Stability analysis and prevention strategies of tobacco smoking model

Abstract

This research work is related to a tobacco smoking model having a significance class of users of tobacco in the form of snuffing. For this purpose, the formulation of the model containing snuffing class is presented; then the equilibrium points as regards being smoking free and smoking positive are discussed. The Hurwitz theorem is used for finding the local stability of the model and Lyaponov function theory is used for the search of global stability. We use different controls for control of smoking and the Pontryagin maximum principle for characterization of the optimal level. For the solution of the proposed model, a nonstandard finite difference (NSFD) scheme and the Runge–Kutta fourth order method are used. Finally, some numerical results are presented for control and without control systems with the help of MATLAB.

1 Introduction

Mathematical biology is a wide field with many applications. In this field, researchers are focusing on the description of different types of diseases with controls in the form of mathematical models. In 1909, Brownlee [1] took the initiative for the development of mathematical biology. He focused on the theory of chance, further in 1912, he presented basic laws for epidemic spreading [2]. In 1927, the details of the epidemic study were discussed by Kermark and McKendrik [3]. Later, many researchers discussed different models of many other diseases; see [4–17]. On the other hand, one of the social habits that is spreading throughout the world rapidly as an infectious disease is smoking, causing many harmful diseases. Smoking is the process by which people inhale smoke of tobacco consisting of particles and gas or simply, smoking is the experience in which smoke is taken into mouth and then released using pipes or cigars. In the sixteenth century, Columbus introduced smoking to Europe [18], but before and after this date, many other exotic species were introduced, with great adverse impact on ecosystems and effect on human habits [19, 20]. Nicot spread tobacco as a cash crop in England, he was the first who used it like a business, and that is why the word nicotine derived from his name. The cigarette making machine was invented at the end of the 19th century and the capability of that machine was producing 200 cigarettes per minute and now cigarette production has increased up to 9000 cigarettes per minute. Smoking can cause different types of diseases including lung cancer, mouth cancer, throat cancer and many other diseases that are harmful to human health [21–30].

For the first time in 1997, Castillo-Garsow et al. [21] formulated a mathematical model for smoking. In this model, they divided the total population in three different classes (potential smokers, chain smokers and permanently quit smokers). In 2008, their model was modified by Sharomi and Gumel [22]. They introduced a new class (temporarily quit smokers). In 2007, Ham [23] identified the different stages and processes of smoking among students through a survey in different vocational technical schools in Korea. Zaman [24] extended the model by introducing a new category (occasional smokers) and presented a dynamical interaction in an integer order. Zeb et al. [28] derived the square root dynamics of a giving up smoking model for the purpose that the system goes to finite time extension. Several others presented the smoking models in integer and fractional order [21–30]. The use of tobacco also occurs in the form of snuffing. Till now, no one has discussed mathematically the snuffing class; by adding the snuffing class in this work, we divided the total population in five classes $X(t)$, $H_{1}(t)$, $H_{2}(t)$, $Y(t)$, $Z(t)$ representing the susceptible smokers, snuffing class, irregular smokers, regular smokers and quit smokers, respectively, at time t. First, we formulate the model according to given assumptions; then, by using the Hurwitz theorem, we find the local stability, and with the help of the Lyaponov function the global stability is discussed. For prevention strategies, the Pontryagin’s maximum principle is used. Finally, some numerical results are presented for control and without control system by using the nonstandard finite difference (NSFD) method and the Runge–Kutta fourth order method.

2 Formation of model

By adding the snuffing class, we divided the total population into five classes $X(t)$, $H_{1}(t)$, $H_{2}(t)$, $Y(t)$, $Z(t)$ standing for susceptible smokers, snuffing class, irregular smokers, regular smokers and quit smokers, respectively at time t. The model is given by

$$ \begin{gathered} \frac{dX}{dt} = \lambda-\beta_{1}XH_{1}- \mu X+\alpha Y, \\ \frac{dH_{1}}{dt} =\beta_{1}XH_{1}-\beta _{2}H_{1}H_{2}-(\rho+\mu)H_{1}, \\ \frac{dH_{2}}{dt} = \beta_{2} H_{1}H_{2}-(d+ \omega+\mu)H_{2}, \\ \frac{dY}{dt}=\omega H_{2}-(\alpha+\gamma+\mu)Y, \\ \frac{dZ}{dt}=\gamma Y-\mu Z,\end{gathered} $$

(1)

where the parameters used in this model are described in Table 1.

Table 1 Parameters and description

Since the first four equations of system (1) are independent of $Z(t)$, without loss of generality, we omit this one and then the system (1) is reduced to the following:

$$\begin{aligned}& \frac{dX}{dt} = \lambda-\beta_{1}XH_{1}- \mu X+\alpha Y, \\ & \frac{dH_{1}}{dt} =\beta_{1}XH_{1}-\beta_{2}H_{1}H_{2}-( \rho+\mu )H_{1}, \\& \frac{dH_{2}}{dt} = \beta_{2} H_{1}H_{2}-(d+ \omega+\mu )H_{2}, \\& \frac{dY}{dt}=\omega H_{2}-(\alpha+\gamma+\mu)Y. \end{aligned}$$

(2)

3 Equilibrium points

3.1 Smoking free equilibrium point

For the smoking free equilibrium point $E_{0}$ we use $H_{1}=H_{2}=Y=0$ in system (2).

So the smoking free equilibrium point $E_{0}$ is

$$E_{0}= \biggl(\frac{\lambda}{\mu},0,0,0 \biggr). $$

The Jacobian of system (2) is given by

$$ J=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\beta_{1} H_{1}-\mu& -\beta_{1}X & 0 & \alpha\\ \beta_{1}H_{1} & \beta_{1}X-\beta_{2}H_{2}-(\rho+\mu) & -\beta _{2}H_{1} & 0 \\ 0 & \beta_{2}H_{2} & \beta_{2}H_{1}-(d+\omega+\mu) & 0 \\ 0 & 0 & \omega& -(\alpha+\gamma+\mu) \end{array}\displaystyle \right ), $$

while the Jacobian at free equilibrium point $E_{0}$ is

$$ J(E_{0})=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\mu& \frac{-\beta_{1}\lambda}{\mu} & 0 & \alpha\\ 0 & \frac{\beta_{1}\lambda}{\mu}-(\rho+\mu) & 0 & 0 \\ 0 & 0 & -(d+\omega+\mu) & 0 \\ 0 & 0 & \omega& -(\alpha+\gamma+\mu) \end{array}\displaystyle \right ). $$

For the reproductive number, we consider the following matrices:

$$\begin{aligned}& F=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} \frac{\beta_{1}\lambda}{\mu} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ), \\& V=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} (\rho+\mu) & 0 & 0 \\ 0 & (d+\omega+\mu) & 0 \\ 0 & -\omega& (\alpha+\gamma+\mu) \end{array}\displaystyle \right ). \end{aligned}$$

The dominant eigenvalue of $FV^{-1}$ is $\frac{\beta_{1}\lambda}{\mu (\rho+\mu)}$, so

$$ R_{0}=\frac{\beta_{1}\lambda}{\mu(\rho+\mu)} $$

(3)

is the required reproductive number [13].

3.2 Smoking present equilibrium point

Theorem 3.1

For$R_{0}>1$, there exists a positive smoking equilibrium point$E^{*}$.

Proof

For smoking present the equilibrium $E^{*}$ using the left side of system (2) is equal to zero, as follows.

The third equation of system (2) implies that

$$ H_{1}^{*}= \frac{(d+\omega+\mu)}{\beta_{2}}, $$

from the second equation of system (2), we have

$$ X^{*}=\frac{\beta_{2}H^{*}_{2}+(\rho+\mu)}{\beta_{1}}, $$

the fourth equation implies that

$$ Y^{*}=\frac{\omega H_{2}^{*}}{(\alpha+\gamma+\mu)}, $$

similarly, the first equation reveals that

$$ X^{*}=\frac{\lambda-\alpha Y^{*}}{\beta_{1} H^{*}_{1}+\mu}. $$

Now, by comparing the values of $X^{*}$ in terms of $H^{*}_{1}$ and $H^{*}_{2}$ we find that

$$ H^{*}_{2}=\frac{(\alpha+\gamma+\mu)(\rho+\mu) [\beta_{2}\mu (R_{0}-1)-\beta_{1}(d+\omega+\mu) ]}{ (\gamma+\mu)(\beta_{1}\beta_{2}\omega)+(\alpha+\gamma+\mu )(\beta_{1}\beta_{2}(d+\mu)+\beta^{2}_{2}\mu)}. $$

We have $\beta_{2}\mu(R_{0}-1)>\beta_{1}(d+\omega+\mu)$ for $R_{0}>1$. Thus, $H^{*}_{2}$ is positive if $R_{0}>1$. So the required positive equilibrium point $E^{*}$ is

$$\begin{aligned} E^{*}\bigl(X^{*},H^{*}_{1},H^{*}_{2},Y^{*} \bigr)={} &\biggl(\frac {\beta_{2}H^{*}_{2}+(\rho+\mu)}{\beta_{1}},\frac{(d+\omega+\mu)}{ \beta_{2}},\frac{\omega H_{2}^{*}}{(\alpha+\gamma+\mu)},\\& \frac {(\alpha+\gamma+\mu)(\rho+\mu)[\beta_{2}\mu(R_{0}-1)-\beta _{1}(d+\omega+\mu)]}{ (\gamma+\mu)(\beta_{1}\beta_{2}\omega)+(\alpha+\gamma+\mu )(\beta_{1}\beta_{2}(d+\mu)+\beta^{2}_{2}\mu)} \biggr).\end{aligned} $$

 □

4 Stability of the model

4.1 Local stability

Theorem 4.1

If$R_{0}<1$, then the system (2) is locally stable and if$R_{0}>1$, then system (2) is unstable.

Proof

For local stability at $E_{0}$, the Jacobian of system (2) is

$$ J(E_{0})=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\mu& \frac{-\beta_{1}\lambda}{\mu} & 0 & \alpha\\ 0 & \frac{\beta_{1}\lambda}{\mu}-(\rho+\mu) & 0 & 0 \\ 0 & 0 & -(d+\omega+\mu) & 0 \\ 0 & 0 & \omega& -(\alpha+\gamma+\mu) \end{array}\displaystyle \right ), $$

from which follow the eigenvalues $\lambda_{1}$, $\lambda_{2}$, $\lambda _{3}$ and $\lambda_{4}$,

$$\begin{aligned}& \lambda_{1}=-\mu< 0, \\& \lambda_{3}=-(d+\omega+\mu)< 0, \\& \lambda_{4}=-(\alpha+\gamma+\mu)< 0, \\& \lambda_{2}=(\rho+\mu) (R_{0}-1), \end{aligned}$$

implying that $\lambda_{2}<0$ for $R_{0}<1$, $\lambda_{2}=0$ for $R_{0}=1$ and $\lambda_{2}>0$ for $R_{0}>1$. □

Theorem 4.2

If$R_{0}>\frac{\beta_{2}\lambda}{(d+\omega+\mu)(\rho+\mu)}$, then the system (2) is locally stable at$E^{*}$, otherwise unstable.

Proof

For local stability at $E^{*}$ the Jacobian of system (2) is

$$\begin{aligned}& J \bigl(E^{*} \bigr)=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\beta_{1} H^{*}_{1}-\mu& -\beta_{1}X^{*} & 0 & \alpha\\ \beta_{1}H^{*}_{1} & \beta_{1}X^{*}-\beta_{2}H^{*}_{2}-(\rho+\mu) & -\beta_{2}H^{*}_{1} & 0 \\ 0 & \beta_{2}H^{*}_{2} & \beta_{2}H^{*}_{1}-(d+\omega+\mu) & 0 \\ 0 & 0 & \omega& -(\alpha+\gamma+\mu) \end{array}\displaystyle \right ), \\& \begin{aligned}J \bigl(E^{*} \bigr)&=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\beta_{1} H^{*}_{1}-\mu& -\beta_{1}X^{*} & 0 & \alpha\\ \beta_{1}H^{*}_{1} & 0 & -\beta_{2}H^{*}_{1} & 0 \\ 0 & \beta_{2}H^{*}_{2} & 0 & 0 \\ 0 & 0 & \omega& -(\alpha+\gamma+\mu) \end{array}\displaystyle \right ) \\ &=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\beta_{1} H^{*}_{1}-\mu& -\beta_{1}X^{*} & 0 & \alpha\\ -\mu& -\beta_{1}X^{*} & -\beta_{2}H^{*}_{1} &\alpha\\ \beta_{1} H^{*}_{1} & \beta_{2}H^{*}_{2} & -\beta_{2}H^{*}_{1} & 0 \\ 0 & 0 & \omega& -(\alpha+\gamma+\mu) \end{array}\displaystyle \right ) \\ &=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\beta_{1} H^{*}_{1}-\mu& -\beta_{1}X^{*} & 0 & \alpha\\ 0& -\beta_{1}X^{*}+\frac{\mu\beta_{2}H^{*}_{2}}{\beta_{1} H^{*}_{1}} & -\beta_{2}H^{*}_{1}-\frac{\mu\beta_{2}H^{*}_{1}}{\beta _{1} H^{*}_{1} } &\alpha\\ \beta_{1} H^{*}_{1} & \beta_{2}H^{*}_{2} & -\beta_{2}H^{*}_{1} & 0 \\ 0 & 0 & \omega& -(\alpha+\gamma+\mu) \end{array}\displaystyle \right ),\end{aligned} \\& J \bigl(E^{*} \bigr)=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\beta_{1} H^{*}_{1}-\mu& -\beta_{1}X^{*} & \frac{\omega\alpha }{(\alpha+\gamma+\mu)} & 0 \\ [3pt] 0& -\beta_{1}X^{*}+\frac{\mu\beta_{2}H^{*}_{2}}{\beta_{1} H^{*}_{1}} & -\beta_{2}H^{*}_{1}-\frac{\mu\beta_{2}H^{*}_{1}}{\beta _{1} H^{*}_{1}}+\frac{\omega\alpha}{(\alpha+\gamma+\mu)} & 0 \\ \beta_{1} H^{*}_{1} & \beta_{2}H^{*}_{2} & -\beta_{2}H^{*}_{1} & 0 \\ 0 & 0 & \omega& -(\alpha+\gamma+\mu) \end{array}\displaystyle \right ). \end{aligned}$$

For simplification, this matrix can also be written as

$$ J \bigl(E^{*} \bigr)= \left ( \textstyle\begin{array}{c@{\quad}c} A & B \\ C & D \end{array}\displaystyle \right ). $$

Here,

$$\begin{gathered} A=\left ( \textstyle\begin{array}{c@{\quad}c} -\beta_{1} H^{*}_{1}-\mu& -\beta_{1}X^{*}\\ 0& -\beta_{1}X^{*}+\frac{\mu\beta_{2}H^{*}_{2}}{\beta_{1} H^{*}_{1}} \end{array}\displaystyle \right ) ,\qquad B=\left ( \textstyle\begin{array}{c@{\quad}c} \frac{\omega\alpha}{(\alpha+\gamma+\mu)} & 0\\ -\beta_{2}H^{*}_{1}-\frac{\mu\beta_{2}H^{*}_{1}}{\beta_{1} H^{*}_{1}}+\frac{\omega\alpha}{(\alpha+\gamma+\mu)}&0 \end{array}\displaystyle \right ) ,\\ C=\left ( \textstyle\begin{array}{c@{\quad}c} \beta_{1} H^{*}_{1} & \beta_{2}H^{*}_{2}\\ 0&0 \end{array}\displaystyle \right ), \qquad D=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}} -\beta_{2} H_{1} & 0\\ \omega& (\alpha+\gamma+\mu) \end{array}\displaystyle S \right ).\end{gathered} $$

Since the eigenvalues of $J(E^{*})$ depend on the eigenvalues of A and D, the eigenvalues of A are given as follows:

$$\begin{aligned}& \lambda_{1} =-\beta_{1}H^{*}_{1}- \mu< 0, \\& \lambda_{2} =-(\rho+\mu)+\frac{\mu\beta_{2}H^{*}_{2}}{\beta _{1}H^{*}_{1}\lambda} \bigl( \lambda-H^{*}_{1}(\rho+\mu)R_{0} \bigr), \end{aligned}$$

if $R_{0}>\frac{\beta_{2}\lambda}{(d+\omega+\mu)(\rho+\mu)}$, then $\lambda_{2}<0$. Now, the eigenvalues of D are

$$\begin{aligned}& \lambda_{3} =-\beta_{2}H^{*}_{1}< 0, \\& \lambda_{4} =-(\alpha+\gamma+\mu)< 0, \end{aligned}$$

which is the required proof. □

4.2 Global stability

Theorem 4.3

If$R_{0}<1$, then the system (2) is globally stable.

Proof

For the proof of this theorem, first we construct the Lyapunov function L as

$$ L=\ln\frac{X}{X_{0}}+\ln\frac{H_{1}}{H_{1_{0}}}+H_{2}+Y. $$

(4)

Differentiating Eq. (4) with respect to time

$$ L'=\frac{\lambda}{X}-\beta_{1}H_{1}+ \frac{\alpha Y}{X}-\mu+\beta _{X}-\beta_{2}H_{2}-( \rho+\mu)-(d+\omega+\mu)H_{2}, $$

using the values of $E_{0}$ in the above equation,

$$\begin{aligned}& L' = \mu-\mu+\frac{\beta_{1}\lambda}{\mu}-(\rho+\mu), \\& L' = R_{0}(\rho+\mu)-(\rho+\mu), \\& L' = (\rho+\mu) (R_{0}-1), \end{aligned}$$

therefore, if $R_{0}<1$, then $L'<0$, which implies that the system (2) is globally stable for $R_{0}<1$. □

5 Numerical method

The NSFD method is used for the numerical solution of the proposed model (1). Basically, NSFD is an iterative method in which we get closer to the solution through iteration [31, 32]. Let the nonstandard ODEs be given by

$$y'_{k}=f[t,y_{1},y_{2}, \ldots,y_{n}], $$

where $k=1,2,\ldots,n$, then by the NFSD method

$$\begin{aligned}& y'_{1}=\frac{y_{1,k+1}-y_{1,k}}{h}, \\& y'_{2}=\frac{y_{2,k+1}-y_{2,k}}{h}, \\& \vdots \\& y'_{n}=\frac{y_{n,k+1}-y_{n,k}}{h}. \end{aligned}$$

Now, using the NSFD method for the numerical solution of system (1) it follows that

$$\begin{aligned}& X_{k+1}=\frac{h(\lambda+\alpha Y_{k})+X_{k}}{1+h(\beta_{1}H_{1,k}+\mu)}, \end{aligned}$$

(5)

$$\begin{aligned}& H_{1,(k+1)}=\frac{H_{1,k}}{1+h(-\beta_{1}X_{k}+\beta _{2}H_{2,k}+(\rho+\mu))}, \end{aligned}$$

(6)

$$\begin{aligned}& H_{2,(k+1)}=\frac{H_{2,k}}{1+h(-\beta_{2}H_{1,k}+(d+\omega+\mu))}, \end{aligned}$$

(7)

$$\begin{aligned}& Y_{k+1}=\frac{h\omega H_{2,k}+Y_{k}}{1+h(\alpha+\gamma+\mu)}, \end{aligned}$$

(8)

and

$$ Z_{k+1}=\frac{h\gamma Y_{k}+Z_{k}}{1+h\mu}. $$

(9)

6 Summary and simulation

In this section, we give approximate values to the parameter of system (1) in Table 2 and with the help of MATLAB we draw the graph of model (1).

Table 2 Values of parameters for numerical solution

Figure 1 shows the result of system (1) graphically. In these graphs, we used the NSFD and RK4 methods. According to these figures, the population of each class gradually decreases, while the population of quit smokers increases gradually.

Figure 1

Plots present the susceptible, irregular, regular and quit smokers

7 Control strategies

For reducing the ratio of smokers to non-smokers in the world, we apply the optimal control scheme on the system (2) presented in this section in a similar way to that used by many authors for different diseases and smoking [33–36]. For this purpose, two control variables $u_{1}$ and $u_{2}$ representing education campaign and anti-nicotine gum/medicine, respectively, are used and by utilizing the Pontryagin maximum principle for the control strategies. Finally, we will show graphically both the systems with control and without control. Using these control variables on system (1), we have

$$ \begin{gathered} \frac{dX}{dt} = \lambda-\beta_{1}XH_{1}- \mu X+\alpha Y, \\ \frac{dH_{1}}{dt} =\beta_{1}XH_{1}-\beta _{2}H_{1}H_{2}-(\rho+\mu)H_{1}+u_{2}Y, \\ \frac{dH_{2}}{dt} = \beta_{2} H_{1}H_{2}-(d+ \omega+\mu )H_{2}-u_{1}H_{2}, \\ \frac{dY}{dt}=\omega H_{2}-(\alpha+\gamma+\mu )Y-u_{2}Y, \\ \frac{dZ}{dt}=\gamma Y-\mu Z+u_{1}H_{2}.\end{gathered} $$

(10)

Now, we construct the objective function for the system (10), which is given by

$$ J(u_{1},u_{2})= \int^{t_{f}}_{0} \biggl[X(t)+H_{1}(t)+H_{2}(t)+Y(t)+Z(t)+ \frac{c_{1}u^{2}_{1}(t)}{2}+\frac {c_{2}u^{2}_{2}(t)}{2} \biggr]\,dt, $$

with initial conditions

$$\begin{gathered} X(0)=X^{0}, \\ H_{1}(0)=H^{0}_{1},\qquad H_{2}(0)=H^{0}_{2},\qquad Y(0)=Y^{0} \quad\mbox{and}\quad Z(0)=Z^{0}.\end{gathered} $$

Now, the Hamiltonian function is defined as

$$\begin{aligned} H =&X(t)+H_{1}(t)+H_{2}(t)+Y(t)+Z(t)+ \frac {c_{1}u^{2}_{1}(t)}{2}+\frac{c_{2}u^{2}_{2}}{2} \\ &{}+ \lambda_{1}[\lambda-\beta_{1}XH_{1}-\mu X+ \alpha Y] \\ &{}+\lambda_{2} \bigl[\beta_{1}XH_{1}- \beta_{2}H_{1}H_{2}-(\rho+\mu )H_{1}+u_{2}Y \bigr] \\ &{}+\lambda_{3} \bigl[\beta_{2} H_{1}H_{2}-(d+ \omega+\mu )H_{2}-u_{1}H_{2} \bigr] \\ &{}+\lambda_{4} \bigl[\omega H_{2}-(\alpha+\gamma+ \mu)Y-u_{2}Y \bigr] \\ &{}+\lambda_{5}[\gamma-\mu Z+u_{1}H_{2}]. \end{aligned}$$

Theorem 7.1

The system (10) satisfies the terminal conditions

$$u^{*}_{1}=\min \biggl(1,\max \biggl(0,\frac{(\lambda_{3}-\lambda _{5})H_{2}}{c_{1}} \biggr) \biggr)\quad \textit{and}\quad u^{*}_{2}=\min \biggl(1,\max \biggl(0, \frac{(\lambda_{4}-\lambda _{2})Y}{c_{2}} \biggr) \biggr). $$

Proof

According to the Pontryagin maximum principle for the control of smoking, put

$$\frac{dX}{dt}=\frac{\partial H}{\partial\lambda},\qquad \frac{\partial H}{\partial U}=0, $$

and

$$\lambda'_{1}=\frac{-\partial H}{\partial X}, \qquad\lambda'_{2}= \frac {-\partial H}{\partial H _{1}}, \qquad\lambda'_{3}=\frac{-\partial H}{\partial H_{2}},\qquad \lambda'_{4}=\frac{-\partial H}{\partial Y }, \qquad\lambda'_{5}= \frac{-\partial H}{\partial Z}. $$

Differentiating the Hamiltonian function with respect to X, $H_{1}$, $H_{2}$, Y and Z, respectively, we get the values of $\lambda '_{1}$, $\lambda'_{2}$, $\lambda'_{3}$, $\lambda'_{4}$, and $\lambda '_{5}$ in the form of

$$\begin{aligned}& \begin{aligned}\lambda'_{1}&=-(1-\lambda_{1} \beta_{1}H_{1}-\lambda_{1}\mu +\lambda_{2} \beta_{1}H_{1}) \\ &=-1+\lambda_{1}\beta_{1}H_{1}+ \lambda_{1}\mu-\lambda_{2}\beta _{1}H_{1},\end{aligned} \\& \begin{aligned}\lambda'_{2}&=- \bigl(1-\lambda_{1} \beta_{1}X+\lambda_{2}\beta _{1}X- \lambda_{2}\beta_{2}H_{2}-\lambda_{2}( \rho+\mu)+\lambda _{3}\beta_{2}H_{2} \bigr) \\ &=-1+\lambda_{1}\beta_{1}X-\lambda_{2} \beta_{1}X+\lambda_{2}\beta _{2}H_{2}+ \lambda_{2}(\rho+\mu)-\lambda_{3}\beta_{2}H_{2},\end{aligned} \\& \begin{aligned}\lambda'_{3}&=- \bigl(1-\lambda_{2} \beta_{2}H_{1}+\lambda_{3}\beta _{2}H_{1}- \lambda_{3}(d+\omega+\mu)-\lambda_{3}u_{1}+\lambda _{5}u_{1} \bigr) \\ &=-1+\lambda_{2}\beta_{2}H_{1}- \lambda_{3}\beta_{2}H_{1}+\lambda _{3}(d+ \omega+\mu)+\lambda_{3}u_{1}-\lambda_{5}u_{1},\end{aligned} \\& \begin{aligned}\lambda'_{4}&=- \bigl(1+\lambda_{1}\alpha+ \lambda_{2}u_{2}-\lambda _{4}(\alpha+\gamma+\mu)- \lambda_{4}u_{2}+\lambda_{5}\gamma \bigr) \\ &=-1-\lambda_{1}\alpha-\lambda_{2}u_{2}+ \lambda_{4}(\alpha+\gamma +\mu)+\lambda_{4}u_{2}- \lambda_{5}\gamma,\end{aligned} \\& \begin{aligned}\lambda'_{5}&=-(1-\lambda_{5}\mu) \\ &=-1+\lambda_{5}\mu.\end{aligned} \end{aligned}$$

For $u^{*}_{1}$ and $u^{*}_{2}$ differentiate the Hamiltonian function with respect to $u_{1}$ and $u_{2}$, respectively, and we have

$$\begin{aligned}& \frac{\partial H}{\partial u_{1}}=0, \\& c_{1}u^{*}_{1}-\lambda_{3}H_{2}+ \lambda_{5}H_{2}=0, \\& c_{1}u^{*}_{1}=\lambda_{3}H_{2}- \lambda_{5}H_{2}, \\& u^{*}_{1}=\frac{(\lambda_{3}-\lambda_{5})H_{2}}{c_{1}}, \\& \frac{\partial H}{\partial u_{2}}=0, \\& c_{2}u^{*}_{2}+\lambda_{2}Y- \lambda_{4}Y=0, \\& c_{2}u^{*}_{2}=\lambda_{4}Y- \lambda_{2}Y, \\& u^{*}_{2}=\frac{(\lambda_{4}-\lambda_{2})Y}{c_{2}}. \end{aligned}$$

The optimality conditions are given as follows:

$$\begin{aligned} u^{*}_{1}=\min \biggl(1,\max \biggl(0,\frac{(\lambda_{3}-\lambda _{5})S_{2}}{c_{1}} \biggr) \biggr), \\ u^{*}_{2}=\min \biggl(1,\max \biggl(0,\frac{(\lambda_{4}-\lambda _{2})Y}{c_{2}} \biggr) \biggr).\end{aligned} $$

The terminal conditions for the system (10) are given as follows:

$$\begin{aligned}& X' = \lambda-\beta_{1}XH_{1}-\mu X+\alpha Y, \\& H'_{1} =\beta_{1}XH_{1}- \beta_{2}H_{1}H_{2}-(\rho+\mu )H_{1}+\min \biggl(1,\max \biggl(0,\frac{(\lambda_{4}-\lambda _{2})Y}{c_{2}} \biggr) \biggr)Y, \\& H'_{2}= \beta_{2} H_{1}H_{2}-(d+ \omega+\mu)H_{2}-\min \biggl(1,\max \biggl(0,\frac{(\lambda_{3}-\lambda_{5})H_{2}}{c_{1}} \biggr) \biggr)H_{2}, \\& Y'=\omega H_{2}-(\alpha+\gamma+\mu)Y-\min \biggl(1,\max \biggl(0,\frac {(\lambda_{4}-\lambda_{2})Y}{c_{2}} \biggr) \biggr)Y, \\& Z'=\gamma Y-\mu Z+\min \biggl(1,\max \biggl(0,\frac{(\lambda_{3}-\lambda _{5})H_{2}}{c_{1}} \biggr) \biggr)H_{2}, \\& \lambda'_{1}=-1+\lambda_{1} \beta_{1}H_{1}+\lambda_{1}\mu-\lambda _{2} \beta_{1}H_{1}, \\& \lambda'_{2}=-1+\lambda_{1} \beta_{1}X-\lambda_{2}\beta _{1}X+ \lambda_{2}\beta_{2}H_{2}+\lambda_{2}( \rho+\mu)-\lambda _{3}\beta_{2}H_{2}, \\& \lambda'_{3}=-1+\lambda_{2} \beta_{2}H_{1}-\lambda_{3}\beta _{2}H_{1}+ \lambda_{3}(d+\omega+\mu)-\lambda_{3}\min \biggl(1,\max \biggl(0, \frac{(\lambda_{3}\lambda_{5})H_{2}}{c_{1}} \biggr) \biggr), \\& \lambda'_{4}=-1-\lambda_{1}\alpha- \lambda_{2}\min \biggl(1,\max \biggl(0,\frac{(\lambda_{4}-\lambda_{2})Y}{c_{2}} \biggr) \biggr) \\& \phantom{\lambda'_{4}=}+\lambda_{4}(\alpha+\gamma+\mu)+\lambda_{4}\min \biggl(1,\max \biggl(0,\frac{(\lambda_{4}-\lambda_{2})Y}{c_{2}} \biggr) \biggr)-\lambda _{5} \gamma, \\& \lambda'_{5}=-1+\lambda_{5}\mu. \end{aligned}$$

 □

8 Numerical solution

This section is concerned with the investigation of a numerical solution of the smoking model with controls $u_{1}$ and $u_{2}$. The NSFD method is used for this purpose and system (10) is presented graphically by using the values of parameters given in Table 2 with $u_{1}=0.7$ and $u_{2}=0.9$. Figure 2 shows the results for both systems with and without control.

Figure 2

Plots present the susceptible, irregular, regular and quit smokers with and without control

9 Conclusion

In this work, the formulation of a model containing a snuffing class is presented; then the equilibrium points that are smoking free and smoking positive are discussed. The Hurwitz theorem is used for finding the local stability of the model and Lyaponov function theory is used for the search of global stability. For control of smoking we use different controls and for a characterization of the optimal level we use the Pontryagin maximum principle. For the solution of the proposed model, a nonstandard finite difference (NSFD) scheme and the Runge–Kutta fourth order method are used. Finally, some numerical results are presented for systems with and without control and by using the nonstandard finite difference (NSFD) method and Runge–Kutta fourth order method with the help of MATLAB.