The most common reported epidemic time series in epidemiological surveillance are the daily or weekly incidence of new cases, the hospital admission count, the ICU admission count, and the death toll, which played such a prominent role in the struggle to monitor the Covid-19 pandemic. We show that pairs of such curves are related to each other by a generalized renewal equation depending on a smooth time varying delay and a smooth ratio generalizing the reproduction number. Such a functional relation is also explored for pairs of simultaneous curves measuring the same indicator in two neighboring countries. Given two such simultaneous time series, we develop, based on a signal processing approach, an efficient numerical method for computing their time varying delay and ratio curves, and we verify that its results are consistent. Indeed, they experimentally verify symmetry and transitivity requirements and we also show, using realistic simulated data, that the method faithfully recovers time delays and ratios. We discuss several real examples where the method seems to display interpretable time delays and ratios. The proposed method generalizes and unifies many recent related attempts to take advantage of the plurality of these health data across regions or countries and time, providing a better understanding of the relationship between them. An implementation of the method is publicly available at the EpiInvert CRAN package.|To monitor an epidemic, it is crucial to understand the relationship between the incidence of new cases, the hospital admission count, the ICU admission count, and the death toll time series. The relationship between any pair of such indicators can be formulated in terms of temporal delays and ratios which evolve across time. Given two such time series, we develop, based on a signal processing approach, an efficient numerical method for computing their time varying delay and ratio curves. Using realistic simulated data, we show that the method faithfully recovers time delays and ratios. In addition, we discuss several applications to real epidemic data where the method seems to output interpretable time delays and ratios. The obtained relationship between these epidemic time series is a key issue in epidemiological surveillance. The proposed technique provides a new tool to visualize, compare, and understand the evolution of key epidemiological time series. An implementation of the method is publicly available at the EpiInvert CRAN package.