Critical dynamics of epidemic processes with Lévy-like diffusion

We investigate the critical behavior of a stochastic one-dimensional lattice model of a diffusion-limited epidemic process with Lévy flights. Particles $A$ represent healthy individuals and particles $B$ are infected. These particles diffuse along the chain with distinct diffusion rates. The hopping distance is assumed to obey a Lévy power-law distribution governed by a characteristic exponent $\alpha$. The epidemic process is governed by the reaction processes $A+B\to 2B$ and $B\to A$ with proper reaction rates. The system presents a non-equilibrium absorbing state phase-transition at a critical total particle density on which the population of $B$ becomes extinct. Using short-time dynamics Monte Carlo simulations, we determine a set of relevant critical exponents for distinct diffusion regimes going from the non-trivial short-range hopping universality classes holding for large values of $\alpha$ towards the mean-field exponents as $\alpha$ approaches unity.