We propose a method that uses a two-species ordinary differential equation (ODE) model to characterize misfolded tau (or simply tau) protein spreading in Alzheimer's disease (AD) and calibrates it from clinical data. The unknown model parameters are the initial condition (IC) for tau and three scalar parameters representing the migration, proliferation, and clearance of tau proteins. Driven by imaging data, these parameters are estimated by formulating a constrained optimization problem with a sparsity regularization for the IC. This optimization problem is solved with a projection-based quasi-Newton algorithm. We investigate the sensitivity of our method to different algorithm parameters. We evaluate the performance of our method on both synthetic and clinical data. The latter comprises cases from the AD Neuroimaging Initiative (ADNI) datasets: 455 cognitively normal (CN), 212 mild cognitive impairment (MCI), and 45 AD subjects. We compare the performance of our approach to the commonly used Fisher-Kolmogorov (FK) model with a fixed IC at the entorhinal cortex (EC). Our method demonstrates an average improvement of 25.7% relative error compared to the FK model on the AD dataset. HFK also achieves an R-squared score of 0.664 for fitting AD data compared with 0.55 from FK model results under the same optimization scheme. Furthermore, for cases that have longitudinal data, we estimate a subject-specific AD onset time.