Inference for stochastic kinetic models from multiple data sources for joint estimation of infection dynamics from aggregate reports and virological data

Before the current pandemic, influenza and respiratory syncytial virus (RSV) were the leading etiological agents of seasonal acute respiratory infections (ARI) around the world. In this setting, medical doctors typically based the diagnosis of ARI on patients' symptoms alone and did not routinely conduct virological tests necessary to identify individual viruses, limiting the ability to study the interaction between multiple pathogens and to 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 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 Potos\'{i}, M\'{e}xico. We interpret the results and make recommendations for future data collection strategies.