,

,

,

with initial condition and initial history functions and .

Here is the level of vitamin D above the basal level in blood, is the number of active bone-resorbing osteoclasts, is the number of active bone-forming osteoblasts, is time, and is the time delay necessary for cells to become active in the bone resorption and formation process. All parameters are positive constants defined in the reference. The time trajectory shows periodic behavior when the delay time is zero and increasing shows damped oscillations leading to a steady state; both of these conditions are observed clinically [1].