control on October 16, 2019, smoking causes more preventable cancer deaths than any other risk factors and in 2017 alone, 2.3 million people worldwide died of smoking, which accounts for 24% of all cancer deaths. On the other hand, based on the WHO global report on trends in the prevalence of tobacco use 2000–2025 [8,9], every year, more than 8 million people die of tobacco use, accounting for about half of its users. More than 7 million of them died of direct smoking, while about 1.2 million nonsmokers died from exposure to second-hand smoke. For this purpose, mathematicians are trying to present the nature of smoking and classify the populations with the help of modeling. From the first smoking model presented by Castillo-Garsow et al. [7], many researchers tried on different aspects to develop smoking models [7,10,11,12,13,14]. In recent years, the authors formulated different forms of smoking models with linear incidence rate [15,16], saturated incidence rate [17,18], square root-type incidence rate [12,19,20] and harmonic mean-type incidence rate [21]. The environmental white noise has great effect on the epidemic models but due to simplicity in deterministic models the authors ignore these terms. For more realistic models, several authors studied the stochastic models by introducing white noise [22,23,24,25,26]. The effects of environment in the AIDS model were studied by Dalal et al. [22] using the method of parameter perturbation. Tornatore et al. [23,24,25] studied the stochastic epidemic models with vaccination. In these studies, they proved the existence, uniqueness and positivity of the solution. A stochastic SIS epidemic model with vaccination is discussed by Zhu and Hu [26]. They obtained the condition of the disease extinction and persistence according to noise and threshold of the deterministic system. Similarly, several authors discussed the same conditions for stochastic models [27,28,29,30,31,32,33 34 35]. In this research, first formulation of a stochastic mathematical smoking model is presented. Then the sufficient conditions for extinction and persistence are determined. Furthermore, the threshold of the proposed stochastic model is discussed, when noises are small or large. Finally, numerical simulations are presented graphically using the software MATLAB.

The rest of the paper is organized as follows: Section 2 is related to the smoking model with random perturbation formulation. Section 3 is related to the unique positive solution of the proposed model. Furthermore, the exponential stability of the proposed model is investigated in Section 4. The persistent conditions are shown in Section 5. Finally, in Section 6 conclusion is given.

2 Model formulation

In this section, a smoking model with random perturbation is formulated as follows:

(1) d P ( t ) d t = Λ − β P ( t ) S ( t ) − μ P ( t ) + δ Q ( t ) − ρ P ( t ) S ( t ) d B ( t ) , d S ( t ) d t = β P ( t ) S ( t ) − ( γ + μ ) S + ρ P ( t ) S ( t ) d B ( t ) , d Q ( t ) d t = γ S ( t ) − μ Q ( t ) − δ Q ( t ) ,

and the description of parameters and variables is given in Table 1.

Table 1

Parameters and description

Notations	Description
P ( t )	Susceptible (potential) smokers
S ( t )	Series (chain) smoker class
Q ( t )	Smokers who quit smoking
Λ	The membership rate to susceptible class by birth or migration
β	Rate at which the potential smokers move to chain smoker class
μ	Natural death
γ	The quitting rate
B ( t )	The standard Brownian motion, with ρ 2 > 0 and with intensity of white noise
δ	The relapse rate

In deterministic form, model (1) is as follows:

(2) d P ( t ) d t = Λ − β P ( t ) S ( t ) − μ P ( t ) + δ Q ( t ) , d S ( t ) d t = β P ( t ) S ( t ) − ( γ + μ ) S , d Q ( t ) d t = γ S ( t ) − μ Q ( t ) − δ Q ( t ) ,

and

(3) d N d t = Λ − μ N ,

where N ( t ) = P ( t ) + S ( t ) + Q ( t ) represents the total population and N ( 0 ) = P ( 0 ) + S ( 0 ) + Q ( 0 ) . Eq. (3) has the exact solution:

(4) N ( t ) = e − μ t N ( 0 ) + Λ μ e μ t .

Also, we have

P ( 0 ) ≥ 0 , S ( 0 ) ≥ 0 , Q ( 0 ) ≥ 0 ⇒ P ( t ) ≥ 0 , S ( t ) ≥ 0 , Q ( t ) ≥ 0 .

So the solution has the positivity property. For stability analysis of model (2), the reproductive number is

(5) R 0 = β γ + μ N .

If R 0 < 1 , then system (2) will be locally stable and will be unstable if R 0 ≥ 1 . Similarly, for Λ = 0 , system (2) will be globally asymptotically stable (see [12]).

3 Existence and uniqueness of the positive solution

Here, first we make the following assumptions:

• Set R + d = { χ i ∈ R d , χ i > 0 , 1 ≤ d } .

• Suppose a complete probability space ( Ω , F , { F } t ≥ 0 , P ) with filtration { F } t ≥ 0 which satisfies the usual conditions.

Generally, consider a stochastic differential equation of n-dimensions as

(6) d x ( t ) = F ( y ( t ) , t ) d t + G ( y ( t ) , t ) d B ( t ) , for t ≥ t 0 ,

with initial value y ( t 0 ) = y 0 ∈ R d . By defining the differential operator L with Eq. (6)

(7) L = ∂ ∂ t + ∑ i = 1 d F i ( y , t ) ∂ ∂ y i + 1 2 ∑ i , j = 1 d G T ( y , t ) G ( y , t ) i j ∂ 2 ∂ y i ∂ y j .

If operator L acts on a function V = ( ℝ d × ℝ ˜ + ; ℝ ˜ + ) , then

(8) L V ( y , t ) = V t ( y , t ) + V y ( y , t ) F ( y , t ) + 1 2 trace G T ( y , t ) V y y ( y , t ) G ( y , t ) .

Theorem 3.1

There is a unique positive solution ( P ( t ) , S ( t ) , Q ( t ) ) of system (1) for t ≥ 0 with ( P ( 0 ) , S ( 0 ) , Q ( 0 ) ) ∈ R + 3 , and solution will be left in R + 3 , with probability 1.

Proof

Since the coefficient of differential equations of system (1) is locally Lipschitz continuous for ( P ( 0 ) , S ( 0 ) , Q ( 0 ) ) ∈ R + 3 , there is a unique local solution ( P ( t ) , S ( t ) , Q ( t ) ) on t ∈ [ 0 , τ e ) , where τ e is time for noise caused by an explosion (see [6]). For demonstration, the solution is global, it is sufficient that τ e = ∞ a.s. Suppose that k 0 ≥ 0 be sufficiently large that ( P ( 0 ) , S ( 0 ) , Q ( 0 ) ) ∈ 1 k 0 , k 0 . For each integer k ≥ k 0 , define the stopping time

τ e = inf t ∈ [ 0 , τ e ) : min ( P ( t ) , S ( t ) , Q ( t ) ) ≤ 1 k 0 or max ( P ( t ) , S ( t ) , Q ( t ) ) ≥ k ,

where we set inf ϕ ( empty set ) = ∞ throughout the paper. For k → ∞ , τ k is clearly increasing. Set τ ∞ = lim k → ∞ τ k where τ ∞ ≤ τ e . If we can show that τ ∞ = ∞ a.s., then τ e = ∞ . If false, then there is a pair of constants T > 0 and ε ∈ ( 0 , 1 ) such that

P { τ ∞ ≤ T } > ε .

So there is an integer k 1 ≥ k 0 , which satisfies

P { τ k ≤ T } ≥ ε for all k ≥ k 1 .

Define a C 2 -function V : ℝ + 3 → ℝ ˜ + by

(9) V ( P , S , Q ) = P − c − c ln P c + ( S − 1 − ln S ) + ( Q − 1 − ln Q ) .

By applying Itô’s formula, we obtain that

(10) d V ( P , S , Q ) = 1 − c P d P + 1 2 P 2 ( d P ) 2 + 1 − 1 S d S + 1 2 S 2 ( d S ) 2 + 1 − 1 Q d Q = L V d t + ρ ( S − P ) d B ( t ) ,

where L V : ℝ + 3 → ℝ ˜ + is defined by

L V = 1 − c P ( t ) ( Λ − β P ( t ) S ( t ) − μ P ( t ) + δ Q ( t ) ) + 1 2 ρ 2 S 2 + 1 − 1 S ( β P ( t ) S ( t ) − ( γ + μ ) S ) + 1 2 ρ 2 P 2 + 1 − 1 Q ( γ S ( t ) − μ Q ( t ) − δ Q ( t ) ) = Λ − β P ( t ) S ( t ) − μ P ( t ) + δ Q ( t ) − c Λ P ( t ) + c β S ( t ) + c μ − c δ Q ( t ) P ( t ) + 1 2 ρ 2 S 2 + β P ( t ) S ( t ) − ( γ + μ ) S ( t ) − β P ( t ) + ( γ + μ ) + 1 2 ρ 2 P 2 + γ S ( t ) − μ Q ( t ) − δ Q ( t ) − γ S ( t ) Q ( t ) + μ + δ ≤ Λ − ( γ + μ ) S + c β S ( t ) + c μ + γ + μ + μ + δ + 1 2 ρ 2 S 2 + 1 2 ρ 2 P 2 .

By choosing c = γ + μ β , it follows

(12) L V ≤ Λ + c μ + γ + μ + μ + δ + 1 2 ρ 2 S 2 + 1 2 ρ 2 P 2 ≜ B .

Further proof follows from Ji et al. [28].□

4 Extinction

In this section, we investigate the condition for extinction of the spread of smoking. Here, we define

(13) 〈 y ( t ) 〉 = 1 t ∫ 0 t y ( s ) d s

and

(14) ℜ ˜ = β Λ μ 1 ( γ + μ ) + 1 2 ρ 2 Λ μ 2 .

A useful lemma concerned to this work is follows.

Lemma 4.1

[26] Let M = { M t } t ≥ 0 be a real value, continuous, local martingale and vanishing at t = 0 . Then

lim t → ∞ 〈 M , M 〉 t = ∞

a.s. implies that

lim t → ∞ M t 〈 M , M 〉 t = 0

and also

lim t → ∞ sup 〈 M , M 〉 t t < ∞ ⇒ lim t → ∞ M t t = 0 .

Theorem 4.1

Let ( P ( t ) , S ( t ) , Q ( t ) ) be the solution of system (1) with initial value ( P ( 0 ) , S ( 0 ) , Q ( 0 ) ) ∈ R + 3 . If

ρ 2 > max β 2 2 ( γ + δ + μ + α ) , β μ Λ or
R ˜ < 1 a n d ρ 2 ≤ β μ Λ .

Then

(15) lim t → ∞ sup log S ( t ) t ≤ − ( γ + μ ) + β 2 ρ 2 < 0 a . s . i f ( 1 ) h o l d s ;

(16) lim t → ∞ sup log S ( t ) t ≤ β Λ μ 1 − 1 ℜ ˜ < 0 a . s . i f ( 2 ) h o l d s .

In addition,

lim t → ∞ P ( t ) = Λ μ = P 0 , lim t → ∞ S ( t ) = 0 , lim t → ∞ Q ( t ) = 0 , a .s .

Proof

Taking integration of system (1)□

P ( t ) − P ( 0 ) t = Λ − β 〈 P ( t ) S ( t ) 〉 − μ 〈 P ( t ) 〉 + δ 〈 Q ( t ) 〉 − ρ P ( t ) S ( t ) d B ( t ) ,

S ( t ) − S ( 0 ) t = β 〈 P ( t ) S ( t ) 〉 − ( γ + μ ) 〈 S ( t ) 〉 + ρ P ( t ) S ( t ) d B ( t ) ,

Q ( t ) − Q ( 0 ) t = γ 〈 S ( t ) 〉 − ( μ + δ ) 〈 Q ( t ) 〉 ,

P ( t ) − P ( 0 ) t + S ( t ) − S ( 0 ) t + δ μ + δ Q ( t ) − Q ( 0 ) t = Λ − β 〈 P ( t ) S ( t ) 〉 − μ 〈 P ( t ) 〉 + δ 〈 Q ( t ) 〉 − ρ P ( t ) S ( t ) d B ( t ) + β 〈 P ( t ) S ( t ) 〉 − ( γ + μ ) 〈 S ( t ) 〉 + ρ P ( t ) S ( t ) d B ( t ) + γ 〈 S ( t ) 〉 − ( μ + δ ) 〈 Q ( t ) 〉 = Λ − μ 〈 P ( t ) 〉 − ( γ + μ ) − δ γ μ + δ 〈 S ( t ) 〉 = Λ − μ 〈 P ( t ) 〉 − ( γ + μ ) ( μ + δ ) − γ δ μ + δ 〈 S ( t ) 〉 ,

〈 P ( t ) 〉 = − 1 μ P ( t ) − P ( 0 ) t + S ( t ) − S ( 0 ) t + δ μ + δ Q ( t ) − Q ( 0 ) t + Λ μ − 1 μ ( γ + μ ) ( μ + δ ) − γ δ μ + δ 〈 S ( t ) 〉 .

By applying lim t → 0

(17) 〈 P ( t ) 〉 = Λ μ − 1 μ ( γ + μ ) ( μ + δ ) − γ δ μ + δ 〈 S ( t ) 〉 ,

(18) d log S ( t ) = β P − ( γ + μ ) − 1 2 ρ 2 P 2 d t + ρ P d B ( t ) ,

(19) log S ( t ) − log S ( 0 ) t = β 〈 P ( t ) 〉 − ( γ + μ ) − 1 2 ρ 2 〈 P ( t ) 2 〉 + ρ t ∫ 0 t P ( r ) d B ( r )

(20) ≤ β 〈 P ( t ) 〉 − ( γ + μ ) − 1 2 ρ 2 〈 P ( t ) 〉 2 + ρ t ∫ t 0 P ( r ) d B ( r ) .

By putting the value of 〈 P ( t ) 〉 from Eq. (17)

log S ( t ) − log S ( 0 ) t ≤ β Λ μ − 1 μ ( γ + μ ) ( μ + δ ) − γ δ μ + δ 〈 S ( t ) 〉 − ( γ + μ ) − 1 2 ρ 2 Λ μ − 1 μ ( γ + μ ) ( μ + δ ) − γ δ μ + δ 〈 S ( t ) 〉 2 + ρ t ∫ 0 t P ( r ) d B ( r ) = β Λ μ − γ + μ μ + δ 〈 S ( t ) 〉 − ( γ + μ ) − 1 2 ρ 2 Λ μ 2 − γ + μ μ + δ 2 〈 S ( t ) 〉 2 + 2 Λ μ γ + μ μ + δ 〈 S ( t ) 〉 + ρ t ∫ 0 t P ( r ) d B ( r ) = β Λ μ − ( γ + μ ) − 1 2 ρ 2 Λ μ 2 − β ( γ + μ ) μ + δ 〈 S ( t ) 〉 + 2 Λ μ γ + μ μ + δ 〈 S ( t ) 〉 − 1 2 ρ 2 − γ + μ μ + δ 2 〈 S ( t ) 〉 2 + ρ t ∫ 0 t P ( r ) d B ( r ) = β Λ μ − ( γ + μ ) + 1 2 ρ 2 Λ μ 2 − β ( γ + μ ) μ + δ 〈 S ( t ) 〉 + 2 Λ μ γ + μ μ + δ 〈 S ( t ) 〉 − 1 2 ρ 2 − γ + μ μ + δ 2 〈 S ( t ) 〉 2 + ρ t ∫ 0 t P ( r ) d B ( r ) = β Λ μ 1 − μ ( γ + μ ) + 1 2 ρ 2 Λ μ 2 β Λ − β ( γ + μ ) μ + δ 〈 S ( t ) 〉 + 2 Λ μ γ + μ μ + δ 〈 S ( t ) 〉 − 1 2 ρ 2 − γ + μ μ + δ 2 〈 S ( t ) 〉 2 + ρ t ∫ 0 t P ( r ) d B ( r ) = β Λ μ 1 − 1 R ˜ − β ( γ + μ ) μ + δ 〈 S ( t ) 〉 + 2 Λ μ γ + μ μ + δ 〈 S ( t ) 〉 − 1 2 ρ 2 − γ + μ μ + δ 2 〈 S ( t ) 〉 2 + ρ t ∫ 0 t P ( r ) d B ( r ) .

If condition (2) is satisfied, then

(21) lim t → ∞ sup log S ( t ) t ≤ β Λ μ 1 − 1 ℜ ˜ < 0 ,

conclusion (16) is proved. Next according to Eq. (20)

log S ( t ) − log S ( 0 ) t ≤ β 〈 P ( t ) 〉 − ( γ + μ ) − 1 2 ρ 2 〈 P ( t ) 〉 2 + ρ t ∫ 0 t P ( r ) d B ( r ) = − 1 2 ρ 2 〈 P ( t ) 〉 − β ρ 2 + β 2 ρ 2 − ( γ + μ ) + ρ t ∫ 0 t P ( r ) d B ( r ) .

If condition (1) is satisfied, then

(22) log S ( t ) t ≤ β 2 ρ 2 − ( γ + μ ) + ρ t ∫ 0 t P ( r ) d B ( r ) + log S ( 0 ) t ,

conclusion (15) is proved

lim t → ∞ log S ( t ) t ≤ − ( γ + μ ) + β 2 ρ 2 < 0 a .s .

According to (15) and (16)

(23) lim t → ∞ S ( t ) = 0 .

Now from the third equation of system (1), it follows that

(24) Q ( t ) = e − ( μ + δ ) t Q ( 0 ) + ∫ 0 t δ S ( r ) e ( μ + δ ) r d r .

By applying L’Hospital’s rule with previous result, we have

(25) lim t → ∞ Q ( t ) = 0 .

As from Eq. (4), it follows that

N ( t ) = e − μ t N ( 0 ) + Λ μ e μ t , P ( t ) + S ( t ) + Q ( t ) = P ( 0 ) + S ( 0 ) + Q ( 0 ) + Λ μ e μ t e μ t , lim t → ∞ P ( t ) = lim t → ∞ { P ( 0 ) + S ( 0 ) + Q ( 0 ) + Λ μ e μ t } e μ t − S ( t ) − Q ( t ) , lim t → ∞ P ( t ) = Λ μ .

Hence, the proof is complete.□

5 Persistence

This section concern the persistence of system (1).

Theorem 5.1

Suppose that μ > ρ 2 2 . Let ( P ( t ) , S ( t ) , Q ( t ) ) be any solution of model (1) with initial conditions ( P ( 0 ) , S ( 0 ) , Q ( 0 ) ) ∈ R + 3 . If R ˆ 0 > 1 , then

(26) lim t → ∞ 〈 P ( t ) 〉 = Λ μ − 1 μ ( γ + μ ) ( μ + δ ) − γ δ μ + δ 〈 S ( t ) 〉 = Λ μ − β Λ μ 1 − 1 R ˜ β − 2 Λ μ ,

(28) lim t → ∞ 〈 S ( t ) 〉 = β Λ μ 1 − 1 R ˜ γ + μ μ + δ β − 2 Λ μ ,

(29) lim t → ∞ 〈 Q ( t ) 〉 = γ γ + μ β Λ μ 1 − 1 R ˜ β − 2 Λ μ .

Proof

As we know that

log S ( t ) ) t ≤ β Λ μ 1 − 1 R ˜ − γ + μ μ + δ β − 2 Λ μ 〈 S ( t ) 〉 + ρ t ∫ 0 t P ( r ) d B ( r ) + log S ( 0 ) t .

By applying the limit

lim t → ∞ 〈 S ( t ) 〉 = β Λ μ 1 − 1 R ˜ γ + μ μ + δ β − 2 Λ μ .

Using Eq. (17) we have

lim t → ∞ 〈 P ( t ) 〉 = Λ μ − 1 μ ( γ + μ ) ( μ + δ ) − γ δ μ + δ lim t → ∞ 〈 S ( t ) 〉 = Λ μ − β Λ μ 1 − 1 R ˜ β − 2 Λ μ .

Further

Q ( t ) − Q ( 0 ) t = γ 〈 S ( t ) 〉 − ( μ + δ ) 〈 Q ( t ) 〉 .

By applying the limit t → ∞ , we have

lim t → ∞ 〈 Q ( t ) 〉 = γ μ + δ lim t → ∞ 〈 S ( t ) 〉 = γ μ + δ β Λ μ 1 − 1 R ˜ γ + μ μ + δ β − 2 Λ μ = γ γ + μ β Λ μ 1 − 1 R ˜ β − 2 Λ μ .

Hence, the proof is complete.□

6 Numerical Simulation

For the illustration of our obtained results, we use the values of parameters and variables given in Table 2.

Table 2

Values of variables and parameters for numerical solution

Variables and parameters	Values of variables and parameters
P ( t )	59
S ( t )	40
Q ( t )	30
Λ	0.08
β	0.003
μ	0.001
γ	0.02011
ρ	0.0075
δ	0.004

Now for the numerical simulation, we use Milstein’s higher order method [34]. The results obtained through this method are shown graphically in Figure 1 for both deterministic and stochastic forms.

$Figure 1 Graphs of (P) potential smokers using a deterministic method (green line) and from a stochastic solution (blue lines), (S) chain smokers using a deterministic method (green line) and from a stochastic solution (blue lines) and (Q) quit smokers using a deterministic method (green line) and from a stochastic solution (blue lines). The stability of stochastic graphs shows best expression than deterministic graphs.$

Figure 1

Graphs of (P) potential smokers using a deterministic method (green line) and from a stochastic solution (blue lines), (S) chain smokers using a deterministic method (green line) and from a stochastic solution (blue lines) and (Q) quit smokers using a deterministic method (green line) and from a stochastic solution (blue lines). The stability of stochastic graphs shows best expression than deterministic graphs.

7 Conclusion

In this work, a formulation of a stochastic mathematical smoking model is presented. The sufficient conditions are determined for extinction and persistence. Furthermore, the threshold of the proposed stochastic model is discussed, when noises are small or large. Finally, numerical simulations are shown graphically with the software MATLAB. The conclusions obtained are as follows: (a) if R 0 < 1 , then system (2) will be locally stable and will be unstable if R 0 ≥ 1 ; (b) if R 0 < 1 , then system (2) will be locally stable and will be unstable if R 0 ≥ 1 ; (c) for Λ = 0 , system (2) will be globally asymptotically stable; (d) smoking will be in control if R ˜ < 1 and ρ 2 ≤ β μ Λ means that white noise is not large; and (e) the value of R ˜ > 1 will lead to the prevalence of smoking.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. D1439-159-130. The authors, therefore, gratefully acknowledge DSR for technical and financial support.

Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this paper.
Authors contribution: The authors equally contributed to the preparation of this manuscript.
Funding: This article was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. D1439-159-130.
Availability of data and materials: The authors confirm that the data supporting the findings of this study are available within the article cited therein.

References

[1] E. A. Smith and R. E. Malone, “Everywhere the soldier will be”: wartime tobacco promotion in the US military, Am. J. Public Health 99 (2009), no. 9, 1595–1602, 10.2105/AJPH.2008.152983.Search in Google Scholar

[2] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, part 1, Proc. Roy. Soc. Edin. Sec. A. Math. 115 (1927), 700–721.Search in Google Scholar

[3] B. Shulgin, L. Stone, and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol. 60 (1998), 1123–1148.10.1016/S0092-8240(98)90005-2Search in Google Scholar

[4] G. Zaman, Y. H. Kang, and I. H. Jung, Stability analysis and optimal vaccination of an SIR epidemic model, BioSyst. 93 (2008), 240–249.10.1016/j.biosystems.2008.05.004Search in Google Scholar

[5] A. A. Lashari and G. Zaman, Global dynamics of vector-borne diseases with horizontal transmission in host population, Comp. Math. Appl. 61c (2011), 745–754.10.1016/j.camwa.2010.12.018Search in Google Scholar

[6] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci. 7 (1976), 335–356.10.1016/0025-5564(76)90132-2Search in Google Scholar

[7] C. Castillo-Garsow, G. Jordan-Salivia, and A. Rodriguez Herrera, Mathematical models for dynamics of tobacco use, recovery and relapse, Technical Report Series BU-1505-M, Cornell University, 2000.Search in Google Scholar

[8] X. K. Zhang, Z. Z. Zhang, J. Y. Tong, and M. Dong, Ergodicity of stochastic smoking model and parameter estimation, Adv. Difference Equ. 2016 (2016), 274, 10.1186/s13662-016-0997-x.Search in Google Scholar

[9] WHO global report on trends in prevalence of tobacco use 2000-2025, third edition, Geneva: World Health Organization, 2019, https://tech.sina.com.cn/roll/2019-12-20/doc-iihnzhfz7080832.shtml [Accessed on February 21, 2020]Search in Google Scholar

[10] O. Sharomi and A. B. Gumel, Curtailing smoking dynamics: A mathematical modeling approach, Appl. Math. Comput. 195 (2008), 475–499.10.1016/j.amc.2007.05.012Search in Google Scholar

[11] G. Zaman, Qualitative behavior of giving up smoking models, Bull. Malays. Sci. Soc. 34 (2011), 403–415.Search in Google Scholar

[12] A. Zeb, G. Zaman, and S. Momani, Square root dynamic of a giving up smoking model, Appl. Math. Model. 37 (2013), no. 7, 5326–5334.10.1016/j.apm.2012.10.005Search in Google Scholar

[13] O. K. Ham, Stages and processes of smoking cessation among adolescents, West J. Nurs. Res. 29 (2007), 301–315.10.1177/0193945906295528Search in Google Scholar PubMed

[14] Z. Alkhudhari, S. Al-Sheikh, and S. Al-Tuwairqi, Global dynamics of a mathematical model on smoking, ISRN Applied Mathematics 2014 (2014), 847075, 10.1155/2014/847075.Search in Google Scholar

[15] G. A. K. van Voorn and B. W. Kooi, Smoking epidemic eradication in a eco-epidemiological dynamical model, Ecol. Complex. 14 (2013), 180–189.10.1016/j.ecocom.2013.01.008Search in Google Scholar

[16] H. F. Huo and C. C. Zhu, Influence of relapse in a giving up smoking model, Abstr. Appl. Anal. 2013 (2013), 525461, 10.1155/2013/525461.Search in Google Scholar

[17] A. Zeb, A. Bano, E. Alzahrani, and G. Zaman, Dynamical analysis of cigarette smoking model with a saturated incidence rate, AIP Advances 8 (2018), no. 4, 045317, 10.1063/1.5028157.Search in Google Scholar

[18] A. Labzai, O. Balatif, and M. Rachik, Optimal control strategy for a discrete time smoking model with specific saturated incidence rate, Discrete Dyn. Nat. Soc. 2018 (2018), 5949303, 10.1155/2018/5949303.Search in Google Scholar

[19] Q. Din, M. Ozair, T. Hussain, and U. Saeed, Qualitative behavior of a smoking model, Adv. Difference Equ. 2016 (2016), 96, 10.1186/s13662-016-0830-6.Search in Google Scholar

[20] Z. Z. Zhang, R. B. Wei, and W. J. Xia, Dynamical analysis of a giving up smoking model with time delay, Adv. Difference Equ. 2019 (2019), 505, 10.1186/s13662-019-2450-4.Search in Google Scholar

[21] G. Rahman, R. P. Agarwal, and Q. Din, Mathematical analysis of giving up smoking model via harmonic mean type incidence rate, Appl. Math. Comput. 354 (2019), 128–148, 10.1016/j.amc.2019.01.053.Search in Google Scholar

[22] N. Dalal, D. Greenhalgh, and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl. 341 (2008), no. 2, 1084–1101, 10.1016/j.jmaa.2007.11.005.Search in Google Scholar

[23] E. Tornatore, S. M. Buccellato, and P. Vetro, On a stochastic disease model with vaccination, Rend. Circ. Mat. Palermo 55 (2006), 223–240, 10.1007/BF02874704.Search in Google Scholar

[24] E. Tornatore, P. Vetro, and S. M. Buccellato, SIVR epidemic model with stochastic perturbation, Neural. Comput. Appl. 24 (2014), 309–315, 10.1007/s00521-012-1225-6.Search in Google Scholar

[25] E. Tornatore, S. M. Buccellato, and P. Vetro, Stability of a stochastic SIR system, Phys. A 354 (2005), 111–126, 10.1016/j.physa.2005.02.057.Search in Google Scholar

[26] L. Zhu and H. X. Hu, A stochastic SIR epidemic model with density dependent birth rate, Adv. Difference Equ. 2015 (2015), 330, 10.1186/s13662-015-0669-2.Search in Google Scholar

[27] Y. Zhao, D. Jiang, and D. O’Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Phys. A 392 (2013), no. 20, 4916–4927, 10.1016/j.physa.2013.06.009.Search in Google Scholar

[28] C. Ji, D. Jiang, and N. Shi, The behavior of an SIR epidemic model with stochastic perturbation, Stoch. Anal. Appl. 30 (2012), no. 5, 755–773, 10.1080/07362994.2012.684319.Search in Google Scholar

[29] A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011), no. 3, 876–902, 10.1137/10081856X.Search in Google Scholar

[30] Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput. 243 (2014), 718–727, 10.1016/j.amc.2014.05.124.Search in Google Scholar

[31] Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett. 34 (2014), 90–93, 10.1016/j.aml.2013.11.002.Search in Google Scholar

[32] C. Ji, D. Jiang, and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Phys. A 390 (2011), no. 10, 1747–1762, 10.1016/j.physa.2010.12.042.Search in Google Scholar

[33] X. Mao, Stochastic Differential Equations and Applications, Ellis Horwood, Chichester, 1997.Search in Google Scholar

[34] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43 (2001), no. 3, 525–546, 10.1137/S0036144500378302.Search in Google Scholar

[35] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probab. Lett. 83 (2013), no. 4, 960–968, 10.1016/j.spl.2012.12.021.Search in Google Scholar

Received: 2020-01-22

Revised: 2020-04-02

Accepted: 2020-07-26

Published Online: 2020-09-26

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

Articles in the same Issue

https://doi.org/10.1515/math-2020-0068

Keywords for this article

stochastic smoking model; Itô’s formula; extinction; persistence; numerical analysis

Creative Commons

BY 4.0