## Section: New Results

### Mathematical models of hematopoietic stem cell dynamics

#### Asymptotic behavior and stability switch for a mature-immature model of cell differentiation

Participants : Mostafa Adimy, Fabien Crauste.

*In collaboration with C. Marquet (University of Pau)*.

We investigated in [3] the stability of a delay differential model describing hematopoietic cell dynamics. The framework we considered was a nonlinear age-structured model describing the dynamics of a cell population divided into mature and immature cells. Immature cells, that can be either proliferating or non-proliferating, differentiate in mature cells, that in turn control the immature cell population through a negative feedback. The initial system was reduced to two delay differential equations, and we investigated the asymptotic stability of the trivial and the positive steady states. By constructing a Lyapunov function, the trivial steady state was proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state is related to a delay-dependent characteristic equation. Existence of a Hopf bifurcation and stability switch for the positive steady state was established. We illustrated the stability with numerical simulations.

**Keywords:** *mature and immature cells, hematopoiesis,
asymptotic stability, lyapunov function, delay-dependent
characteristic equation, stability switch, Hopf bifurcation*.

#### Stability and Hopf bifurcation for a cell population model with state-dependent delay

Participants : Mostafa Adimy, Fabien Crauste.

*In collaboration with H. Hbid (University of Marrakech), R. Qesmi (University of Toronto, Canada)*.

We proposed in [2] a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduced the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis was performed. A sufficient condition for the global asymptotic stability of the trivial steady state was obtained using a Lyapunov-Razumikhin function. A unique positive steady state was shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concluded the existence of a Hopf bifurcation and gave criteria for stability switches. A numerical analysis confirmed the results and stressed the role of each parameter involved in the system on the stability of the positive steady state.

**Keywords:** *hematopoietic stem cells, functional differential
equation, state-dependent delay, Lyapunov-Razumikhin function, Hopf bifurcation*.

#### Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics

Participants : Mostafa Adimy, Fabien Crauste.

*In collaboration with A. El Abdllaoui (University of Pau)*.

We investigated in [1] a system of nonlinear differential equations with distributed delays, arising from a model of hematopoietic stem cell dynamics. We stated uniqueness of a global solution under a classical Lipschitz condition. Sufficient conditions for the global stability of the population were obtained, through the analysis of the asymptotic behavior of the trivial steady state and using a Lyapunov function. Finally, we gave sufficient conditions for the unbounded proliferation of a given cell generation.

**Keywords:** *hematopoiesis, time-delay systems, Lyapunov function*.