Influenza and respiratory syncytial virus (RSV) are the leading etiological agents of seasonal acute respiratory infections (ARI) around the world. Medical doctors typically base the diagnosis of ARI on patients symptoms alone and do not always conduct virological tests necessary to identify individual viruses, which limits the ability to study the interaction between multiple pathogens and make public health recommendations. We consider a stochastic kinetic model (SKM) for two interacting ARI pathogens circulating in a large population and an empirically motivated background process for infections with other pathogens causing similar symptoms. An extended marginal sampling approach based on the Linear Noise Approximation to the SKM integrates multiple data sources and additional model components. We infer the parameters defining the pathogens dynamics and interaction within a Bayesian hierarchical model and explore the posterior trajectories of infections for each illness based on aggregate infection reports from six epidemic seasons collected by the state health department, and a subset of virological tests from a sentinel program at a general hospital in San Luis Potosi, Mexico. We interpret the results based on real and simulated data and make recommendations for future data collection strategies. Supplementary materials and software are provided online.
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