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Virus development on scale-free networks.
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Authors
T. De Coster*
W. Gins*
H. Hooyberghs
* = equal authorship
Background
The study of complex networks, particularly the epidemic models pertaining to them, would seem to be more in place in studies performed in theoretical informatics and biology, seeing as most examples and applications find themselves in these fields. However, the mathematical study began with Paul Erdos, and nowadays an approach called the mean field approach, commonly used in statistical physics, gives very accurate results, thereby confirming its place as a physics-subject.
The focus will be on a specific type of network, namely a scale-free network. This type of network is quite accurate in representing actual, real-life networks, ranging from financial networks, to social circles and sexual contacts. For this reason, scale-free networks enjoy a position of prime interest in the research for applications of network theory.
Figure 1: Graphical representation of the application areas of network theory. The size of the node is an indication of the number of papers in this area that use network theory, the thickness of the links represents the strength of the cohesion of the areas.
Our research centered on the spreading of viruses, for which we considered both the SIS- and SIR-models. These models are based on observed disease patterns, and allow the prediction of things like the spreading of an economic crisis in the financial market. Surprisingly, the results obtained are in excellent agreement with actual data.
Figure 2: Graphical representation of both the SIS- and SIR-model.
The spreading of a virus is studied, under both the SIS and SIR assumptions. Both numerical results and analytical expressions are given and analyzed. We conclude that, if a virus is more infectious than a critical parameter, it cannot be removed from a complex network. In very large networks, this threshold drops to zero, enabling any virus to continue to spread.
Figure 3: In the SIR model, the severity of the infection depends on the degree of the originally infected node. The averaging over 1000 iterations shows that an infection with a high original grade is more severe, and more consistent, than a low original grade. However, even at a low degree an infection can be very severe, according to the maximum observation (see inset).
Figure 4:
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