  Many systems arising in biological applications are subject to periodic
forcing. In these systems the forcing parameter is not only time-varying but
also known to have a periodic structure. We present an approach to estimating
periodic, time-varying parameters that imposes periodic structure by treating
the time-varying parameter as a piecewise function with unknown coefficients.
This method allows the resulting parameter estimate more flexibility in shape
than prescribing a specific functional form (e.g., sinusoidal) to model its
behavior, while still maintaining periodicity. We employ nonlinear filtering,
more specifically, a version of the augmented ensemble Kalman filter (EnKF), to
estimate the unknown coefficients comprising the piecewise approximation of the
periodic, time-varying parameter. This allows for straightforward comparison of
the proposed method with an EnKF-based parameter tracking algorithm, where
periodicity is not guaranteed.
  We demonstrate the effectiveness of the proposed approach on two biological
examples: a synthetic example with data generated from the nonlinear
FitzHugh-Nagumo system, modeling the excitability of a nerve cell, to estimate
the external voltage parameter, and a case study using reported measles
incidence data from three locations during the pre-vaccine era to estimate the
seasonal transmission parameter. The formulation of the proposed approach also
allows for simultaneous estimation of initial conditions and other static
system parameters, such as the reporting probability of measles cases, which is
vital for predicting under-reported incidence data.
