We propose a mathematical framework, based on conic geometric programming, to control a susceptible-infected-susceptible viral spreading process taking place in a directed contact network with unknown contact rates. We assume that we have access to time series data describing the evolution of the spreading process observed by a collection of sensor nodes over a finite time interval. We propose a data-driven robust convex optimization framework to find the optimal allocation of protection resources (e.g., vaccines and/or antidotes) to eradicate the viral spread at the fastest possible rate. In contrast to current network identification heuristics, in which a single network is identified to explain the observed data, we use available data to define an uncertainty set containing all networks that are coherent with empirical observations. Our characterization of this uncertainty set of networks is tractable in the context of conic geometric programming, recently proposed by Chandrasekaran and Shah, which allows us to efficiently find the optimal allocation of resources to control the worst-case spread that can take place in the uncertainty set of networks. We illustrate our approach in a transportation network from which we collect partial data about the dynamics of a hypothetical epidemic outbreak over a finite period of time.