Detectable sensation of a stochastic smoking model

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:

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

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

and

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:

Also, we have

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

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

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

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

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

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

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

By applying Itô’s formula, we obtain that

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

By choosing c = γ + μ β , it follows

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

and

A useful lemma concerned to this work is follows.

a.s. implies that

and also

Then

In addition,

By applying lim t → 0

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

If condition (2) is satisfied, then

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

If condition (1) is satisfied, then

conclusion (15) is proved

According to (15) and (16)

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

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

As from Eq. (4), it follows that

Hence, the proof is complete.□

5 Persistence

This section concern the persistence of system (1).

6 Numerical Simulation

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

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.

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.