Statistical and Biological Physics

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Subdiffusive Activity Spreading in the Diffusive Epidemic Process

Borislav Polovnikov, Patrick Wilke, Erwin Frey


Absorbing state phase transitions are an important class of stochastic systems which are of fundamental importance for the field on nonequilibrium and statistical physics, while being directly or conceptually relevant to a variety of natural processes. The diffusive epidemic process (DEP) is a paradigmatic model in this class, introduced as a minimal model for the stochastic spreading of an epidemic. Unlike directional percolation, the process involves two different types of individuals/particles (healthy and diseased individuals) that diffuse with different diffusion constants and where the total number of individuals is conserved. Interestingly, it has also been discussed as a minimal model for the emergence of protein pattern in cell polarity systems.

While it has previously been shown by renormalization group analysis that there are three distinct universality classes depending on the relative magnitude of the diffusion constants, the nature of the critical behavior has remained unresolved and a matter of continuing debate. Neither perturbative or nonperturbative nor numerical simulations have led to a conclusive picture.

Here, we present a comprehensive computational study of the DEP complemented by scaling analyses and heuristic arguments that resolves the nature of the critical dynamics and significantly advances the field by discovering new phenomena:

We discover that for the controversial and biologically more relevant case in which sick individuals diffuse more slowly than healthy individuals, the spread of clusters of sick individuals is subdiffusive. This is a highly surprising result, which contrasts with the previously presumed purely diffusive dynamics obtained from perturbative renormalization group studies.

We show that the critical dynamics is governed by two qualitatively distinct dynamic processes: subdiffusive propagation of infection clusters and diffusive fluctuations in the healthy population. This can be described by a novel generalized scaling law for cluster spreading with two distinct dynamic exponents.

Finally, our large-scale computer simulations enabled us to obtain highly accurate results of all critical exponents, which provide a solid basis for future analytical work and, in particular, resolve previously contradictory numerical results.

Our results suggest that the critical dynamics of the DEP are determined by a strong coupling fixed point that allows for two different time scales. We speculate that this puts the DEP in the same interesting class as growth models such as the Kardar-Parisi-Zhang (KPZ) equation, which also exhibits strong coupling behavior. Based on this, we expect that our work will stimulate mathematical, possibly non-perturbative approaches that would help to unravel the observed anomalous dynamics. Since the DEP is only the simplest representative of a much broader class of mass-conserving systems, we also expect that our work will stimulate further investigations that will, at their core, yield similar results in more complex models, which are inspired by biology.