Mathematical expressions for epidemics and immunization in small-world networks

In this paper we propose 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. The obtained results suggest that the network's structure is the key factor to influence the time evolution law of the number of all infected vertices when without immune strategy. The infecting percentage with immune strategy has different maximum for different immune probability and triggering time (at which the control strategy is triggered), but for different values of them the infecting percentage reaches each maximum almost at same time. Our analytic model is more convenient and efficient than the previous simulation method.