Optimal control of spatial diseases spreading in networked reaction–diffusion systems

Infectious diseases have long been acknowledged as significant public health menaces by both the general public and health authorities, emphatically underscoring the crucial necessity for highly efficacious prevention and control strategies. Within the realm of statistical physics and complex systems, optimal control theory emerges as a fundamental and indispensable framework for formulating these preventive measures. Simultaneously, networked reaction–diffusion systems have emerged as essential tools for comprehensively understanding the complex dynamics of infectious disease transmission. These systems integrate diverse and essential aspects of human spatial behavior, including habitat distribution, small-world network properties, and large-scale movement patterns, key elements in the in-depth study of complex systems. Consequently, there is a rapidly burgeoning interest in exploring the optimal control problems associated with these reaction–diffusion equations. However, study on the complex dynamics and optimal control of network infectious disease models remains limited, especially in the context of higher-order networks that introduce additional layers of complexity. This article reviews recent advances in the dynamics and optimal control of networked reaction–diffusion systems, underscoring their vital and irreplaceable role in disease prevention and control. We deep dive into the dynamics within both regular and complex networks, investigating how network structure and diffusion parameters influence disease transmission. Furthermore, we comprehensively expound upon several optimal control strategies, including sparse and local optimal control, and propose a comprehensive approach that integrates both reaction and diffusion terms. Finally, we outline future research directions, emphasizing the great potential of integrated strategies to effectively tackle spatial disease transmission, thereby providing a solid theoretical foundation and practical guidance for related fields within the expansive domain of statistical physics and complex systems.