Mathematical Modeling and Optimal Control of Artificial Intelligence–Human Intelligence Dynamics for Sustainable Development, Dr Abdullahi Baba

Dr Abdullahi Mohammed Baba
Department of Mathematics/Statistics, School of Mathematics and Computing
Kampala International University
Kampala
Uganda

Abstract: The rapid spread of artificial intelligence (AI) across economic and social systems is fundamentally transforming the way societies work, produce, and make decisions. While AI-driven technologies offer considerable opportunities for improving productivity, efficiency, and innovation, their unchecked or poorly governed expansion may weaken human skills, deepen socio-economic inequalities, and intensify ethical and institutional risks. Understanding how AI interacts with human intelligence (HI) over time is therefore essential for achieving sustainable and inclusive development. This study develops a deterministic mathematical model to examine the dynamic co-evolution of artificial intelligence and human intelligence within a socio-economic system. The model captures key processes, including human capital accumulation, AI-driven automation, AI–human collaboration, and the emergence of ethical and systemic risks. An optimal control framework is employed to assess how policy interventions can guide these interactions toward desirable long-term outcomes. The control variables represent investments in education and skills development, AI research and deployment, ethical regulation and governance, and the promotion of effective collaboration between humans and AI systems. Using Pontryagin’s Maximum Principle, the necessary conditions for optimality are derived, and the qualitative structure of the optimal policies is analyzed. The results indicate that sustainable development is best achieved when AI growth is balanced with sustained investment in human intelligence and supported by strong regulatory and ethical oversight. In contrast, development paths dominated by automation without adequate human-centered policies lead to higher risks and weaker long-term sustainability. By integrating technological progress, human capital dynamics, and governance considerations into a unified analytical framework, this study provides a rigorous and policy-relevant foundation for designing strategies that align AI-driven transformation with inclusive and sustainable societal development.

Keywords: Artificial Intelligence; Human Intelligence; Mathematical Modeling; Optimal Control; Sustainable Development; AI Governance; Human-centered Systems.