Rare Event Extinction on Stochastic Networks

We consider the problem of extinction processes on random networks with a given structure. For sufficiently large well-mixed populations, the process of extinction of one or more state variable components occurs in the tail of the quasi-stationary probability distribution, thereby making it a rare event. Here we show how to extend the theory of large deviations to random networks to predict extinction times. In particular, we use the theory to find the most probable path leading to extinction. We apply the methodology to epidemic models and discover how mean extinction times scale with epidemiological and network parameters in Erdős-Rényi networks. The results are shown to compare quite well with Monte Carlo simulations of the network in predicting both the most probable paths to extinction and mean extinction times. In many models of finite populations, random fluctuations occur due to internal interactions between individuals or agents, and/or external stochastic forces. Such fluctuations are evident in the modeling of well mixed populations that support epidemics of disease spread [1], as well as ecological bio-diversity of species [2,3], among others. Typically, the dynamics performs small random fluctuations about an attracting state. However, in almost all populations of finite size, fluctuations may organize in such a way to drive one or more components of the population to extinction. Mechanisms conjectured to play a role in extinction processes include small population size, low contact frequency for frequency-dependent transmission, competition for resources, and evolutionary pressure [4], as well as heterogeneity in populations and transmission in coupled population models [5, 6]. In epidemic models where the population is well-mixed, extinction of infectious individuals has been shown to be affected by noise intensity [7], peak infectious population size [8], and seasonal phase occurrence [9]. Moreover, since the extinct state is typically unstable in the deterministic mean field and is an absorbing state of a stochastic process, time scales for extinction may be exponentially long [10]; i.e., the probability of extinction is a decreasing exponential function [11]. Vaccination and treatment programs have been studied to speed up the extinction of disease in well-mixed populations [10, 12]. For example, although most vaccination schedules are designed to be administered periodically (deterministic) [13–15], Poisson distributed scheduling was recently shown to be more efficient than regular treatment schedules [16]. In network populations, outbreak extinction probabilities have been predicted for early times when an infection has just been introduced [17,18]. Other studies of extinction on networks attempt to predict whether a persistent non-extinct state exists, such as for computer viruses in growing networks [19] and epidemics in various network geometries (e.g., [20]). This is equivalent to finding the bifurcation point where the extinct state becomes unstable and thus can frequently be predicted using deterministic approximations (e.g., [19]). Extinction times for an endemic disease in a network have occasionally been studied, mostly numerically. In [21], extinction times were measured in simulations of an epidemic on an adaptive network, and the log of the extinction time was observed to increase with a power of the distance from a bifurcation point. In [22], lifetimes were measured in a similar model with the addition of pulsed vaccination. Similar lifetimes were obtained in simulations by adding stochastic pulsed vaccination to a deterministic mean-field model based on a pair approximation, suggesting that it is not necessary to model the full stochastic network dynamics in detail to capture the extinction time. Here we will present a method for finding the most probable path to extinction and the extinction time in networks that can be described using a pair approximation. p-1 ar X iv :1 41 1. 00 17 v2 [ qbi o. PE ] 8 D ec 2 01 4 Lindley, Shaw, and Schwartz Here we consider the problem of epidemic extinction in stochastic networks, and we find that the extinction process depends not just on the nodes of the network, but also on how the links change as the system evolves. That is, along the most probable path to extinction, we have derived a new approximate model showing that both nodes and links play an important role in the mean time to disease eradication. We have introduced a novel mathematical tool so that the path is derived constructively. The specific example we will consider here is a network of N nodes and K links, with an average degree of 2K/N . The dynamics on the network is an SIS epidemic model, where susceptibles capable of acquiring the disease become infectious through a contact with an infective individual, and become susceptible again after a recovery period. Here we divide the population into two groups, with S as the number of susceptible individuals and I as the number of infected individuals, such that S + I = N . The population is closed, and there are no births and deaths. To quantify how the network topology compares with a well-mixed global, or all-to-all, coupling structure, we allow for two modes of disease spread. Specifically, global disease transmission occurs in a well-mixed population when the disease transmits from any infected person to any susceptible person. In contrast to global transmission models, local disease transmission occurs when a network of connections between individual persons is considered, and diseases can only spread along links between infected and susceptible individuals. In the case of transmission via only the network, we let p be the infection rate per SI link. When transmission is only global, we define a global contact rate of p2K/N , selected so that the epidemic threshold occurs at p∗ = rN 2K in both the globally coupled and network transmission cases. We will introduce a homotopy parameter which continuously transforms the system between the local and global transmission models. Finally, r is a recovery rate. We approximate the dynamics of the network by considering transitions in nodes and links, similar to the pairbased proxy model of [23]. We let X denote the vector with components containing both the number of nodes and number of links of the network. Specifically, we let X = [S, I,NSS , NSI , NII ], where S, I denotes the numbers of susceptibles and infectives, respectively, and NAB denotes the number of connections between nodes of type A and B. We assume large but finite population size, N , and we suppose the dynamics proceeds as a Markov process. To complete the formulation for the network dynamics, we suppose there existM transition events with transition rates W (X,νk) having increments νk. The dynamics of the probability density, ρ(X, t), can now be modeled as a master equation [11,24,25]: