  Delay differential equations form the underpinning of many complex dynamical
systems. The forward problem of solving random differential equations with
delay has received increasing attention in recent years. Motivated by the
challenge to predict the COVID-19 caseload trajectories for individual states
in the U.S., we target here the inverse problem. Given a sample of observed
random trajectories obeying an unknown random differential equation model with
delay, we use a functional data analysis framework to learn the model
parameters that govern the underlying dynamics from the data. We show existence
and uniqueness of the analytical solutions of the population delay random
differential equation model when one has discrete time delays in the functional
concurrent regression model and also for a second scenario where one has a
delay continuum or distributed delay. The latter involves a functional linear
regression model with history index. The derivative of the process of interest
is modeled using the process itself as predictor and also other functional
predictors with predictor-specific delayed impacts. This dynamics learning
approach is shown to be well suited to model the growth rate of COVID-19 for
the states that are part of the U.S., by pooling information from the
individual states, using the case process and concurrently observed economic
and mobility data as predictors.
