Sharp Thresholds for a SIR Model on One-Dimensional Small-World Networks

03/30/2021
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by   Luca Becchetti, et al.
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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 𝒒_n,c/n ErdΕ‘s-RΓ©nyi 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) -1β‰ˆ .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 π’ͺ(log n) nodes, and that above the threshold it affects Ξ©(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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