A convex framework to control spreading processes in directed networks

We propose a convex optimization framework to compute the optimal distribution of protection resources in order to control a spreading process propagating throughout a network of contacts. The spreading process under consideration is an extension of the popular SIS model of viral infection in a network with non-identical nodes and directed edges. We assume we have a limited budget available to invest on three types of network protection resources: (i) Edge control resources, (ii), preventative resources and (iii) corrective resources. Edge control resources are employed to impose restrictions on the contact rates across directed edges in the contact network. Preventative resources are allocated to nodes in order 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 with numerical simulations in a real-world air transportation network.