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Local-density dependent Markov processes on graphons with epidemiological applications

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  • Keliger, Dániel
  • Horváth, Illés
  • Takács, Bálint

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.

Suggested Citation

  • Keliger, Dániel & Horváth, Illés & Takács, Bálint, 2022. "Local-density dependent Markov processes on graphons with epidemiological applications," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 324-352.
    • Handle: RePEc:eee:spapps:v:148:y:2022:i:c:p:324-352
    DOI: 10.1016/j.spa.2022.03.001
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    References listed on IDEAS

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    1. repec:cup:cbooks:9780511771576 is not listed on IDEAS
    2. Kurtz, Thomas G., 1978. "Strong approximation theorems for density dependent Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 6(3), pages 223-240, February.
    3. Easley,David & Kleinberg,Jon, 2010. "Networks, Crowds, and Markets," Cambridge Books, Cambridge University Press, number 9780521195331, October.
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    Cited by:

    1. Lücke, Marvin & Heitzig, Jobst & Koltai, Péter & Molkenthin, Nora & Winkelmann, Stefanie, 2023. "Large population limits of Markov processes on random networks," Stochastic Processes and their Applications, Elsevier, vol. 166(C).

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