Pattern dynamics and practical application of infectious disease reaction–diffusion systems based on higher-order networks and physics-informed neural networks

Traditional epidemic models have limitations in capturing spatial heterogeneity, describing population-oriented migration, and integrating real data. This study constructs reaction-diffusion and reaction-advection-diffusion systems based on higher-order networks and triangular lattice torus networks and further extends them to reaction-advection-diffusion systems with periodic diffusion coefficients. Through theoretical derivation and numerical simulation, the necessary conditions for Turing instability are analyzed, and the key roles of diffusion coefficients, advection mechanisms, higher-order interactions, and periodic diffusion in inducing pattern formation and pattern morphological evolution are revealed. In terms of practical application, this paper applies physics-informed neural networks to the dengue fever epidemic data of 49 countries in the Pan-American region. Based on the actual data, we construct a weighted comprehensive network that integrates geographical proximity, global air route data, and maritime port data. This network is used to define the higher-order Laplacian matrix describing diffusion, as well as an advective Laplacian matrix characterizing population flows driven by the gross domestic product of each country. The training results show that the model achieves high-accuracy fitting to the historical epidemic data, with the errors mainly concentrated in a few countries with a large number of cases during the epidemic peak period. In addition, the model also predicts the epidemic data for the next 52 weeks.