Delay Linear Chains in Mathematical Biology: Migratory Birds, Stem Cell Maturation, and Intracellular Chlamydia Infection

We consider a simple linear chain, which represents how organisms pass through a number of intermediate stages before reaching a final phase, allowing time delays between consecutive compartments. The final state of this linear system is determined explicitly as a function of initial data and transition rates. Three biological applications are provided: 1. First, we consider a spatial dynamic model of seasonal bird migration with stopovers along the migratory routes. Time delays arise from the large distance between stopovers. A whole season is tracked from the departure patch through spring migration, summer breeding, and eventually to the wintering ground, to determine the overall yearly growth of the population. 2. We consider a stem cell population that differentiates into progenitor and eventually mature cells. In the absence of regulatory feedback, this system can be modeled by a linear chain, where the time delays arise from the length of the cell cycle to complete cell division. We conclude that the eventual size of a developing organ can be finite even in the absence of any regulation, if the self-renewal rates are not too large. In this case, the size of the developed organ is independent of the division rates. 3. We model a laboratory experiment of chlamydia bacteria infecting human cells, taking into account the intracellular development cycle of the pathogen. Our simple model can replicate an important empirical finding of the chlamydia literature and allows us to infer biological parameters that cannot be directly measured.