Epidemic Dynamics in Complex Networks

Epidemic Dynamics in Complex Networks

Complex networks are currently being studied across many fields of science. Undoubtedly, many systems in nature can be described by models of complex networks, which are structures consisting of nodes or vertices connected by links or edges. Examples are numerous. The Internet is a network of routers or domains. The World Wide Web (WWW) is a network of websites. The brain is a network of neurons. An organization is a network of people.

Considering of the epidemic dynamics in Complex Networks, the AIDS propagation network is quite typical. When AIDS first emerged as a disease about twenty years ago, few people could have predicted how the epidemic would evolve, and even fewer could have been able to describe with certainty the best way of fighting it. Unfortunately, according to estimates from the Joint United Nations Programme on HIV/AIDS (UNAIDS) and the World Health Organization (WHO), 21.8 million people around the world had died of AIDS up to the end of 2000 and 36.1 million people were living with the human immunodeficiency virus (HIV) by the same time.

As another example, in spite of technological progress and great investments to ensure a secure supply of electric energy, blackouts of the electric transmission grid are not uncommon. Cascading failures in large-scale electric power transmission systems are an important cause of the catastrophic blackouts. Most well known is the cascading series of failures in power lines in August 1996, leading to blackouts in 11 US states and two Canadian provinces. This incident left about 7 million customers without power for up to 16 hours, and cost billions of dollars in total damage. There is an urgent need for developing innovative methodologies and conceptual breakthroughs for analysis, planning,operation, and protection of the complex and dynamical electric power networks. In yet another example, the ILOVEYOU computer virus spread over the Internet in May 2000 and caused a loss of nearly 7 billion dollars in facility damage and computer down-time.

How do diseases, jokes, and fashions spread out over the social networks? How do cascading failures propagate through large-scale power grids? How do computer viruses spread out through the Internet? Serious issues like these are attracting much attention these days. Clearly, the topology of a network has great influence on the overall behaviour of an epidemic spreading in the network.

Recently, some researchers have started to study such spreading phenomena, for example on small-world and scale-free networks. A notable attempt of Pastor-Satorras and Vespignani was to study both analytically and numerically a large-scale dynamical model on the spreading of epidemics in complex networks. The standard susceptible infected- susceptible (SIS) epidemiological model was used for investigation. 

Each node of the network represents an individual, and each link is a connection along which the infection can spread from one individual to some others. It is natural to assume that each individual can only exist in one of two discrete states—susceptible and infected. At every time step, each susceptible node is infected with probability υ if it is connected to at least one infected node. At the same time, infected nodes are cured and become again susceptible with probability δ. They together define an effective spreading rate, λ = υ/δ.

The updating can be performed with both parallel and sequential dynamics. The main prediction of the SIS model in homogeneous networks (including lattices, random graphs, and small-world models) is the presence of a non-zero epidemic threshold, λc > 0. If λ ≥ λc, the infection spreads and becomes persistent in time; yet if λ < λc, the infection dies out exponentially fast.(Fig 13(a))

It was found that, while for exponential networks the epidemic threshold is a positive constant, for a large class of scale-free networks the critical spreading rate tends to zero (Fig. 13(b)). In other words, scale-free networks are prone to the spreading and the persistence of infections, regardless of the spreading rate of the epidemic agents. It implies that computer viruses can spread far and wide on the Internet by infecting only a tiny fraction of the huge network. Fortunately, this is balanced by exponentially small prevalence and by the fact that it is true only for a range of very small spreading rates (λ << 1) that tend to zero.