Modeling Cyclic Waves of Circulating T Cells in Autoimmune Diabetes

Abstract

Type 1 diabetes (T1D) is an autoimmune disease in which immune cells, notably T lymphocytes, target and kill the insulin‐secreting pancreatic beta cells. Elevated blood‐sugar levels and full‐blown diabetes result once a large enough fraction of these beta cells has been destroyed. Recent investigation of T1D in animals, namely nonobese diabetic (NOD) mice, has revealed large cyclic fluctuations in the levels of T cells circulating in the blood, weeks before the onset of diabetes [J. D. Trudeau, C. Kelly‐Smith, C. B. Verchere, J. F. Elliott, J. P. Dutz, D. T. Finegood, P. Santamaria, and R. Tan, J. Clin. Invest., 111 (2003), pp. 217–223], but the mechanism for these oscillations is unclear. We here describe a mathematical model for the immune response that suggests a possible explanation for the cyclic pattern of behavior. We show that cycles similar to those observed experimentally can occur when activation of T cells is an increasing function of self‐antigen level, whereas the production of memory cells declines with that level. Our model extends previous theoretical work on T‐cell dynamics in T1D [A. F. M. Marée, P. Santamaria, and L. Edelstein‐Keshet, Int. Immunol., 18 (2006), pp. 1067–1077], and leads to interesting nonlinear dynamics, including Hopf and homoclinic bifurcations in biologically reasonable regimes of parameters. The model leads to the following explanation for cycles: High rates of beta‐cell death, and corresponding elevation of self‐antigen, shut off memory‐cell production, leading to a gap in the population of activated T cells. Once peptide has been cleared by nonspecific mechanisms, the memory pool is renewed, and the cyclic behavior results.