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 estimate periodic, time-varying parameters using nonlinear Bayesian filtering. Most parameter estimation methodology in the literature is aimed at estimating constant parameters or parameters whose values drift over time with no imposed structure. The proposed technique imposes periodic structure by treating the time-varying parameter as a piecewise function with unknown coefficients , estimated using the ensemble Kalman filter (EnKF) . 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 compare the proposed method to an EnKF-based parameter drift algorithm, where periodicity is not guaranteed , using synthetic data generated from the  FitzHugh-Nagumo system which models the spiking dynamics of a neuron. We further demonstrate the proposed method by estimating the seasonal transmission parameterin an epidemic model for the spread of measles. Results are obtained using time-series data of measles case reports from three locations during the pre-vaccine era , in particular the weekly reported cases in England and Wales (1948-1967) and monthly reported cases in New York City (1945-1964) and Baltimore (1928-1960). The augmented EnKF implementation 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.
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.
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