An efficient curing policy for epidemics on graphs

We provide a dynamic policy for the rapid containment of a contagion process modeled as an SIS epidemic on a bounded degree undirected graph with n nodes. We show that if the budget $r$ of curing resources available at each time is ${\Omega}(W)$, where $W$ is the CutWidth of the graph, and also of order ${\Omega}(\log n)$, then the expected time until the extinction of the epidemic is of order $O(n/r)$, which is within a constant factor from optimal, as well as sublinear in the number of nodes. Furthermore, if the CutWidth increases only sublinearly with n, a sublinear expected time to extinction is possible with a sublinearly increasing budget $r$

Optimal curing policy for epidemic spreading over a community network with heterogeneous population

The design of an efficient curing policy, able to stem an epidemic process at an affordable cost, has to account for the structure of the population contact network supporting the contagious process. Thus, we tackle the problem of allocating recovery resources among the population, at the lowest cost possible to prevent the epidemic from persisting indefinitely in the network. Specifically, we analyze a susceptible-infected-susceptible epidemic process spreading over a weighted graph, by means of a first-order mean-field approximation. First, we describe the influence of the contact network on the dynamics of the epidemics among a heterogeneous population, that is possibly divided into communities. For the case of a community network, our investigation relies on the graph-theoretical notion of equitable partition; we show that the epidemic threshold, a key measure of the network robustness against epidemic spreading, can be determined using a lower-dimensional dynamical system. Exploiting the computation of the epidemic threshold, we determine a cost-optimal curing policy by solving a convex minimization problem, which possesses a reduced dimension in the case of a community network. Lastly, we consider a two-level optimal curing problem, for which an algorithm is designed with a polynomial time complexity in the network size.Comment: to be published on Journal of Complex Network

When is a network epidemic hard to eliminate?

We consider the propagation of a contagion process (epidemic) on a network and study the problem of dynamically allocating a fixed curing budget to the nodes of the graph, at each time instant. For bounded degree graphs, we provide a lower bound on the expected time to extinction under any such dynamic allocation policy, in terms of a combinatorial quantity that we call the resistance of the set of initially infected nodes, the available budget, and the number of nodes n. Specifically, we consider the case of bounded degree graphs, with the resistance growing linearly in n. We show that if the curing budget is less than a certain multiple of the resistance, then the expected time to extinction grows exponentially with n. As a corollary, if all nodes are initially infected and the CutWidth of the graph grows linearly, while the curing budget is less than a certain multiple of the CutWidth, then the expected time to extinction grows exponentially in n. The combination of the latter with our prior work establishes a fairly sharp phase transition on the expected time to extinction (sub-linear versus exponential) based on the relation between the CutWidth and the curing budget

A lower bound on the performance of dynamic curing policies for epidemics on graphs

We consider an SIS-type epidemic process that evolves on a known graph. We assume that a fixed curing budget can be allocated at each instant to the nodes of the graph, towards the objective of minimizing the expected extinction time of the epidemic. We provide a lower bound on the optimal expected extinction time as a function of the available budget, the epidemic parameters, the maximum degree, and the CutWidth of the graph. For graphs with large CutWidth (close to the largest possible), and under a budget which is sublinear in the number of nodes, our lower bound scales exponentially with the size of the graph

The Cost of Uncertainty in Curing Epidemics

Motivated by the study of controlling (curing) epidemics, we consider the spread of an SI process on a known graph, where we have a limited budget to use to transition infected nodes back to the susceptible state (i.e., to cure nodes). Recent work has demonstrated that under perfect and instantaneous information (which nodes are/are not infected), the budget required for curing a graph precisely depends on a combinatorial property called the CutWidth. We show that this assumption is in fact necessary: even a minor degradation of perfect information, e.g., a diagnostic test that is 99% accurate, drastically alters the landscape. Infections that could previously be cured in sublinear time now may require exponential time, or orderwise larger budget to cure. The crux of the issue comes down to a tension not present in the full information case: if a node is suspected (but not certain) to be infected, do we risk wasting our budget to try to cure an uninfected node, or increase our certainty by longer observation, at the risk that the infection spreads further? Our results present fundamental, algorithm-independent bounds that tradeoff budget required vs. uncertainty.Comment: 35 pages, 3 figure

Halting viruses in scale-free networks

The vanishing epidemic threshold for viruses spreading on scale-free networks indicate that traditional methods, aiming to decrease a virus' spreading rate cannot succeed in eradicating an epidemic. We demonstrate that policies that discriminate between the nodes, curing mostly the highly connected nodes, can restore a finite epidemic threshold and potentially eradicate a virus. We find that the more biased a policy is towards the hubs, the more chance it has to bring the epidemic threshold above the virus' spreading rate. Furthermore, such biased policies are more cost effective, requiring less cures to eradicate the virus

Path-Based Epidemic Spreading in Networks

Conventional epidemic models assume omnidirectional contact-based infection. This strongly associates the epidemic spreading process with node degrees. The role of the infection transmission medium is often neglected. In real-world networks, however, the infectious agent as the physical contagion medium usually flows from one node to another via specific directed routes ( path-based infection). Here, we use continuous-time Markov chain analysis to model the influence of the infectious agent and routing paths on the spreading behavior by taking into account the state transitions of each node individually, rather than the mean aggregated behavior of all nodes. By applying a mean field approximation, the analysis complexity of the path-based infection mechanics is reduced from exponential to polynomial. We show that the structure of the topology plays a secondary role in determining the size of the epidemic. Instead, it is the routing algorithm and traffic intensity that determine the survivability and the steady-state of the epidemic. We define an infection characterization matrix that encodes both the routing and the traffic information. Based on this, we derive the critical path-based epidemic threshold below which the epidemic will die off, as well as conditional bounds of this threshold which network operators may use to promote/suppress path-based spreading in their networks. Finally, besides artificially generated random and scale-free graphs, we also use real-world networks and traffic, as case studies, in order to compare the behaviors of contact- and path-based epidemics. Our results further corroborate the recent empirical observations that epidemics in communication networks are highly persistent