  A common theme among the proposed models for network epidemics is the
assumption that the propagating object, i.e., a virus or a piece of
information, is transferred across the nodes without going through any
modification or evolution. However, in real-life spreading processes, pathogens
often evolve in response to changing environments and medical interventions and
information is often modified by individuals before being forwarded. In this
paper, we investigate the evolution of spreading processes on complex networks
with the aim of i) revealing the role of evolution on the threshold,
probability, and final size of epidemics; and ii) exploring the interplay
between the structural properties of the network and the dynamics of evolution.
In particular, we develop a mathematical theory that accurately predicts the
epidemic threshold and the expected epidemic size as functions of the
characteristics of the spreading process, the evolutionary dynamics of the
pathogen, and the structure of the underlying contact network. In addition to
the mathematical theory, we perform extensive simulations on random and
real-world contact networks to verify our theory and reveal the significant
shortcomings of the classical mathematical models that do not capture
evolution. Our results reveal that the classical, single-type bond-percolation
models may accurately predict the threshold and final size of epidemics, but
their predictions on the probability of emergence are inaccurate on both random
and real-world networks. This inaccuracy sheds the light on a fundamental
disconnect between the classical bond-percolation models and real-life
spreading processes that entail evolution. Finally, we consider the case when
co-infection is possible and show that co-infection could lead the order of
phase transition to change from second-order to first-order.
