Understanding the impact of non-linearity in the SIS model

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The SIS model is a classic model from epidemiology that formalizes a variety of diffusion processes on networks, such as biological infections and information dissemination. In this model, vertices are either infected or susceptible to an infection. Infected vertices infect their neighbors independently at a rate λ>0, and each infected vertex becomes susceptible at a rate of 1. Overall, these dynamics imply that each susceptible vertex with exactly m infected neighbors becomes infected at rate λm, that is, linear in m. However, it has been observed that various processes exhibit a non-linear scaling of the infection rate with respect to m. For these kinds of processes, no fully rigorous guarantees exist so far.We address this shortcoming by considering a variant of the SIS model in which vertices get infected at a rate that scales polynomially in the number of their infected neighbors, weighted by the infection coefficient λ. We give the first fully rigorous results for thresholds of λ at which the expected survival time becomes super-polynomial. For cliques we show that when the infection rate scales sub-linearly, the threshold only shifts by a poly-logarithmic factor, compared to the standard SIS model. In contrast, super-linear scaling changes the process considerably and shifts the threshold by a polynomial term. For stars, sub-linear and super-linear scaling behave similar and both shift the threshold by a polynomial factor. Our bounds are almost tight, as they are only apart by at most a poly-logarithmic factor from the lower thresholds, at which the expected survival time is logarithmic.