Understanding the epidemiological patterns in spatial networks

Self-organized patterns with abundant structure characteristics in epidemiology are of great significance. The reaction-diffusion (RD) equations in continuous space have made plentiful achievements in exploration of this direction, but their spatiotemporal dynamics are limitedly supported in essence by the RD equations defined on a class of regular lattices as their counterparts discretized in space. However, patterns in complex spatial networks beyond lattice networks remain largely unexplored. In this paper, we creatively develop an epidemic reaction-diffusion model defined on our well-designed basic and modified spatially embedded networks to investigate the epidemiological patterns in spatial networks. We apply some basic properties of the Kronecker product to determine the eigenvalues and their corresponding eigenvectors of a high-dimensional matrix, which leads us to derive the necessary and sufficient conditions for Turing instability. With series and groups of comparative simulations, we systematically study the influence of factors including network size, nonlocal connectivity, asymmetrical connectivity, degree heterogeneity and randomly connected links on the pattern formations in spatial networks, and obtain some scarcely documented results deepening and broadening our understanding about the epidemiological patterns in space and networks. Especially, we find that the degree heterogeneity in spatial networks whose degrees of nodes follow even the power law distribution does not trigger the essential change of pattern types. Remarkably, the randomly connected links in spatial networks act as a mechanism to induce irregular stationary patterns being substitutes for regular ones, and narrow prevalence difference of diseases in the whole spatial networks, even prevent the occurrence of Turing instability.

Automated summary

This study investigates the epidemiological patterns in spatial networks through the development of an epidemic reaction-diffusion model based on spatially embedded networks. By systematically studying factors such as network size, connectivity, and degree heterogeneity, the study provides new insights into the formation of patterns in spatial networks. It is found that degree heterogeneity does not trigger fundamental changes in pattern types, and randomly connected links in spatial networks act as a mechanism to induce irregular stationary patterns, narrowing prevalence differences and preventing Turing instability.