Age-structured population models in cyclical neutropenia: a numerical investigation.

Blood is composed of a variety of cells which play important roles in the health of an organism. Among these cells are white blood cells which are responsible for the body's immune response. An important type of white blood cell is the neutrophil. In this thesis, we investigate a model of cyclical neutropenia, a hematological disease characterized by abnormal oscillations in the neutrophil count of an organism. A standard treatment for this disease is to inject an apoptosis-inhibiting hormone, G-CSF, at periodic intervals. Mathematical models to simulate the dynamics of neutrophil populations with and without G-CSF treatment were developed by C. Foley, [4]. These models include the populations in the cell line from stem cells to neutrophils, and consist of a nonlinear hyperbolic system of coupled integro-differential equations. The author then reduces the model to a system of delay differential equations which are then discretized to yield approximate solutions. In this thesis, we first provide a quick overview of age-structured population models. We then discuss the origin of the PDE (partial differential equation) models in [4], and highlight some of their features which render their simulation very challenging. We describe some numerical approximation strategies employed by other authors for age-structured population models which did not converge for our model, and provide some insight into the reasons. We then discuss the modification of a splitting strategy, which does provide a convergent method for the system of PDE. We finally provide some numerical results, and compare our findings to those obtained in [4] on the DDE (delay differential equation) model.