Diffusive epidemic process: theory and simulation

TL;DR

This paper investigates the phase transition in a one-dimensional diffusive epidemic model using mean-field theory and Monte Carlo simulations, revealing complex critical behavior and universality classes.

Contribution

It provides a combined theoretical and computational analysis of the diffusive epidemic process, highlighting nonmonotonic critical rates and universality classes based on diffusion rates.

Findings

No evidence of discontinuous transition.

Critical recovery rate depends nonmonotonically on diffusion rate.

Existence of three universality classes based on diffusion rates.

Abstract

We study the continuous absorbing-state phase transition in the one-dimensional diffusive epidemic process via mean-field theory and Monte Carlo simulation. In this model, particles of two species (A and B) hop on a lattice and undergo reactions B -> A and A + B -> 2B; the total particle number is conserved. We formulate the model as a continuous-time Markov process described by a master equation. A phase transition between the (absorbing) B-free state and an active state is observed as the parameters (reaction and diffusion rates, and total particle density) are varied. Mean-field theory reveals a surprising, nonmonotonic dependence of the critical recovery rate on the diffusion rate of B particles. A computational realization of the process that is faithful to the transition rates defining the model is devised, allowing for direct comparison with theory. Using the quasi-stationary simulation method we determine the order parameter and the survival time in systems of up to 4000 sites. Due to strong finite-size effects, the results converge only for large system sizes. We find no evidence for a discontinuous transition. Our results are consistent with the existence of three distinct universality classes, depending on whether A particles diffusive more rapidly, less rapidly, or at the same rate as B particles.