Universality classes in the time evolution of epidemic outbreaks on complex networks

We investigate the full temporal evolution of epidemic outbreaks in complex networks, focusing on the susceptible-infected (SI) model of disease transmission. Combining theoretical analysis with large-scale numerical simulations, we uncover two universal patterns of epidemic growth, determined by the structure of the underlying network. In small-world networks, the prevalence follows a Gompertz-like curve, while in fractal networks it evolves according to Avrami-type dynamics, typical of spatially constrained systems. These regimes define distinct universality classes that remain robust across arbitrary transmission rates. Notably, our approach provides explicit analytical formulas for the global epidemic prevalence and class-specific scaling relations capturing its dependence on the transmission rate. We show that the commonly assumed early exponential growth occurs only in small-world networks, where it corresponds to the short-time approximation of the Gompertz function. In contrast, this exponential phase is entirely absent in fractal networks, where spreading is markedly slower and governed by different mechanisms. Our approach clarifies the structural origins of these contrasting behaviors and offers a unified framework for understanding epidemic dynamics across diverse network topologies.