Discrete-time Markov chain approach to contact-based disease spreading in complex networks

¹  Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili - 43007 Tarragona, Catalonia, Spain, EU
²  Department of Informatics and Automation, University of Rome “Roma Tre” - Via della Vasca Navale, 79, Rome 00146, Italy, EU
³  Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza Corona de Aragón 42, 50009 Zaragoza, Spain, EU
⁴  Department of Theoretical Physics, University of Zaragoza - 50009 Zaragoza, Spain, EU

Corresponding author: alexandre.arenas@urv.cat

Received: 23 October 2009
Accepted: 19 January 2010

Abstract

Many epidemic processes in networks spread by stochastic contacts among their connected vertices. There are two limiting cases widely analyzed in the physics literature, the so-called contact process (CP) where the contagion is expanded at a certain rate from an infected vertex to one neighbor at a time, and the reactive process (RP) in which an infected individual effectively contacts all its neighbors to expand the epidemics. However, a more realistic scenario is obtained from the interpolation between these two cases, considering a certain number of stochastic contacts per unit time. Here we propose a discrete-time formulation of the problem of contact-based epidemic spreading. We resolve a family of models, parameterized by the number of stochastic contact trials per unit time, that range from the CP to the RP. In contrast to the common heterogeneous mean-field approach, we focus on the probability of infection of individual nodes. Using this formulation, we can construct the whole phase diagram of the different infection models and determine their critical properties.