Novel exponents control the quasi-deterministic limit of the extinction transition

TL;DR

This paper investigates how universal exponents govern the behavior of epidemic models near extinction, revealing new scaling laws in the quasi-deterministic limit across different models.

Contribution

It introduces novel universal exponents that describe the scaling of correlation lengths with population size in epidemic models near extinction.

Findings

Exponents scale with population size in SIS and SIR models.

Different exponents characterize SIS (directed percolation) and SIR (dynamic percolation).

Proposes a classification framework for stochastic processes near extinction.

Abstract

The quasi-deterministic limit of the generic extinction transition is considered within the framework of standard epidemiological models. The susceptible-infected-susceptible (SIS) model is known to exhibit a transition from extinction to spreading, as the infectivity is increased, described by the directed percolation equivalence class. We find that the distance from the transition point, and the prefactor controlling the divergence of the (perpendicular) correlation length, both scale with the local population size, $N$, with two novel universal exponents. Different exponents characterize the large $N$ behavior of the susceptible-infected-recovered (SIR) model, which belongs to the dynamic percolation class. Extensive numerical studies in a range of systems lead to the conjecture that these characteristics are generic and may be used in order to classify the high density limit of any stochastic process on the edge of extinction.