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This paper proposed a control-strategies for nodes to control the spread of an epidemic outbreak in arbitrary directed graphs by optimally allocating their resources throughout the network. Epidemic propagation is well modeled as a networked version of the Susceptible-Exposed-Infected-Susceptible (SEIS) epidemic process. Using the Kolmogorov forward equations and mean-field approximation, we present a mean-field model to describe the spreading dynamics and prove the existence of a necessary and sufficient condition for global exponential stability. Based on this stability condition, we can derive another condition to control the spread of an epidemic outbreak in terms of the eigenvalues of a matrix that depends on the network structure and the parameters of the model. According to different control purposes and conditions, two types of control-theoretic decision can be considered: 1)given a fixed budget, find the optimal resource allocation to achieve the highest level of containment, 2)given a decay rate of epidemic, find the minimum cost to control the spreading process at a desired decay rate. A geometric program can be formulated to solve the optimal problems and the existence of solutions is also proved. Numerical simulations can illustrated our results.

A geometric programming approach for optimal resource allocation to control epidemic outbreaks in arbitrary networks