2021/06/24 by Jean‐Jil Duchamps, Duchamps, Jean-Jil, Félix Foutel‐Rodier +3
Mathematics · Physics and Astronomy · #60J85 (Primary) #60K35 (Secondary) #92D30 #COVID-19 epidemiological studies #Complex Network Analysis Techniques #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.2106.13135
openalex publication_date 2021/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 R0 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.