Optimal Resource Allocation for Competitive Spreading Processes on Bilayer Networks

Nicholas J. Watkins, Cameron Nowzari, Victor M. Preciado, George J. Pappas

Abstract: We consider a competitive epidemic process in a bilayer network, and develop a framework to find an optimal allocation of control resources to eliminate one of the epidemics. We consider the SI <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> SI <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> S model, a recent generalization of the popular SIS model to the case of two competitive epidemics. We start our analysis by extending the standard SI <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> SI <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> S formulation with homogeneous parameters to a heterogeneous setting with edge-dependent infection rates, and node-dependent recovery rates. We then find necessary and sufficient conditions under which the mean-field approximation of a chosen epidemic process stabilizes to extinction exponentially quickly. Leveraging this result, we develop a framework for the solution of two optimization problems. In the first, we find an optimal allocation of control resources in order to eradicate the chosen epidemic at a minimum cost. In the second, we are given a fixed budget and propose a method which provably attains the extinction condition when sufficient capital is available, and otherwise mitigates the spread of the unwanted epidemic as much as possible. We explore the efficacy of our methods through extensive simulation.