Mathematical Expressions For Epidemics And Immunization In Small-world Networks

Complex networks describe a wide range of systems in nature and society. Frequently cited examples include internet, WWW, a network of chemicals linked by chemical reactions, social relationship networks, et al. Studies show that these complex networks exhibit some characters, such as Small-World effect and scale free property and so on.We summarize the development of complex networks, and we especially introduce some main epidemical models and the spreading behavior of epidemical model on complex networks, and we pay more attention to the newest investigation of disease spreading in small world networks. On the basis of the results that Newman researched the disease spreading process on the infinite small-world networks, considered that all the realistic networks is finite, we amend that results through propounding a critical point, and give out the criterion, according that we proposed an analytic model for describing the epidemic spreading action on the one-dimensional small-world network. Based on the model, the epidemic spreading behaviors without immune strategy and with immune strategy are studied under different conditions. For the immune process, the time when triggering the control action affects the analytic model, it also can be considered as a critical point.According to the analytic solutions we discussed the effect of adding shortcuts probabilityφ, network structure parameter k , infecting probability p , the time when triggering the control actionτand immune probabilityq on spreading action(that is the infected density); and also discussed the influence of these parameters on the biggest infected density.Besides, in order to check the reasonability of our analytic model, we compared the dynamic behavior in the model with the simulation result of"P-A model", having the same results. Although the results of our analytic model are similar to the previous work, by comparing with the simulation method, our analytic model is more convenient and efficient. The research of these parameters'relation and how they influence the spreading behavior with immune strategy more deeply are further studies.