Takeover times for a simple model of network infection

Abstract

We study a stochastic model of infection spreading on a network. At each time step a node is chosen at random, along with one of its neighbors. If the node is infected and the neighbor is susceptible, the neighbor becomes infected. How many time steps T does it take to completely infect a network of N nodes, starting from a single infected node? An analogy to the classic "coupon collector" problem of probability theory reveals that the takeover time T is dominated by extremal behavior, either when there are only a few infected nodes near the start of the process or a few susceptible nodes near the end. We show that for N 1, the takeover time T is distributed as a Gumbel distribution for the star graph, as the convolution of two Gumbel distributions for a complete graph and an Erdős-Rényi random graph, as a normal for a one-dimensional ring and a two-dimensional lattice, and as a family of intermediate skewed distributions for d-dimensional lattices with d 3 (these distributions approach the convolution of two Gumbel distributions as d approaches infinity). Connections to evolutionary dynamics, cancer, incubation periods of infectious diseases, first-passage percolation, and other spreading phenomena in biology and physics are discussed.

Evolutionary dynamics of incubation periods

“…Yet under Death-birth dynamics, the distribution stays Gumbel for all nonzero values of ( Figure 4d,e,f ). The fact that birth-death dynamics returns a normal for whereas Death-birth still returns a Gumbel can be rationalized via various convergence theorems ( Baum and Billingsley, 1965 ; Ottino-Löffler et al, 2017 ; Pósfai, 2010 ). However, the fact that similar update rules behave so differently under a reasonable perturbation should caution us to be mindful of our choice of models.…”

“…Proof: The proof of this claim simply involves calculating the characteristic functions and taking a limit. We have presented the details elsewhere ( Ottino-Löffler et al, 2017 ).…”

“…The time to complete this collection (which is also well approximated by the sum of exponentials) has been the subject of much historical study. In fact, an exact distribution for in the limit of large is known ( Ottino-Löffler et al, 2017 ; Pósfai, 2010 ; Feller, 1968 ; Erdős and Rényi, 1961 ; Baum and Billingsley, 1965 ; Rubin and Zidek, 1965 ), and is given in normalized form by where is the Euler-Mascheroni constant. Therefore, if we can connect our fixation time to the coupon collector’s time , we would know its distribution.…”

Universal Statistics of Incubation Periods and Other Detection Times via Diffusion Models

“…One reason of the universal behavior in our works comes from modeling with SDEs. Their big advantage (in comparison with models of the kind considered in [BOL17], [OLSS17]) is that each SDE defines a whole universality class such that the macroscopic behavior of the models in the class can be effectively described by the exemplar SDE. This includes discrete and continuous random dynamics.…”

“…The new approach of [BOL17] and a companion paper [OLSS17] is to model random incubation periods as stopping times for certain probabilistic models of the disease spread within an individual infected organism. In this approach, an organism is modeled by a network of nodes connected to each other by edges and the spread of the infection or disease is modeled by random evolution of labeling of the network nodes.…”