Epidemic processes

Various phenomena in our daily world exhibit similarities to epidemic spreading. Fashion or technical trends are referred to as viral when they reach a large part of a society. In general, trends, opinions, diseases and failure in social or technical networks are characterized by the fact that they spread from an ``infected'' source to a ``non-infected'' state. We use the term epidemic process to refer to the described variety of different spreading phenomena. In addition to infection processes, external influences such as media in the case opinion spreading among people also play a major role. In contrast to a simple contagion, where one source is sufficient to sustain spreading, there also exist complex contagion phenomena where multiple sources are required. Social opinion formation is often described by the latter process. Until now, different spreading models have been developed and applied to account for different numbers of sources. We propose a general contagion model which accounts for both external influences and different numbers of ``infectious'' sources. We show that our contagion model exhibits a novel characteristic critical behavior which we describe using mathematical concepts that have their origin in catastrophe theory. Besides the stationary critical behavior, we also verify dynamical universal behavior by analyzing the corresponding correlation and response functions. The identification of universal critical behavior allows to demonstrate similarities between epidemic models that appear different at first sight. Based on those similarities, it is possible to classify spreading models. This way of classifying models is closely related to the study of phase transitions in statistical physics. Typically, one studies an order parameter (e.g. the number of infected) as a function of a control parameter (e.g. infection rate). In many situations, one finds a transition point or critical point of the control parameter at which the order parameter starts growing. At this point, there exists a characteristic behavior of the order parameter as a function of the control parameter. Considerations like these allow to apply methods from statistical physics which are important for describing phase transitions of percolation and magnetic systems. We consider different applications after defining and classifying our general modeling framework. We use the gained insights from the analysis of our general contagion model to propose a model for political campaigns. The suggested campaign model accounts for the complex interplay between activists, persuadables, political clout, budget and costs. Furthermore, we also study the opinion formation process in online petitions by applying different time series analysis methods. The last two examples have their origin in the field of disease spreading. We describe a very effective method of targeted recovery of infected individuals, which makes it possible to drastically reduce the number of infected cases. This protocol is also applicable to other spreading processes. Another application deals with the spatio-temporal correlations of the spreading of dengue