Trend filtering deconvolution methods in epidemiology

Reported cases provide a delayed and incomplete view of infection incidence, and hence make it difficult to recover the true timing and magnitude of an epidemic. To use such surveillance data to estimate latent infections, this dissertation develops a trend filtering deconvolution framework. Trend filtering uses discrete difference-based regularization of the signal to allow for locally adaptive reconstruction of infections over time, while deconvolution uses estimated probabilities to statistically reconstruct a signal from convolved data. These two modelling frameworks are thus useful for modelling epidemics whereby changes in infection counts can occur rapidly and belated case data are provided. We first establish our trend filtering deconvolution framework and use it to retrospectively estimate daily COVID-19 infections across U.S. states prior to the Omicron period, incorporating state-specific delay distributions as well as variant-specific incubation periods to obtain state-variant-specific deconvolved case estimates. We then scale the deconvolved estimates for unreported infections by using seroprevalence data in an antibody prevalence model. To extend the approach into the Omicron era, we introduce wastewater concentration data that enable infection estimation when case and seroprevalence surveillance data become less reliable. We then study the statistical properties of the estimator for convolution settings by extending classical trend filtering theory to the case where the design arises from delay distributions as a Toeplitz convolution operator. Using this approach, we establish an error bound that captures the extent that estimation accuracy depends on the smoothness of the signal and the conditioning of the convolution operator. We also develop a backfitting approach for multi-component deconvolution problems in additive settings and show how it can be used in practice to separate and estimate variant-specific latent infection signals.