The paper devoted to the regulation of the production of blood cells. By using denotation \(x(t)\) and reduction to the barest of descriptions it can be described most simply by a differential delay equation of the following form \[ \frac{dx}{dt} = -\gamma x(t) + f(x(t - \tau)) \] with constant \(\gamma > 0\), where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is monotone decreasing. In this work, the nonlinearity \(f\) is replaced by a piecewise constant function which permits to compute solutions explicitly and to analytically study their behaviour. The response of the solutions to perturbation means to represent the effect of cytokine administration.
The goal of this paper is to study the effect of such a perturbation on the solution of a given equation. At first, the authors describe the model and formulate it in a mathematically convenient form providing required basic facts. Then they introduce the pulse-like perturbations (a perturbation of constant amplitude lasting for a finite period of time) of the model which correspond to the effect of medication in the sense that during a finite time interval the production of blood cells is increased. It is shown that the response of the system to such perturbations is continuous provided the latter are not too large. This includes a continuity result for the cycle length map, which assigns to each onset time of increased production a time of return to the periodic orbit, after the end of increased production. The bulk of derived results are presented in Section 5 where the authors examine the effect of cytokine perturbation when the perturbation away from the stable periodic orbit begins at different points in the cycle. In particular, they look at phase resetting properties of the system (in terms of the cycle length map) and at the minima and maxima compared to the amplitudes of the periodic solution. It is presented how the derived results may be potentially used to tailor therapy to achieve certain results. Finally, there is a simple extension in which a pulse-like perturbation may decrease the nadir of the limit cycle as is noted clinically.