Epidemic thresholds with external agents

Abstract

We study the effect of external infection sources on phase transitions in epidemic processes. In particular, we consider an epidemic spreading on a network via the SIS/SIR dynamics, which in addition is aided by external agents -sources unconstrained by the graph, but possessing a limited infection rate or virulence. Such a model captures many existing models of externally aided epidemics, and finds use in many settings -epidemiology, marketing and advertising, network robustness, etc. We provide a detailed characterization of the impact of external agents on epidemic thresholds. In particular, for the SIS model, we show that any external infection strategy with constant virulence either fails to significantly affect the lifetime of an epidemic, or at best, sustains the epidemic for a lifetime which is polynomial in the number of nodes. On the other hand, a random external-infection strategy, with rate increasing linearly in the number of infected nodes, succeeds under some conditions to sustain an exponential epidemic lifetime. We obtain similar sharp thresholds for the SIR model, and discuss the relevance of our results in a variety of settings.

The behavior of epidemics under bounded susceptibility

“…This being true, we have that the expected time to absorption for Z(t) is less than that of N (t). We note that this proof structure is similar to that of Theorem 4.1 from [8]; however it is much simplified via the ergodic embedding technique of [10].…”

“…Epidemic processes on networks have been studied across many disciplines; readers interested in more details regarding epidemics are referred to several excellent books on the subject [1,12,21,22]. Specifically, there is a vast literature to characterize spreading time in various contexts for SI processes [1,2,3,4], and phase transitions/extinction time for SIS processes [6,7,8,9,10,11]. Phase transitions for SIR processes are available in [13].…”

“…The SI/SIS Dynamics The SI, SIS dynamics models we use are standard in literature [8,10]. We consider a graph G(V, E) with n nodes (vertices).…”

“…standard Markov chain machinery, along with careful use of measure concentration. Some of the ideas we use are of fairly recent origin, in particular, the sharp expressions for extinction time in SIS models via embedding in an ergodic Markov chain (Lemma 1, which we adapt from Lemma 8 in [10]), and characterization of the SI spreading time in clique-like graphs, which is similar to results for first-passage percola-tion in random graphs [4]. However, unlike [4], our results (Theorems 1,3, 4 and 6) do not focus on a particular graph or generative model, but rather, relate the epidemic behavior to structural properties of the network.…”

“…A small modification of this model gives the popular SIS or 'Susceptible-Infected-Susceptible' dynamics [5,6,7,8,9,10,11] for epidemic processes. This is identical to the SI dynamics, except that now an infected node can revert to being susceptible at a (normalised) rate of '1'.…”