Epidemic Thresholds with External Agents

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.Comment: 12 pages, 2 figures (to appear in INFOCOM 2014

The Behavior of Epidemics under Bounded Susceptibility

We investigate the sensitivity of epidemic behavior to a bounded susceptibility constraint -- susceptible nodes are infected by their neighbors via the regular SI/SIS dynamics, but subject to a cap on the infection rate. Such a constraint is motivated by modern social networks, wherein messages are broadcast to all neighbors, but attention spans are limited. Bounded susceptibility also arises in distributed computing applications with download bandwidth constraints, and in human epidemics under quarantine policies. Network epidemics have been extensively studied in literature; prior work characterizes the graph structures required to ensure fast spreading under the SI dynamics, and long lifetime under the SIS dynamics. In particular, these conditions turn out to be meaningful for two classes of networks of practical relevance -- dense, uniform (i.e., clique-like) graphs, and sparse, structured (i.e., star-like) graphs. We show that bounded susceptibility has a surprising impact on epidemic behavior in these graph families. For the SI dynamics, bounded susceptibility has no effect on star-like networks, but dramatically alters the spreading time in clique-like networks. In contrast, for the SIS dynamics, clique-like networks are unaffected, but star-like networks exhibit a sharp change in extinction times under bounded susceptibility. Our findings are useful for the design of disease-resistant networks and infrastructure networks. More generally, they show that results for existing epidemic models are sensitive to modeling assumptions in non-intuitive ways, and suggest caution in directly using these as guidelines for real systems

Optimal Resource Allocation Over Time and Degree Classes for Maximizing Information Dissemination in Social Networks

We study the optimal control problem of allocating campaigning resources over the campaign duration and degree classes in a social network. Information diffusion is modeled as a Susceptible-Infected epidemic and direct recruitment of susceptible nodes to the infected (informed) class is used as a strategy to accelerate the spread of information. We formulate an optimal control problem for optimizing a net reward function, a linear combination of the reward due to information spread and cost due to application of controls. The time varying resource allocation and seeds for the epidemic are jointly optimized. A problem variation includes a fixed budget constraint. We prove the existence of a solution for the optimal control problem, provide conditions for uniqueness of the solution, and prove some structural results for the controls (e.g. controls are non-increasing functions of time). The solution technique uses Pontryagin's Maximum Principle and the forward-backward sweep algorithm (and its modifications) for numerical computations. Our formulations lead to large optimality systems with up to about 200 differential equations and allow us to study the effect of network topology (Erdos-Renyi/scale-free) on the controls. Results reveal that the allocation of campaigning resources to various degree classes depends not only on the network topology but also on system parameters such as cost/abundance of resources. The optimal strategies lead to significant gains over heuristic strategies for various model parameters. Our modeling approach assumes uncorrelated network, however, we find the approach useful for real networks as well. This work is useful in product advertising, political and crowdfunding campaigns in social networks.Comment: 14 + 4 pages, 11 figures. Author's version of the article accepted for publication in IEEE/ACM Transactions on Networking. This version includes 4 pages of supplementary material containing proofs of theorems present in the article. Published version can be accessed at http://dx.doi.org/10.1109/TNET.2015.251254

The impact of the network topology on the viral prevalence: a node-based approach

This paper addresses the impact of the structure of the viral propagation network on the viral prevalence. For that purpose, a new epidemic model of computer virus, known as the node-based SLBS model, is proposed. Our analysis shows that the maximum eigenvalue of the underlying network is a key factor determining the viral prevalence. Specifically, the value range of the maximum eigenvalue is partitioned into three subintervals: viruses tend to extinction very quickly or approach extinction or persist depending on into which subinterval the maximum eigenvalue of the propagation network falls. Consequently, computer virus can be contained by adjusting the propagation network so that its maximum eigenvalue falls into the desired subinterval