Epidemic spreading survey

Lots of real word problems can be motivated and modeled as spreads of epidemics through a network. Prominent examples include the spread of worms and email viruses over the Internet, the spread of disease among the population, and the spread of harmful gossip or panic in a social network. We want to protect computer networks from viruses, prevent disease spreads, and control the leakage of sensitive information and unpleasant gossip. At the same time our resources (anti-virus software, vaccination, influence) are costly and limited, so we are interested in achieving the best possible effect, while allocating the minimum possible resources. Two epidemic models are commonly used when studying the spreads of epidemics through a network, the susceptible-infected-susceptible (SIS) model and the susceptible-infected-recovered (SIR) model. The main difference between these two models is, a recovered individual can be infected again in the SIS model, but not in the SIR model which assume recovered individuals have life long immunity to the disease. And this difference makes them suitable for different kinds of problems. For instance, when studying childhood diseases which individuals can have long-lasting immunity, either naturally or from vaccination, it is more appropriate to use SIR model; when studying viruses transmitting over the Internet, often times it is more reasonable to use SIS model, since most of the viruses mutate, in which case even if a computer is cured by anti-virus software and therefore not susceptible to the original virus, it is still susceptible to a mutated virus. Within each model, quite amount of research work has been done, touching different aspects of the problems. This survey is trying to give a comprehensive overview of the research work that has been done in the spreads of epidemics, and discover open problems and further directions. The survey is going to be structured as follows. In section 1, we introduce SIR model formally, browsing different kinds of problems and techniques studied in such a model, and proposing some open problems and further directions in this line of work. In section 2, we use similar approach to introduce SIS model. In section 3, we analyze different kinds of contact graphs, since the contact graph plays an important roll in epidemic problems. Different kinds of graphs have different set of properties, which may help us get different results under SIR/SIS models. More importantly, we need to know for different problems, what kind of contact graph models the real underlying epidemic network well.