In the face of persistent threats posed by infectious diseases, despite remarkable medical advancements, understanding and efficiently controlling their spatial spread through mathematical modeling remain imperative. Networked reaction-diffusion systems offer a promising avenue to effectively delineate population discrete distribution and individual movement heterogeneity. However, the dynamics of spatial diseases within these systems and the formulation of optimal control strategies are currently undergoing vigorous development. In this letter, we illustrate the dynamics of spatial disease spread in networked reaction-diffusion systems through the lens of optimal control, considering various network complexities from pairwise networks to higher-order networks. It then emphasizes their applicability in designing effective spatial disease control strategies across diverse network complexities. Finally, we discuss the existing challenges.

Networked Turing patterns often manifest as groups of nodes distributed on either side of the homogeneous equilibrium, exhibiting high and low density. These pattern formations are significantly influenced by network topological characteristics, such as the average degree. However, the impact of clustering on them remains inadequately understood. Here, we investigate the relationship between clustering and networked Turing patterns using classical prey–predator models. Our findings reveal that when nodes of high and low density are completely distributed on both sides of the homogeneous equilibrium, there is a linear decay in Turing patterns as global clustering coefficients increase, given a fixed node size and average degree; otherwise, this linear decay may not always hold due to the presence of high-density nodes considered as low-density nodes. This discovery provides a qualitative assessment of how clustering coefficients impact the formation of Turing patterns and may contribute to understanding why using refuges in ecosystems could enhance the stability of prey–predator systems. The results link network topological structures with the stability of prey–predator systems, offering new insights into predicting and controlling pattern formations in real-world systems from a network perspective.

We introduce a control scheme able to steer the evolution of networked reaction-diffusion systems toward any intended dynamics. Specifically, we consider simplicial complexes of reaction-diffusion systems that can be collapsed into weighted complex networks by leveraging the linear relationship between the Laplacian matrix of the simplicial complex and the Laplacian matrix of each order simplex. The method integrates reaction and network-based diffusion controls and, as far as effectiveness and cost are concerned, features a significantly improved performance over traditional single-variable control methods. Numerical simulations using the FitzHugh-Nagumo and susceptible-infected-removed models demonstrate the resilience of the proposed mixed control across diverse systems, highlighting the potential of the method for complex system management and intervention strategies.

Current research on pattern formations in networked reaction–diffusion (RD) systems predominantly focuses on the impacts of diffusion heterogeneity between nodes, often overlooking the contact heterogeneity between individuals within nodes in the reaction terms. In this paper, we establish a networked RD model incorporating infection through higher-order interaction in simplicial complexes in the reaction terms. Through theoretical and numerical analysis, we find that these higher-order interactions may induce Turing instability in the system. Notably, the relationship between the size of the Turing instability range and the average 2-simplices degree within nodes can be approximated by a quadratic function. Additionally, as the average 2-simplices degree increases, the amplitude of the patterns exhibits three distinct trends: increasing, decreasing, and initially increasing then decreasing, while the average infection density increases consistently. We then provide a possible explanation for these observations. Our findings offer new insights into the effects of contact heterogeneity within nodes on networked pattern formations, thereby informing the development of epidemic prevention and control measures.