Eur. Phys. J. B (2018) 91: 58
https://doi.org/10.1140/epjb/e2018-80343-3

Regular Article

Sudden spreading of infections in an epidemic model with a finite seed fraction

¹ Department of Mathematics and Informatics, Ibaraki University, 2-1-1, Bunkyo, Mito 310-8512, Japan
² Department of Physics, Hokkaido University, Kita 10 Nishi 8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan

^a e-mail: takehisa.hasegawa.sci@vc.ibaraki.ac.jp

Received: 14 June 2017
Received in final form: 25 September 2017
Published online: 29 March 2018

Abstract

We study a simple case of the susceptible-weakened-infected-removed model in regular random graphs in a situation where an epidemic starts from a finite fraction of initially infected nodes (seeds). Previous studies have shown that, assuming a single seed, this model exhibits a kind of discontinuous transition at a certain value of infection rate. Performing Monte Carlo simulations and evaluating approximate master equations, we find that the present model has two critical infection rates for the case with a finite seed fraction. At the first critical rate the system shows a percolation transition of clusters composed of removed nodes, and at the second critical rate, which is larger than the first one, a giant cluster suddenly grows and the order parameter jumps even though it has been already rising. Numerical evaluation of the master equations shows that such sudden epidemic spreading does occur if the degree of the underlying network is large and the seed fraction is small.