  This paper studies the SI1SI2S spreading model of two competing behaviors
over a bilayer network. We address the problem of determining resource
allocation strategies which design a spreading network so as to ensure the
extinction of a selected process. Our discussion begins by extending the
SI1SI2S model to edge-dependent infection and node-dependent recovery
parameters with generalized graph topologies, which builds upon prior work that
studies the homogeneous case. We then find conditions under which the
mean-field approximation of a chosen epidemic process stabilizes to extinction
exponentially quickly. Leveraging this result, we formulate and solve an
optimal resource allocation problem in which we minimize the expenditure
necessary to force a chosen epidemic process to become extinct as quickly as
possible. In the case that the budget is not sufficient to ensure extinction of
the desired process, we instead minimize a useful heuristic. We explore the
efficacy of our methods by comparing simulations of the stochastic process to
the mean-field model, and find that the mean-field methods developed work well
for the optimal cost networks designed, but suffer from inaccuracy in other
situations.
