We develop a mathematical model that describes concentration dynamics of PrP C and PrP Sc prion proteins at the neuron scale and includes the effect of the Unfolded Protein Response (UPR). We first introduce a single neuron model taking the UPR mechanism into account. We investigate it and propose a stability study among which a bifurcation analysis with respect to three of its parameters. Then, we generalize it to two neurons showing PrP Sc proteins interaction. Stability results are given when neurons exhibit identical parameters but interact differently (strong, weak or no interaction).

Many studies have shown that periodic erythrocytic (red blood cell linked) diseases are extremely rare in humans. To explain this observation, we develop here a simple model of erythropoiesis in mammals and investigate its stability in the parameter space. A bifurcation analysis enables us to sketch stability diagrams in the plane of key parameters. Contrary to some other mammal species such as rabbits, mice or dogs, we show that human specific parameter values prevent periodic oscillations of red blood cells levels. In other words, human erythropoiesis seems to lie in a region of parameter space where oscillations exclusively concerning red blood cells cannot appear. Further mathematical analysis show that periodic oscillations of red blood cells levels are highly unusual and if exist, might only be due to an abnormally high erythrocytes destruction rate or to an abnormal hematopoietic stem cell commitment into the erythrocytic lineage. We also propose numerical results only for an improved version of our approach in order to give a more realistic but more complex approach of our problem.