Analysis of the hate speech and racism co-existence dissemination model with optimal control strategies

Hate speech, racism, and their co-existence are the human mind infections that are major factors that affect the living conditions of people, societal well-being, and political stability, economic and social disturbance throughout the world especially in underdeveloped nations of the world. The main objective of this study is to formulate and analyze the hate-speech and racism co-existence model with optimal control strategies and investigate the optimal effects of protection and improvement (rehabilitation) strategies to minimize and tackle the hate speech and racism co-existence dissemination in the community. In this study, we have computed the hate speech sub-model, the racism sub-model, and the hate speech and racism co-existence model equilibrium points and analyzed their stabilities, using the next generation operator approach all the models' effective reproduction numbers are computed, examined the phenomenon of backward bifurcation whenever the model effective reproduction number is less than one where at which both the positive co-existence free equilibrium point and the positive co-existence dominance equilibrium point exist simultaneously. To achieve the objective minimizing the hate speech and racism co-existence dissemination dynamics by implementing the efforts towards the protection and improvement (rehabilitation) strategies, the optimal control problem is re-formulated, and optimal control analysis is performed for the co-existence model using Pontryagin's maximum principle. Numerical simulations using MATLAB ode45 solver with fourth-order Runge-Kutta numerical methods to investigate the behavior of the co-existence model solutions and to explore the optimal effects of the protections and improvements (rehabilitation) strategies for the hate speech and racism co-existence dynamics are then performed. From the results, we observed that implementing the protection and improvement (rehabilitation) strategies simultaneously is the most effective approach to minimize and tackle the speech and racism co-existence propagation in the community.