Estimating Consensus Epidemic Trajectories via a Constrained Power Fr\'echet Mean with Functional Registration

Abstract

In infectious disease modeling during the early phase of a pandemic, SEIR-type compartmental models are standard tools, and they require epidemiological parameters as inputs. Because these parameters are subject to uncertainty, different research groups often report different epidemic curves, and obtaining a representative curve that captures the characteristics of these trajectories is important for decision-making. However, simple pointwise summaries can attenuate epidemic peaks under temporal misalignment and generally do not preserve the dynamical structure of the underlying compartmental models. To address these limitations, we propose a method for summarizing multiple solutions to SEIR-type compartmental models on a functional space by computing a constrained power Fr\'echet mean with temporal shift registration. In our method, we regard the pairs of exposed and infectious compartments as objects in a Hilbert space, and the consensus curve is defined as the solution to a constrained optimization problem. Differential equation constraints and population constraints are incorporated in the optimization to preserve a partially mechanistic interpretation regarding the infectious compartment. We develop an implementable block-optimization algorithm based on basis function expansion. In simulation studies based on early COVID-19 parameter estimates, the proposed method produced consensus curves with a single epidemic peak, whereas pointwise summaries exhibited attenuated or multiple peaks. We further applied the method to six literature-derived parameter sets from early COVID-19 studies and obtained a representative trajectory with interpretable epidemiological parameters. The proposed approach provides a generalized trajectory-summarization method that includes mean- and median-type estimators and preserves mechanistic interpretability.