Influence of time delay on the dynamics of a mathematical model of CAR-T cell therapy with logistic tumor growth

Figure 1.  Illustration of the processes that tumor cells (red cells) and CAR-T cells (green cells) undergo according to Eqs. (1). The left panel: tumor cell can proliferate at a rate of $ \rho_T $ or die by interactions with CAR-T cells at a rate of $ \alpha_T $. The right panel: CAR-T cells can be stimulated to division when encountering tumor cell at a time $ t-\tau $ with a rate $ a $, can be inactivated by tumor cells at a rate $ b $, or can undergo natural death

Figure 2.  Phase portraits for Eqs. (1) without delay and with the logistic tumor growth for parameter values as follows: (1) $ a \leqslant 1 $; (2) $ a > 1 $ and $ g_{\min}>0 $ and (2a) $ K<K_{cr} $ (no positive steady state), (2b) $ K = K_{cr} $ (one positive steady state); (2c) $ K>K_{cr} $ and the first positive steady state being unstable; (2d) $ K>K_{cr} $ and the first positive steady state being stable; (3) $ a > 1 $ and $ g_{\min}<0 $ (one positive steady state)

Figure 3.  Numerical simulation of a homoclinic bifurcation for Eqs. (1) without delay and with the logistic tumor growth with $ \rho_T = 5 $, $ \alpha_T = 0.45 $, $ a = 3 $, $ b = 5.7 $ and $ K = 10.5 $

Figure 4.  Solution of Eqs. (1) with $ T(t) = 4\exp(5 t) $ for $ t \in [-\tau,0] $ and $ C(t) = 0 $ for $ t \in [-\tau,0) $ and $ C(0) = 3 $ for (1) $ \tau = 0.1 $ (2) $ \tau = 0.9 $ with parameters $ a : = 3; b : = 1; \alpha_T : = 0.45; \rho_T : = 5; K : = 5 $

Figure 5.  Solution of Eqs. (1) with $ T(t) = 2\exp(5t) $ for $ t \in [-\tau,0] $ and $ C(t) = 0 $ for $ t \in [-\tau,0) $ and $ C(0) = 10 $ (1) the steady state $ (\bar T, \bar C) $ with $ \bar T< T_{min} $ is a stable focus, $ \tau = 0.1 $ (2) the steady state $ (\bar T, \bar C) $ with $ \bar T< T_{min} $ is unstable focus, $ \tau = 0.3 $. Other parameters $ a = 3; b = 1; \alpha_T = 0.45; \rho_T = 5; K = 5 $