Optimal Control and Spectral Collocation Method for Solving Smoking Models

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.


Introduction
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 , wherein the idea that mathematical modeling helps in discovering diseases' spread [17]. In 1766, Islamic scholars presented for the first time what was considered the beginning of modern epidemiology discovery to be since followed. Model smoking is one of the most famous and important model commonly used by researchers [8][9][10][11][12][13][14], [18][19][20][21], [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. According to the World Health Organization [14], the global tobacco pandemic killed several million people. The cost of health care is increasing and its budgets are decreasing as developing economies are shrinking [17].
Smoking negatively affects overall health as well [1][2][3][4]. According to [1][2][3], more than five million individuals are killed every year from smoking tobacco, as such it is the main cause of preventable deaths. Serious illnesses caused by it include asthma, lung cancer, heart disease, and mouth ulcers. In [8][9], we discussed the smoking dynamics of the integer {of the integer what}. Moreover, [13] analyzed the above using the Homotopy analysis method. Four kinds of polynomials, Chebyshev [35], [57][58][59]: Chebyshev's model includes four kinds of polynomials. The outcomes of each and their applications can be found in several books. Moreover, many researchers use several of these polynomials in a single model [33,35]. However, the shifted Chebyshev polynomial of the second kind Z Ã n ðyÞ still needs further study [35]. The model in [56] has proven the equilibrium points.
In the present manuscript, we discuss the analytical solutions for alpha = 1, for the following order of smoking models [4,5,56]: The initial conditions are as follows: In Fig. 1, a schematic proposition graph is a tool that is applied to display the interrelation between model states and enables the application of graph-theoretic tools to discover novel features of the model.
Tab. 1 describes the parameters of the model.
where X=U-0, Y=L-0, Z=S-0, V=Q-0, W=R-0, and The stability of (2.1) is the same as the linearized stability (2.3). The stability of (2.3) depends on eigenvalues B. The point P1 is asymptotically locally stable if eigenvalues are real negative parts; conversely, P1 is unstable if at least one eigenvalue of B is a nonnegative real part. For details on stability see [56,63].
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 [28][29][30][31][32]: 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 (3.5) and (3.7), the conditions, necessary and sufficient for OPC are where λ k , k = 1, 2, 3, 4, 5 are Lagrange multipliers. Eqs. (3.9)-(3.10) clarify the conditions of the Hamiltonian for the OPC.

Second Kind of Shifted Chebyshev Polynomials
On the interval y ∈ [0, 1], Z Ã n ðyÞis defined via substituting a change variable z = 2y − 1. Hence, Z Ã n ðyÞis defined as [33][34][35]  The following formula represents the analytical The solution of this model can be written as Z Ã ðyÞ.

Style Integral Collocation for Resolving the Nonlinear Smoking Model
In this section, we present the model's implementation through the following steps: i) We first approximate the function using Eqs.

Conclusions
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.