Cigarettes and alcohol are detrimental to human health and are among the leading causes of death today. Both cigarette smoking and alcoholism are addictive and need to be understood and controlled effectively. This paper presents a mathematical framework involving bifurcation analysis and multiobjective nonlinear model predictive control for two models, the first involving cigarette smoking and the second involving alcoholism.  Bifurcation analysis is a powerful mathematical tool used to address the nonlinear dynamics of any process. Several factors must be taken into account, and multiple objectives must be achieved simultaneously. The MATLAB program MATCONT was utilized to conduct the bifurcation analysis. The MNLMPC calculations were performed using the optimization language PYOMO in conjunction with the advanced global optimization solvers IPOPT and BARON. The bifurcation analysis revealed the presence of limit and branch points in the two models. These limit and branch points are advantageous as they allow the multiobjective nonlinear model predictive control calculations to converge to the Utopia point, which represents the most beneficial solution. The combination of bifurcation analysis and multiobjective nonlinear model predictive control for models involving cigarette smoking and alcoholism is the main contribution of this paper. 

Lahrouz et al (2011) [1] discussed the Deterministic and Stochastic Stability of a Mathematical Model of Smoking. Bhunu (2012) [2] developed a mathematical analysis of alcoholism. Mulonea and Straughan (2012) [3] developed and tested a model about binge drinking. Wang et al (2016) [4] provided optimal control strategies in an alcoholism model. Ullah et al (2016) [5] discussed the dynamical Features of a Mathematical Model on Smoking. Sikander et al. (2017) [6] produced Optimal Solutions for a bio-mathematical model of the Evolution of smoking habits. Ur Rahman et al (2018) [7], discussed the threshold dynamics and optimal control of an age-structured model for stopping smoking. Mu'tamar Khozin(2018) [8] developed an optimal control strategy for the alcoholism model with two infected compartments. Uçar et al (2018) [9] conducted a mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative. Sun and Jiav (2019) [10], discussed the optimal control of a delayed smoking model with immigration. Mahdy et al (2020) [11] studied the dynamical characteristics and signal flow graph of nonlinear fractional smoking mathematical model. Zhang et al (2020) [12] studied the harmonic mean type dynamics of a delayed giving up smoking model and optimal control strategy via legislation. Mahdy et al (2020) [13] developed an approximate solution for solving nonlinear fractional order smoking model. Ilmayasinta and Purnawan, H. (2021) [14] performed Optimal Control in a Mathematical Model of Smoking. 

This paper aims to perform bifurcation analysis in conjunction with multiobjective nonlinear model predictive control (MNLMPC) for the smoker model (Ilmayasinta and Purnawan, H. (2021) [14]) and the alcoholism model with two infected compartments Mu'tamar Khozin(2018) [8]. This paper is organized as follows. First, the model equations are presented. The numerical procedures (bifurcation analysis and multiobjective nonlinear model predictive control (MNLMPC) are then described. This is followed by the results and discussion, and conclusions.

In this section, details of the smoker model (Ilmayasinta and Purnawan, H. (2021) [14]) and the alcoholism model with two infected compartments, Mu'tamar Khozin(2018) [8] are presented.

Smoker model 

The MATLAB software MATCONT is used to perform the bifurcation calculations. Bifurcation analysis deals with multiple steady-states and limit cycles. Multiple steady states occur because of the existence of branch and limit points. Hopf bifurcation points cause limit cycles . A commonly used MATLAB program that locates limit points, branch points, and Hopf bifurcation points is MATCONT(Dhooge Govearts, and Kuznetsov, 2003[15]; Dhooge Govearts, Kuznetsov, Mestrom and  Riet, 2004[16] ). This program detects Limit points(LP), branch points(BP), and Hopf bifurcation points(H) for an ODE system 

Flores Tlacuahuaz et al (2012) [20] developed a multiobjective nonlinear model predictive control (MNLMPC) method that is rigorous and does not involve weighting functions or additional constraints. This procedure is used 
 

Bifurcation analysis for the Smoker model revealed the existence of a limit point at 

(Pv, Ov, Sv, Qt, Qp,u2 ) values of ( 1.663894, 0, 0, 5.834, 992.5019, -0.001 ). This is shown in Fig. 1. Here u2 is chosen as the bifurcation variable. 

For the alcoholics model the bifurcation analysis revealed the existence of a branch point at

(s,a1,ar,r,u) values of ( 1.0,0.0, 0.0, 0.0, 0.4 ). This is shown in Fig. 2. u is chosen as the bifurcation parameter.

subject to the equations governing the model. This led to a value of zero (the Utopia solution).  The MNLMPC control values obtained for u1 u2 u3 and u4 were 0.2318, 0.01082, 0.4571, and 0.010441. 

The various profiles for this MNLMPC calculation are shown in Figs. 3a,3b,3c and 3d. The obtained control profile of u1 u2 u3 and u4  exhibited noise (Fig. 3e and 3f.). This issue was addressed using the Savitzky-Golay Filter. The smoothed version of the profiles are shown in Fig. 3g and 3h. The MNLMPC calculations converged to the Utopia solution, validating the analysis by Sridhar (2024) [24], which demonstrated that the presence of a limit point/branch point enables the MNLMPC calculations to reach the optimal (Utopia) solution. 

Fig. 1 Bifurcation Diagram for the Smoker Model

Fig. 2 Bifurcation Diagram for the Alcoholics Model

Fig. 3a MNLMPC smoker model Pv, Ov vs t

Fig. 3b MNLMPC smoker model Sv vs t

Fig. 3c MNLMPC smoker model Qt vs t

Fig. 3d MNLMPC smoker model Qp vs t

Fig. 3e MNLMPC smoker model u1 u2 (noise exhibited)

Fig. 3f MNLMPC smoker model u3 u4 (noise exhibited)

Fig. 3g MNLMPC smoker model u1 u2 (with Sazitzky Golay filter) vs t (noise eliminated)

Fig. 3h MNLMPC smoker model u3 u4 (with Sazitzky Golay filter) vs t (noise eliminated)

Fig. 4a MNLMPC alcoholics models, r vs t

Fig. 4b MNLMPC alcoholics model a1, a2 vs t

Fig. 4c MNLMPC alcoholics model u vs t (noise exhibited)

Fig. 4d u vs t (with Sazitzky Golay filter) vs t (noise eliminated)

Bifurcation analysis and Multiobjective nonlinear model predictive control calculations were performed on a cigarette smoking and alchholics models.  The bifurcation analysis revealed the existence of a limit and a branch point. The limit and branch points (which causes multiple steady-state solutions from a singular point) is very beneficial because it enables the Multiobjective nonlinear model predictive control calculations to converge to the Utopia point (the best possible solution) in the models. A combination of bifurcation analysis and Multiobjective Nonlinear Model Predictive Control(MNLMPC) for a dynamic models involving cigarette smoking and alcoholism is the main contribution of this paper.

All data used is presented in the paper

The author, Dr. Lakshmi N Sridhar has no conflict of interest.

Dr. Sridhar thanks Dr. Carlos Ramirez and Dr. Suleiman for encouraging him to write single-author papers.