  Estimating model parameters is a crucial step in mathematical modelling and
typically involves minimizing the disagreement between model predictions and
experimental data. This calibration data can change throughout a study,
particularly if modelling is performed simultaneously with the calibration
experiments, or during an on-going public health crisis as in the case of the
COVID-19 pandemic. Consequently, the optimal parameter set, or maximal
likelihood estimator (MLE), is a function of the experimental data set. Here,
we develop a numerical technique to predict the evolution of the MLE as a
function of the experimental data. We show that, when considering perturbations
from an initial data set, our approach is significantly more computationally
efficient that re-fitting model parameters while resulting in acceptable model
fits to the updated data. We use the continuation technique to develop an
explicit functional relationship between fit model parameters and experimental
data that can be used to measure the sensitivity of the MLE to experimental
data. We then leverage this inverse sensitivity analysis to select between
model fits with similar information criteria, \textit{a priori} determine the
experimental measurements to which the MLE is most sensitive, and suggest
additional experiment measurements that can resolve parameter uncertainty.
