Mean field limit for survival probability of the high-dimensional contact process

Abstract

In this paper we are concerned with the contact process on the squared lattice. The contact process intuitively describes the spread of the infectious disease on a graph, where an infectious vertex becomes healthy at a constant rate while a healthy vertex is infected at rate proportional to the number of infectious neighbors. As the dimension of the lattice grows to infinity, we give a mean field limit for the survival probability of the process conditioned on the event that only the origin of the lattice is infected at t = 0. The binary contact path process is a main auxiliary tool for our proof.

Survival probabilities of high-dimensional stochastic SIS and SIR models with random edge weights

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In this paper, we are concerned with the stochastic SIS (susceptible-infectedsusceptible) and SIR (susceptible-infected-recovered) models on high-dimensional lattices with random edge weights, where a susceptible vertex is infected by an infectious neighbor at rate proportional to the weight on the edge connecting them. All the edge weights are assumed to be i.i.d.. Our main result gives mean field limits for survival probabilities of the two models as the dimension grows to infinity, which extends the main conclusion given in [13] for classic stochastic SIS model.

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“…The initial motivation of the study in this paper is to extend the main result in [13], which gives the mean field limit for survival probability of high-dimensional classic contact process, to the case where the contact process is with random edge weights. We find out that the SIR model is a useful auxiliary tool for us to accomplish our objective and similar conclusion holds for the SIR model simultaneously according to our proof.…”