In this manuscript, we solve the ordinary model of nonlinear smoking mathematically by using the second kind of shifted Chebyshev polynomials. The stability of the equilibrium point is calculated. The schematic of the model illustrates our proposition. We discuss the optimal control of this model, and formularize the optimal control smoking work through the necessary optimality cases. A numerical technique for the simulation of the control problem is adopted. Moreover, a numerical method is presented, and its stability analysis discussed. Numerical simulation then demonstrates our idea. Optimal control for the model is further discussed by clarifying the optimal control through drawing before and after control. Fractional request differential equations (FDEs) are usually used to display frameworks that have memory and exist in a few thermoelasticity models and organic standards. FDEs show the realistic biphasic decline of infection of diseases but at a slower rate. FDEs are more suitable than integer order ones in modeling complex systems, such as biological systems.

Infectious diseases have a tremendous influence on human life. Every year billions of people suffer from or die due to various infectious diseases. Mathematical modeling is of considerable importance in epidemiology because it may provide understanding of the underlying mechanisms that influence the spread of a disease and thus may be used to offer control strategies. Many scientists explored the detection of illnesses and pests [

In the present manuscript, we discuss the analytical solutions for alpha = 1, for the following order of smoking models [

The initial conditions are as follows:

In

T(t) | Total population with time |

P(t) | Probable smokers |

L(t) | Occasional smokers |

S(t) | Heavy smokers |

Q(t) | Temporary quitters |

R(t) | Smokers who quit permanently |

Rate of natural death | |

b | Contact rate between probable and occasional smokers |

c | Contact rate between occasional and smokers heavy |

d | Average number of [x] who quit smoking |

e | The fraction of remaining smokers |

f | Contact rate between smokers and temporary quitters |

The organization of the remainder of this paper is as follows. The stability of equilibrium points is studied in Section 2. Optimal control for the smoking model is discussed in Section 3. In Section 4, the second kind of Chebyshev polynomials and their properties are given. In Section 5, the numerical implementation is given. Finally, a conclusion is presented in Section 6.

For the stability behavior of this model at E0=(1, 0, 0, 0, 0), we use transformations [

U = 1-P, P = 1-U, L=L, S=S, Q=Q, R=R,

U*= −P*, L*=L*, S*=S*, Q*=Q*, R*=R*.

It follows that

The system

P1=(0, 0, 0, 0, 0), P2=(U*, L*, S*, Q*, R*), P3=(U**, L**, S**, Q**, R**),

P4=(U***, L***, S***, Q***, R***) and P5=(U****, L****, S****, Q****, R****),

where, U*=1-P*, U**=1-P**, U***=1-P*** and U****=1-P****.

Through Taylor approximation, the linearized form of the model is

where X=U-0, Y=L-0, Z=S-0, V=Q-0, W=R-0, and

The stability of

Consider the state presented (1.2), in R^{5}, with control functions admissible [

where T_{f} is the final time, and u_{V}(.) and u_{R}(.) are functions controls.

The objective function is defined as

where A, B, and C represent the number of occasional smokers, the rate of contact between smokers and quitters, and temporarily who regain support to smoking, and rate of smoking quitting.

We minimize the objective function as follows [

which is subjected to the constraint

The following initial conditions are satisfied:

OCP is defined, and we consider the following modified objective (cost) function:

where the Hamiltonian and control smoking objective functions are defined as

From

where λ_{k}, k = 1, 2, 3, 4, 5 are Lagrange multipliers.

We construct a theorem similar to that presented in [

If u_{V} and u_{R} are optimal controls with states corresponding to

Co-state equation

With condition transversality:

Optimality conditions

Therefore, the function controls

For more on optimal controls for solving models, see [

Chebyshev polynomials Z_{n}(y) of the second kind are rectangular polynomials of stage n in x presented on the interval [−1, 1] [

Polynomials have rectangular with rating to the products indoor

where

Z_{n}(y) can be generated by recurring relations

Z_{n}(y) = 2_{n−1}(y) − Z_{n−2}(y), n = 2, 3, …, n, with Z_{0}(y) = 1. Z_{1}(y) = 2

The analytical form Z_{n}(y) of stage n is given by [

On the interval y ∈ [0, 1],

The following inner product is orthogonal on the interval [0, 1]:

with weight function

The following formula represents the analytical

The solution of this model can be written as

Let g(y) be a square integral in [0, 1], and then the second kind of shifted Chebyshev polynomials can be represented as follows:

The coefficients a_{j}, j = 0, 1, …, are expressed as

or

We use only the (r + 1) terms. Then,

Using the practice to construct an integral collocation style then

By integrating

From

We presently register ^{(r)}, Ω^{(r−1)}, …, Ω^{(0)} are combined matrices.

In this section, we present the model's implementation through the following steps:

We first approximate the function using

where

Then, the nonlinear smoking model

Now, we collocate _{y}, y = 0 − 5; as follows:

The roots of shifted Chebyshev polynomial

(ii) By putting the initial

(iii) We use Newton's iteration to resolve the system and solve for the unknowns.

In this manuscript, a mathematical nonlinear smoking model is studied. The optimal control of this smoking model is discussed. The stability of the equilibrium point is calculated, and a schematic of the proposed model is presented. Moreover, the integral collocation style is used to obtain the approximate solutions of the model. ICM using the shifted Chebyshev polynomials of the second kind is a new technique for solving these problems. Under the application with the necessary optimality conditions, we have studied the problem control with numerical techniques for the simulation. Moreover, a numerical method and its stability is discussed.

The authors are grateful for Taif University. Taif University researchers support project number (TURSP-2020/160), Taif University, Taif, Saudi Arabia.