Mathematical Models of T-cell Population Dynamics in Aging and Immune Disease
In this dissertation, we use birth-death-immigration systems of ordinary differential equations to study the dynamics of human naive T-cell populations in healthy aging and disease of the immune system. We derive a model that tracks both total cell counts and counts of clones (groups of genetically identical cells) of a particular size; from the latter, we compute the total clone count, or ``diversity", which provides a quantitative measure of the extent to which the T-cell pool can cope with invading pathogens. We first formulate a nonautonomous model of T-cell birth and replenishment throughout an individual's lifetime, and use it to assess the relationship between the immune tissue damage, T-cell loss and dysfunction, and weakened immune response all observed simultaneously in aging. We identify tissue loss in the thymus as a fundamental cause of diminished T-cell counts, and also that diversity loss can only underlie weakened immune effectiveness in aging if small clones are ineffective against pathogen. Using a short-time, autonomous version of the same ODE, we then study changes to the T-cell pool during an instance of acute thymic atrophy and recovery. We identify equilibrium solutions that arise at different rates of T-cell production, and derive analytic approximations to the eigenvalues and eigenvectors of the linearization around the equilibria. From the forms of the eigenvalues and eigenvectors, we are able to estimate rates at which different size-segregated groups of clones converge to equilibria--that is, ``adjust" to the changing rate of T-cell production. Finally, we formulate a total cell count model of populations of HIV-immune and HIV-susceptible cells of the T-cell lineage in the bone marrow, thymus, and peripheral blood after transplant of HIV-immune bone marrow into an HIV-infected individual. We show that independent of assumptions about the homeostatic mechanism in the periphery, the ratio of HIV-immune to HIV-susceptible T-cells in each immune region should asymptotically approach the ratio of HIV-immune to HIV-susceptible stem cells that results after bone marrow transplant with no infection present. We show that with virus introduced into the system, the asymptotic ratio of HIV-immune to HIV-susceptible peripheral T-cells is highly sensitive to parameters affecting the body's ability to clear the virus.