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Duchamps, Jean-Jil; Foutel-Rodier, Félix; Schertzer, Emmanuel

General epidemiological models: law of large numbers and contact tracing. (English) Zbl 1520.92061

Electron. J. Probab. 28, Paper No. 98, 37 p. (2023).
Summary: We study a class of individual-based, fixed-population size epidemic models under general assumptions, e.g., heterogeneous contact rates encapsulating changes in behavior and/or enforcement of control measures. We show that the large-population dynamics are deterministic and relate to the Kermack-McKendrick PDE. Our assumptions are minimalistic in the sense that the only important requirement is that the basic reproduction number of the epidemic \(R_0\) be finite, and allow us to tackle both Markovian and non-Markovian dynamics. The novelty of our approach is to study the “infection graph” of the population. We show local convergence of this random graph to a Poisson (Galton-Watson) marked tree, recovering Markovian backward-in-time dynamics in the limit as we trace back the transmission chain leading to a focal infection. This effectively models the process of contact tracing in a large population. It is expressed in terms of the Doob \(h\)-transform of a certain renewal process encoding the time of infection along the chain. Our results provide a mathematical formulation relating a fundamental epidemiological quantity, the generation time distribution, to the successive time of infections along this transmission chain.

MSC:

92D30 Epidemiology
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J85 Applications of branching processes
05C80 Random graphs (graph-theoretic aspects)

Keywords:

contact-tracing; Crump-Mode-Jagers process with interaction; functional law of large numbers; infection graph; Kermack-McKendrick model; local weak convergence; non-Markovian epidemic model

Software:

EpiEstim
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\textit{J.-J. Duchamps} et al., Electron. J. Probab. 28, Paper No. 98, 37 p. (2023; Zbl 1520.92061)
Full Text: DOI arXiv Link

References:

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