Velocity of front propagation in the epidemic model $A+B\to2A$

TL;DR

This paper analyzes the propagation speed of an epidemic front in a one-dimensional irreversible model with diffusing particles, providing analytic estimates and observing a crossover in velocity dependence on diffusion rates.

Contribution

It introduces a systematic analytic approach to estimate front velocity in the $A+B o 2A$ model considering different diffusion rates, highlighting fluctuation effects.

Findings

Analytic estimates agree with simulations.

Identifies a crossover from linear to square root velocity dependence on $D_A$.

Shows deviations from mean field predictions due to fluctuations.

Abstract

We study front propagation in the irreversible epidemic model $A+B\to 2A$ in one dimension. Here, we allow the particles $A$ and $B$ to diffuse with rates $D_A$ and $D_B$, which, in general, may be different. We find analytic estimates for the front velocity by writing truncated master equation in a frame moving with the rightmost $A$ particle. The results obtained are in reasonable agreement with the simulation results and are amenable to systematic improvement. We also observe a crossover from the linear dependence of front velocity $V$ on $D_A$ for smaller values of $D_A$ to $V\propto \sqrt{D_A}$ for larger $D_A$, but numerically still significantly different from the mean field value. The deviations reflect the role of internal fluctuations which is neglected in the mean field description.