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Deterministic rate equations are widely used in the study of stochastic,interacting particles systems. This approach assumes that the inherent noise,associated with the discreteness of the elementary constituents, may beneglected when the number of particles $N$ is large. Accordingly, it failsclose to the extinction transition, when the amplitude of stochasticfluctuations is comparable with the size of the population. Here we present ageneral scaling theory of the transition regime for spatially extended systems.Two fundamental models for out-of-equilibrium phase transitions are considered:the Susceptible-Infected-Susceptible (SIS) that belongs to the directedpercolation equivalence class, and the Susceptible-Infected-Recovered (SIR)model belonging to the dynamic percolation class. Implementing the Ginzburgcriteria we show that the width of the fluctuation-dominated region scales like$N^{-\kappa}$, where $N$ is the number of individuals per site and $\kappa =2/(d_u-d)$, $d_u$ is the upper critical dimension. Other exponents that controlthe approach to the deterministic limit are shown to depend on $\kappa$. Thetheory is supported by the results of extensive numerical simulations forsystems of various dimensionalities.

A scaling theory for the quasi-deterministic limit