In this study, the authors have calculated the equilibrium points of the hate speech sub-model, the racism sub-model, and the hate speech and racism co-existence model and analyzed their stabilities. The phenomenon of backward bifurcation was also examined. It occurs whenever the model effective reproduction number is less than one and at which both the positive co-existence free equilibrium point and the positive co-existence dominance equilibrium point exist simultaneously. All of these calculations were made using the next generation operator approach. The optimal control problem is reformulated, and optimal control analysis is carried out for the co-existence model using Pontryagin’s maximum principle in order to achieve the goal of minimizing the hate speech and racism co-existence dissemination dynamics by putting the efforts towards the protection and improvement (rehabilitation) strategies into practice. The behavior of the co-existence model solutions is then investigated, as well as the best ways to improve and protect (rehabilitation) against hate speech and racism. Numerical simulations using MATLAB ode45 solver and fourth-order Runge-Kutta numerical methods are then carried out. Based on the findings, the investigators found that the best way to reduce and address the co-existence of racism and speech in the community is to simultaneously implement improvement (rehabilitation) and protection strategies.

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