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Time Evolution of Disease Spread on Networks with Degree Heterogeneity

Two crucial elements facilitate the understanding and control of communicable disease spread within a social setting. These components are, the underlying contact structure among individuals that determines the pattern of disease transmission; and the evolution of this pattern over time. Mathematical models of infectious diseases, which are in principle analytically tractable, use two general approaches to incorporate these elements. The first approach, generally known as compartmental modeling, addresses the time evolution of disease spread at the expense of simplifying the pattern of transmission. On the other hand, the second approach uses network theory to incorporate detailed information pertaining to the underlying contact structure among individuals. However, while providing accurate estimates on the final size of outbreaks/epidemics, this approach, in its current formalism, disregards the progression of time during outbreaks. So far, the only alternative that enables the integration of both aspects of disease spread simultaneously has been to abandon the analytical approach and rely on computer simulations. We offer a new analytical framework based on percolation theory, which incorporates both the complexity of contact network structure and the time progression of disease spread. Furthermore, we demonstrate that this framework is equally effective on finite- and "infinite"-size networks. Application of this formalism is not limited to disease spread; it can be equally applied to similar percolation phenomena on networks in other areas in science and technology.