Epidemiological models for the spread of pathogens in a population are usually only able to describe a single pathogen. This makes their application unrealistic in cases where multiple pathogens with similar symptoms are spreading concurrently within the same population. Here we describe a method which makes possible the application of multiple single-strain models under minimal conditions. As such, our method provides a bridge between theoretical models of epidemiology and data-driven approaches for modeling of influenza and other similar viruses. Our model extends the Susceptible-Infected-Recovered model to higher dimensions, allowing the modeling of a population infected by multiple viruses. We further provide a method, based on an overcomplete dictionary of feasible realizations of SIR solutions, to blindly partition the time series representing the number of infected people in a population into individual components, each representing the effect of a single pathogen. We demonstrate the applicability of our proposed method on five years of seasonal influenza-like illness (ILI) rates, estimated from Twitter data. We demonstrate that our method describes, on average, 44\% of the variance in the ILI time series. The individual infectious components derived from our model are matched to known viral profiles in the populations, which we demonstrate matches that of independently collected epidemiological data. We further show that the basic reproductive numbers ($R0$) of the matched components are in range known for these pathogens. Our results suggest that the proposed method can be applied to other pathogens and geographies, providing a simple method for estimating the parameters of epidemics in a population.