When emergencies like a pandemic occur, public opinion goes berserk, and a lot of uncertain and unreliable information is generated and spreads uncontrollably. This paper presents a mathematical framework for understanding and controlling the dissemination mechanism of uncertain information on online social platforms. Bifurcation analysis and Multiobjective nonlinear model predictive control calculations were performed on a dynamic information dissemination model triggered after major emergencies.  Bifurcation analysis is a powerful mathematical tool used to deal with the nonlinear dynamics of any process. Several factors must be considered, and multiple objectives must be met simultaneously.  The MATLAB program MATCONT was used to perform the bifurcation analysis. The MNLMPC calculations were performed using the optimization language PYOMO   in conjunction with the state-of-the-art global optimization solvers IPOPT and BARON. The bifurcation analysis revealed the existence of a branch point. The branch point is beneficial because it enables the multiobjective nonlinear model predictive control calculations to converge to the Utopia point, which is the most beneficial solution. A combination of bifurcation analysis and multiobjective nonlinear model predictive control for an information dissemination model is the main contribution of this paper. 

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This paper aims to perform bifurcation analysis in conjunction with multiobjective nonlinear model predictive control (MNLMPC) for the uncertain information dissemination model described in Li et al (2024) [34].    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.

Equations of the Uncertain Information Dissemination Model Li et al (2024)

Based on different decision-making behaviors, netizens are categorized into eight groups: unknowns, sval; thinkers eval, uncertain information publishers fval, clarifiers of uncertain information tval, internet users who believe uncertain information fb, internet users who do not believe any online information mval, internet users who only believe true information tb,   and information immunizers rval.

Figure 1: Bifurcation Diagram indicating branch point

Figure 2: sval, eval, fval profiles for MNLMPC calculations

Figure 3: Tval, fb profiles for MNLMPC calculations

Figure 4: mval, tb rval profiles for MNLMPC calculations

Figure 5: Control profiles (showing noise) for MNLMPC calculations

Figure 6:  Control profiles (noise eliminated with Savitzky Golay filter)

For the MNLMPC calculation,  were minimized individually, and each led to a value of 0. The multiobjective optimal control problem will involve the minimization of  subject to the equations governing the model. This led to a value of zero (the Utopia solution).   The MNLMPC control values obtained for ;; and   were (1.271,  1.667, 0.6769,1.1667). 

The various profiles for this MNLMPC calculation are shown in Figs. 2, 3, 4. The obtained control profile of ; and   exhibited noise (Fig. 5). This issue was addressed using the Savitzky-Golay Filter. The smoothed version of this profile is shown in Fig. 6 . The MNLMPC calculations converged to the Utopia solution, validating the analysis by Sridhar (2024) [44], which demonstrated that the presence of a limit point/branch point enables the MNLMPC calculations to reach the optimal (Utopia) solution. 

Conclusions 

Bifurcation analysis and Multiobjective nonlinear model predictive control calculations were performed on a dynamic information dissemination model triggered after major emergencies.   The bifurcation analysis revealed the existence of a branch point. The branch point (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 this model.  A combination of bifurcation analysis and Multiobjective Nonlinear Model Predictive Control (MNLMPC) for a dynamic information dissemination model triggered after major emergencies is the main contribution of this paper. 

Data Availability Statement

All data used is presented in the paper

Conflict of interest 

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

Acknowledgement 

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

The differential equations representing the model