Stability analysis of an equation with two delays and application to the production of platelets

Figure 2.1.  Region of stability for the null solution of (2.1) with $ \tau = 1 $. The numbers indicate the number of pairs of eigenvalues with positive real parts. The graph is the same for any positive $ \tau $

Figure 2.2.  Region of stability for the null solution of (2.1) with $ A = 0.2 $. The numbers indicate the number of pairs of eigenvalue with positive real parts

Figure 2.3.  Region of stability for the null solution of (2.1) with $ A = 0.55 $

Figure 2.4.  Region of stability for the null solution of (2.1) with A = 1

Figure 2.5.  Region of stability for the null solution of (2.1) with A = 2

Figure 4.1.  Parametric portraits for the phase portraits near the double Hopf bifurcation (from [1], Figure 3.3)

Figure 4.2.  Numerical simulation of (1.1) for $ \tau_2 = 4.75 \times \tau_1 $ in the pseudo-phase plane $ (x(t), x(t-\tau_1-\tau_2)) $.]{Numerical simulation of (1.1) for $ \tau_2 = 4.75 \times \tau_1 $ in the pseudo-phase plane $ (x(t), x(t-\tau_1-\tau_2)) $, corresponding to the lowest wedge of the $ \mu_1<0, \mu_2>0 $ quadrant of Figure 4.1. Once the transient dynamic is lost, a stable limit cycle appears

Figure 4.3.  Numerical simulation of (1.1) for $ \tau_2 = 4.24 \times \tau_1 $ in the pseudo-phase space $ (x(t), x(t-\tau_1), x(t-\tau_1-\tau_2)) $, corresponding to the lowest interior wedge of the $ \mu_1<0 $, $ \mu_2>0 $ quadrant of Figure 4.1. Once the transient dynamic is lost, a stable torus appears

Figure 4.4.  Numerical simulation of (1.1) for $ \tau_2 = 4.24 \times \tau_1 $ in the pseudo-phase plane $ (x(t), x(t-\tau_1-\tau_2)) $, corresponding to the $ \mu_1>0 $, $ \mu_2>0 $ quadrant of Figure 4.1. Once the transient dynamic is lost, a stable limit cycle appears

Figure 5.1.  Stability as $ \tau_2 $ or $ \gamma $ are varied and other parameters are fixed. Blue dotted lines represent the values in healthy patients, and red dotted lines mark the limits after which the equilibrium is unstable. We see that when $ \tau_2 $ decreases of one day (to $ \tau_2 = 7.2 $), then the system loses its stability. Furthermore, if $ \gamma $ is multiplied more than 12 times (to $ \gamma = 0.625 $) then the system also loses its stability

Figure 5.2.  The evolution of the platelet count with time (blue line) for different values of $ \tau_2 $ and $ \gamma $, after the transient phase. The green doted line represents the average platelet count of healthy patients, $ 20 \times 10^9 $ platelets/kg, and the two red dotted lines represent the healthy range of platelet count, $ 11 \times 10^9 $ - $ 32 \times 10^9 $