Local-density dependent Markov processes on graphons with epidemiological applications

Dániel Keliger, Illés Horváth and Bálint Takács

Stochastic Processes and their Applications, 2022, vol. 148, issue C, 324-352

Abstract: We investigate local-density dependent Markov processes on a class of large graphs sampled from a graphon, where the transition rates of the vertices are influenced by the states of their neighbors. We show that as the average degree converges to infinity (faster than a given threshold), the evolution of the process in the transient regime converges to the solution of a set of non-local integro-partial differential equations depending on the limit graphon. We also provide rigorous derivation for the epidemic threshold in the case of the Susceptible–Infected–Susceptible (SIS) process on such graphons.

Date: 2022
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DOI: 10.1016/j.spa.2022.03.001

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