Modeling epidemic spread using time-dependent graph diffusion equations on complex networks

Abstract

Epidemic spread in real populations is shaped by changing contact patterns, uneven connectivity, and localized transmission, so static and uniformly mixed models are often not sufficient. Complex networks provide a stronger mathematical basis for representing these evolving interactions because they capture both structural heterogeneity and temporal variation. Existing research has examined temporal networks, multilayer epidemic systems, and graph-based transmission models, but many formulations still do not fully unify node-wise epidemic states, weighted graph diffusion, and time-dependent transmission in one solvable framework. A clear gap therefore remains in developing a mathematically consistent model that can represent epidemic propagation on evolving complex networks with both analytical and numerical clarity. This study addresses that gap by proposing a time-dependent graph diffusion model for epidemic spread. The paper focuses on developing the graph-mathematical formulation, deriving the governing equations, implementing an explicit numerical solution method, and testing the model on various network types, diffusion strengths, temporal transmission patterns, and intervention scenarios. Results demonstrate the significant impact of structure and time on epidemic spread, showing faster initial spread in scale-free networks, shifts in peak timing in evolving-contact graphs, and reduced outbreak severity under early intervention. The proposed framework offers a solid and practical basis for predicting epidemics on changing networks.