Study Of Epidemic Threshold And Immunization On Generalized Networks

Complex network is a research object based on graph theories. As an effective modeling tool, it is applied in biology, sociology, physics, etc. Epidemic models can describe the dynamics of epidemic spreading among individuals. The susceptible-infected-susceptible (SIS) model is one of the most widely used epidemic models in complex networks. The significances of the research on epidemic spreading in complex networks are as follows: (1) It provides a mathematical approach for epidemic spreading; (2) Different from micro-scope strategies such as drug immunization and individual isolation, it provides a macro-vision on immunization; (3) The global strategy it provided can help the government make scientific decisions; (4) The similarity between epidemics and computer virus can make this research help control the computer virus.When SIS model is applied on contact networks, these networks mostly consist of nodes connected by undirected and unweighted edges following certain statistical properties, whereas this dissertation considers the threshold and immunization problem for the SIS model on generalized networks that may contain different kinds of nodes and edges which are very possible in the real situation. These works include:(1) It is proposed and proved that an epidemic will become extinct if and only if the spectral radius of the corresponding parameterized adjacent matrix (PAM) is smaller than 1, which is verified by simulation result.(2) By taking uniform immunization, random immunization, targeted immunization and acquaintance immunization as examples, this proposition shows its ability of evaluating the efficiency of immune strategies.(3) Several methods are developed that can precisely find the optimal immune strategies, and quickly find the approximated optimal immunization, for networks with the given PAM.The contribution of this dissertation can be used in numerical computation and symbolic derivation in relative areas, and can promote the design of immune strategies in computer networks. Finally in this dissertation, the summary is provided and the future direction of the complex network research on epidemics is discussed. In the future, the generalized networks may developed to become dynamical social networks, so that to be a new trend in the research of this area.