Epidemic outbreaks in human populations are facilitated by the underlying transportation network. We consider strategies for containing a viral spreading process by optimally allocating a limited budget to three types of protection resources: (i ) Traffic control resources, (ii ), preventative resources and (iii ) corrective resources. Traffic control resources are employed to impose restrictions on the traffic flowing across directed edges in the transportation network. Preventative resources are allocated to nodes to reduce the probability of infection at that node (e.g. vaccines), and corrective resources are allocated to nodes to increase the recovery rate at that node (e.g. antidotes). We assume these resources have monetary costs associated with them, from which we formalize an optimal budget allocation problem which maximizes containment of the infection. We present a polynomial time solution to the optimal budget allocation problem using Geometric Programming (GP) for an arbitrary weighted and directed contact network and a large class of resource cost functions. We illustrate our approach by designing optimal traffic control strategies to contain an epidemic outbreak that propagates through a real-world air transportation network. * The authors are with the Department of Electrical and Systems Engineering at the University of Pennsylvania, Philadelphia PA 19104.

We consider the problem of controlling the propagation of an epidemic outbreak in an arbitrary contact network by distributing vaccination resources throughout the network. We analyze a networked version of the Susceptible-Infected-Susceptible (SIS) epidemic model when individuals in the network present different levels of susceptibility to the epidemic. In this context, controlling the spread of an epidemic outbreak can be written as a spectral condition involving the eigenvalues of a matrix that depends on the network structure and the parameters of the model. We study the problem of finding the optimal distribution of vaccines throughout the network to control the spread of an epidemic outbreak. We propose a convex framework to find cost-optimal distribution of vaccination resources when different levels of vaccination are allowed. We also propose a greedy approach with quality guarantees for the case of all-or-nothing vaccination. We illustrate our approaches with numerical simulations in a real social network.

Most existing work uses dual decomposition and subgradient methods to solve network optimization problems in a distributed manner, which suffer from slow convergence rate properties. This paper proposes an alternative distributed approach based on a Newton-type method for solving minimum cost network optimization problems. The key component of the method is to represent the dual Newton direction as the solution of a discrete Poisson equation involving the graph Laplacian. This representation enables using an iterative consensus-based local averaging scheme (with an additional input term) to compute the Newton direction based only on local information. We show that even when the iterative schemes used for computing the Newton direction and the stepsize in our method are truncated, the resulting iterates converge superlinearly within an explicitly characterized error neighborhood. Simulation results illustrate the significant performance gains of this method relative to subgradient methods based on dual decomposition.

Abstract-Dual descent methods are commonly used to solve network optimization problems because their implementation can be distributed through the network. However, their convergence rates are typically very slow. This paper introduces a family of dual descent algorithms that use approximate Newton directions to accelerate the convergence rate of conventional dual descent. These approximate directions can be computed using local information exchanges thereby retaining the benefits of distributed implementations. The approximate Newton directions are obtained through matrix splitting techniques and sparse Taylor approximations of the inverse Hessian. We show that, similarly to conventional Newton methods, the proposed algorithm exhibits superlinear convergence within a neighborhood of the optimal value. Numerical analysis corroborates that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods. A connection with recent developments that use consensus iterations to compute approximate Newton directions is also presented.

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