Dynamical Behavior of Biological Healthy Steady State in Leukemia Using a New Leukemic & Healthy Stem Cells Cohabitation Model with Distributed Delay

Acute Myeloid Leukemia (AML) treatment protocol from clinical point of view, aims to maintain a normal amount of healthy cells and to eradicate all malignant cells. This particular objective is biologically qualified as a positive healthy situation. In this paper, we give sufficient and necessary conditions for the global stability of such a healthy situation. To this end, we first propose a new distributed delay model of AML. The latter is an improvement of an existing delayed coupled model describing the dynamics of hematopoesis stem cells in AML. We modify the PDEs equations and transform them into a set of distributed delay equations. The proposed model is more suitable for biological phenomena than constant delay models as the proliferation time differs from a cell type to another. Furthermore in the proposed model, we consider the sub-population of cells that have lost their capacity of self-renewal and became progenitors. In second, we derive sufficient and necessary conditions for the global stability of healthy steady state. For this, the positivity of the obtained model and sequences of functions theory are used to construct new Lyapunov function candidates. Finally, we conduct numerical simulations to show that the obtained results complete and generalize those published in the literature.