Most adult organs retain a population of stem cells that can respond to changes in physiological conditions or injury by producing new cells to maintain tissue homeostasis or repair damage. Upon division, stem cells can produce stem cells or progenitor cells, i.e. cells that can then differentiate into more specialized cells. Mathematical modeling is a tool that can help understand how stem cells decide between self-renewal and differentiation, and predict the number and fraction of cell types in tissues. It has been found experimentally that in neural stem cells, quiescence, i.e. a state in which the stem cell is inactive (or ``resting"), increases with age. Mathematical models could help to identify underlying reasons why this increase in quiescence with age seems to be the case. Therefore, I consider how including switching between quiescent and active stem cell states affects the neural stem cell (NSC) population count in the aging brain. I consider this from a perspective of considering no progenitor cells at all, a single progenitor state, and then finally multiple progenitor states, with and without depletion of quiescent stem cells. I find that time-dependent activation rate provides the best fit to the activated cell fraction (ACF) of NSCs over time, but that other model variants with constant parameter values can better fit the total number of NSCs over time. I also consider an alternate model for NSCs with nonlinear feedback from progenitor cells that affect NSC parameters, and compare all models to experimental stem cell and progenitor data. However, all of the feedback models considered provide a worse fit to the experimental data. This suggests that when switching between active and quiescent stem cells is considered, a time-dependent linear model outperforms the integral feedback mechanism considered by other models of stem cell lineages. Fitting progenitor data for both the time varying and feedback models indicates that four or five intermediate transit amplifying progenitor states are necessary, depending on whether or not depletion of stem cells is considered in the model. My results suggests that in order to determine whether an increase in age-related neural stem cell quiescence is determined by a decreasing stem cell activation rate or an increased stem cell depletion rate, additional experiments should be designed to explore whether or not depletion of the stem cell pool is occurring, and that a higher resolution time series for activated cell fraction (ACF) would be best to resolve this issue. I also study a simplified two component nonlinear cell lineage models with several different assumptions including, (1) the impact of allowing progenitor cells to divide, (2) the impact of including cellular death, (3) the impact of forcing progenitor division to be dependent on stem cell population (i.e. positive feedback), and (4) the impact of including symmetric and asymmetric cell divisions. I find that including progenitor division can lead to emergence of unrealistic stem cell free steady state. I also find that the inclusion of death of progenitors is necessary (though not sufficient) for nontrivial steady states to exist. I find that Hopf bifurcations may be possible in some of these models, though the parameter ranges at which they occur are unrealistic. I also show that while assuming asymmetric cell division does not have a major impact on model output, having a nonzero death rate of stem cells and allowing progenitor cells to divide has a major impact.  Models, where progenitor division is modified to depend on stem cells, are found to have significantly smaller populations of stem and progenitor cells compared to the original variants. A stochastic simulation of these nonlinear stem-progenitor models using the Gillespie algorithm is also performed. I find that if the number of stem cells at steady state is small, then there is a high probability of extinction in the stochastic version of the model.