Modeling and Analysis of the Spread of Racism in the Community with Optimal Control Strategy

Tesfaye Worku Gutema

doi:10.53941/ams.2026.100005

Abstract

The spread of racism in society affects all aspects of their lives. Hence, this study aims to analyze the control measures to reduce the spread of racism using a non-linear deterministic model with an optimal control strategy. To ensure the model is biologically and mathematically sound, we confirmed that its solutions are always non-negative and bounded under defined initial conditions. Additionally, we determined the basic reproductive number by applying the next-generation matrix method. Moreover, we analyze the stability of the equilibrium point using the Jacobian matrix and Lyapunov function techniques. Our analysis confirms that the racism-free equilibrium is locally asymptotically stable when R₀ < 1, meaning the behavior effectively dies out over time if the initial levels are low enough. Conversely, if R₀ > 1, the system shifts to a locally asymptotically stable endemic equilibrium, where the behavior persists and settles into a steady state within the population. Furthermore, the sensitivity analysis of the parametric value of the model is illustrated using the normalized forward sensitivity method. The optimal control strategy employed the application of Pontryagin’s maximum principle, which was used to test the effectiveness of proposed control measures. The model was extended to an optimal control method, with the use of two time-dependent controls to assess the spread of racism in the community, namely, mass education campaigns using social media and teaching through religious institutions. Finally, numerical simulation of the optimal control model shows that a combination of mass education campaigns and religious teaching is the most effective strategy for reducing the spread of racism.

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