Amyloid fibrils are a common component in many debilitating human neurological diseases such as Alzheimer's (AD), Parkinson's, and Creutzfeldt-Jakob, and in animal diseases such as BSE. The role of fibrillar ?? proteins in AD has stimulated interest in the kinetics of ?? fibril formation. Kinetic models that include reaction pathways and rate parameters for the various stages of the process can be helpful towards understanding the dynamics on a molecular level. Based upon experimental data, we have developed a mathematical model for the reaction pathways and determined rate parameters for peptide secondary structural conversion and aggregation during the entire fibrillogenesis process from random coil to mature fibrils, including the molecular species that accelerate the conversions. The model and the rate parameters include different molecular structural stages in the nucleation and polymerization processes and the numerical solutions yield graphs of concentrations of different molecular species versus time that are in close agreement with experimental results. The model also allows for the calculation of the time-dependent increase in aggregate size. The calculated results agree well with experimental results, and allow differences in experimental conditions to be included in the calculations. The specific steps of the model and the rate constants that are determined by fitting to experimental data provide insight on the molecular species involved in the fibril formation process.