Eur. Phys. J. B 7, 137-145 (1999)
https://doi.org/10.1007/s100510050596

Lévy-flight spreading of epidemic processes leading to percolating clusters

¹ Institut für Theoretische Physik III, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
² Laboratoire de Physique Théorique et Hautes Énergies (Laboratoire associé au Centre National de la Recherche Scientifique -URA D00 63.) , Université de Paris-Sud, 91405 Orsay Cedex, France

Received: 17 July 1998
Revised: 20 July 1998
Accepted: 28 July 1998
Published online: 15 January 1999

Abstract

We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as . By means of Wilson's momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an ε-expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection .