# Improved Bounds on the Epidemic Threshold of Exact SIS Models on Complex Networks

###### Abstract

The SIS (susceptible-infected-susceptible) epidemic model on an arbitrary network, without making approximations, is a -state Markov chain with a unique absorbing state (the all-healthy state). This makes analysis of the SIS model and, in particular, determining the threshold of epidemic spread quite challenging. It has been shown that the exact marginal probabilities of infection can be upper bounded by an -dimensional linear time-invariant system, a consequence of which is that the Markov chain is “fast-mixing” when the LTI system is stable, i.e. when (where is the infection rate per link, is the recovery rate, and is the largest eigenvalue of the network’s adjacency matrix). This well-known threshold has been recently shown not to be tight in several cases, such as in a star network. In this paper, We provide tighter upper bounds on the exact marginal probabilities of infection, by also taking pairwise infection probabilities into account. Based on this improved bound, we derive tighter eigenvalue conditions that guarantee fast mixing (i.e., logarithmic mixing time) of the chain. We demonstrate the improvement of the threshold condition by comparing the new bound with the known one on various networks and epidemic parameters.

## I Introduction

The mathematical modeling and analysis of epidemic spread is of great importance in understating dynamical processes over complex networks (e.g. social networks) and has attracted significant interest from different communities in recent years. The study of epidemics plays a key role in many areas beyond epidemiology [bailey1975mathematical], such as viral marketing [phelps2004viral, richardson2002mining], network security [alpcan2010network, acemoglu2013network], and information propagation [jacquet2010information, cha2009measurement]. Although there is a huge body of work on epidemic models, classical ones mostly neglect the underlying network structure and assume a uniformly mixed population, which is obviously far from reality. However, in recent years more realistic networked models have been introduced, and many interesting results are now known [nowzari2016analysis, pastor2015epidemic].

In the simplest case (the binary-state or SIS model) each node is in one of two different states: susceptible (S) or infected (I). During any time interval, each susceptible (healthy) node has a chance of being independently infected by any of its infected neighbors (with probability ). Further, during any time interval, each infected node has a chance of recovering (with probability ) and becoming susceptible again. For a network with nodes, this yields a Markov chain with states, which is referred to as the exact or “stochastic” model. Since analyzing this model is quite challenging, most researchers have resorted to -dimensional linear and nonlinear approximations (the most common being the âmean-fieldâ approximation), which are sometimes called “deterministic” models. This paper focuses on improving known bounds on the exact model.

The spreading process can be considered either as a discrete-time Markov chain or a continuous-time one. Although the discrete-time model is sometimes argued to be more realistic [arenas2010discrete, ahn2014random], there is no fundamental difference between the two, and similar results have been shown for both. We focus on the discrete-time Markov chain here.

It is known that these epidemic models exhibit a phase transition behavior at a certain threshold [castellano2010thresholds, barrat2008dynamical] , i.e., once the effective infection rate approaches a critical value [nowzari2016analysis] the epidemic appears not to die out. We should remark that the Markov chain has a unique absorbing state, which is the all-healthy state, because once the system reaches this state it remains there forever since there are no infected nodes to propagate infections. This means that if we wait long enough the epidemic will eventually die out, which may seem to be odd at first. However, what this means is that the question of the epidemic dying out is not interesting; what is interesting is the question of how long it takes for the epidemic to die out? In particular, if the mixing time of the Markov chain is exponentially large one will not see it die out in any reasonable time. Therefore the right question to ask is what is the mixing time of the Markov chain (or, equivalently, its mean-time-to-absorption); it turns out that the threshold corresponds to the phase transition between “slow mixing” (exponential time) and “fast mixing” (logarithmic time) of the MC [draief2010epidemics, van2013decay, van2014upper].

The epidemic threshold (critical value) of general networks is still an open problem. However, lower- and upper-bounds have been found using different techniques [draief2010epidemics, van2014upper]. The best known lower-bound is , i.e. the inverse of the leading eigenvalue of the adjacency matrix, which is derived by upper-bounding the marginal probabilities of infection and using a linear dynamical system. In fact, this method relies on keeping track of variables which are upper bounds on the marginal probability of infection for any of the nodes. In this paper, we focus on improving this upper-bound on the infection probabilities and ultimately the lower-bound on the epidemic threshold. The key idea, is to maintain the “pairwise” probabilities of nodes’ infections, in addition to the marginals. This comes at the cost of increased, yet still perfectly feasible, computation. There is a trade-off between the tightness of the bound and the complexity, and in theory if one takes into account all marginals, pairs, triples, and higher order terms, we get back to the original -state Markov chain.

We first briefly review the known bound with marginals, and show a simple alternative approach for deriving it. We then move on to pairs and use the machinery developed in Section II to derive tighter bounds on the probabilities and connect it with the mixing time of the Markov chain (Sections LABEL:sec:p_ij and LABEL:sec:q_ij). Finally, we demonstrate the improvement of the bounds through extensive simulations (Section LABEL:sec:simulations), and conclude with future directions.

## Ii The Markov Chain and Marginal Probabilities of Infection

Let be an arbitrary connected undirected network with nodes, and with adjacency matrix . Each node can be in a state of health, represented by “0”, or a state of infection, represented by “1”. The state of the entire network can be represented by a binary n-tuple , where each of the entries represents the state of a node at time , i.e. is infected if and it is healthy if .

Given the current state , the infection probability of each node in the next step is determined independently, and therefore the transition matrix of this Markov Chain has elements of the following form:

(1) |

for any two state vectors .

As mentioned before, a healthy node can receive infection from any of its infected neighbors independently with probability per infected link, and an infected node can recover from the disease with probability . That is