Sharp Thresholds for a SIR Model on One-Dimensional Small-World Networks
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We study epidemic spreading according to a Susceptible-Infectious-Recovered (for short, SIR) network model known as the Reed-Frost model, and we establish sharp thresholds for two generative models of one-dimensional small-world graphs, in which graphs are obtained by adding random edges to a cycle. In 3-regular graphs obtained as the union of a cycle and a random perfect matching, we show that there is a sharp threshold at .5 for the contagion probability along edges. In graphs obtained as the union of a cycle and of a 𝒢_n,c/n Erdős-Rényi random graph with edge probability c/n, we show that there is a sharp threshold p_c for the contagion probability: the value of p_c turns out to be √(2) -1≈ .41 for the sparse case c=1 yielding an expected node degree similar to the random 3-regular graphs above. In both models, below the threshold we prove that the infection only affects 𝒪(log n) nodes, and that above the threshold it affects Ω(n) nodes. These are the first fully rigorous results establishing a phase transition for SIR models (and equivalent percolation problems) in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative phenomena emerge from our analysis, including the spread of the disease through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.
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