| Citation: |
Renxiang Shi. THE INFLUENCE OF DELAYS AND INJECTION TREATMENTS ON THE DYNAMICS OF TUMOR-IMMUNE SYSTEM[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1312-1350. doi: 10.11948/20260107
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THE INFLUENCE OF DELAYS AND INJECTION TREATMENTS ON THE DYNAMICS OF TUMOR-IMMUNE SYSTEM
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Abstract
We propose a tumor-CD4+-cytokine-host system with two delays and injection treatments, where two delays denote activation delays for CD4+ T cells and cytokines respectively and host cells compete with tumor cells. First we prove the positivity and boundedness of solutions without injection treatments, the positivity means that tumor cells always exist, the boundedness means that immune cells can control the increase of tumor cells, then we discuss the influence of two delays on the dynamics of system, that is as any of the delays is sufficiently small, the corresponding interior positive equilibrium point is locally stable, as the delay increases to critical value, Hopf bifurcation occurs and periodic solution bifurcates from the equilibrium point, which means the relapse of tumor cells occurs, furthermore base on center manifold and normal form, we study the property of Hopf bifurcation, at last, in absence of delay, we study the influence of relative parameters on the volume of tumor cells and feasibility of injection treatments of CD4+ T cells and cytokines on eradication of tumor cells, that is the tumor cells can be eradicated if the dose of injection treatment of CD4+ T cells or cytokines is sufficiently large. All the results are supported by simulations.
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Keywords:
- Immunotherapy /
- tumor-immune-host system /
- delays /
- Hopf bifurcation /
- stability /
- cytokine
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References
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Renxiang Shi. THE INFLUENCE OF DELAYS AND INJECTION TREATMENTS ON THE DYNAMICS OF TUMOR-IMMUNE SYSTEM[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1312-1350. doi: 10.11948/20260107
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Figure 1.
With ode45 solver, when
$ \tau_1=\tau_2=0 $ $ x $ $ y $ $ z $ $ w $ -
Figure 2.
With ode23 solver, when
$ \tau_1=\tau_2=0 $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 3.
With dde23 solver, when
$ \tau_1=0,\tau_2=1<\tau_{20} $ $ x $ $ y $ $ z $ $ w $ -
Figure 4.
With dde23 solver, when
$ \tau_1=0,\tau_2=1<\tau_{20} $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 5.
With dde23 solver, when
$ \tau_1=0,\tau_2=3.95>\tau_{20} $ $ x $ $ y $ $ z $ $ w $ -
Figure 6.
With dde23 solver, when
$ \tau_1=0,\tau_2=3.95>\tau_{20} $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 7.
With dde23 solver, when
$ \tau_2=0,\tau_1=4<\tau_{10} $ $ x $ $ y $ $ z $ $ w $ -
Figure 8.
With dde23 solver, when
$ \tau_2=0,\tau_1=4<\tau_{10} $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 9.
With dde23 solver, when
$ \tau_2=0,\tau_1=13.7>\tau_{10} $ $ x $ $ y $ $ z $ $ w $ -
Figure 10.
With dde23 solver, when
$ \tau_2=0,\tau_1=13.7>\tau_{10} $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 11.
With dde23 solver, when
$ \tau_1=7\in(0,\tau_{10}),\tau_2=0.2<\tau_{20}^* $ $ x $ $ y $ $ z $ $ w $ -
Figure 12.
With dde23 solver, when
$ \tau_1=7\in(0,\tau_{10}),\tau_2=0.2<\tau_{20}^* $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 13.
With dde23 solver, when
$ \tau_1=7\in(0,\tau_{10}),\tau_2=2.1>\tau_{20}^* $ $ x $ $ y $ $ z $ $ w $ -
Figure 14.
With dde23 solver, when
$ \tau_1=7\in(0,\tau_{10}),\tau_2=2.1>\tau_{20}^* $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 15.
With dde23 solver, when
$ \tau_2=2\in(0,\tau_{20}),\tau_1=3<\tau_{10}^* $ $ x $ $ y $ $ z $ $ w $ -
Figure 16.
With dde23 solver, when
$ \tau_2=2\in(0,\tau_{20}),\tau_1=3<\tau_{10}^* $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 17.
With dde23 solver, when
$ \tau_2=2\in(0,\tau_{20}),\tau_1=7>\tau_{10}^* $ $ x $ $ y $ $ z $ $ w $ -
Figure 18.
With dde23 solver, when
$ \tau_2=2\in(0,\tau_{20}),\tau_1=7>\tau_{10}^* $ $ x $ $ y $ $ z $ $ x $ $ y $ $ w $ -
Figure 19.
With dde23 solver, when
$ \tau_1=0,\tau_2\in(0,4) $ $ \tau_2 $ $ \tau_2 $ -
Figure 20.
With dde23 solver, when
$ \tau_1=0,\tau_2\in(0,4) $ $ \tau_2 $ $ \tau_2 $ -
Figure 21.
With dde23 solver, when
$ \tau_2=0,\tau_1\in(0,14) $ $ \tau_1 $ $ \tau_1 $ -
Figure 22.
With dde23 solver, when
$ \tau_2=0,\tau_1\in(0,14) $ $ \tau_1 $ $ \tau_1 $ -
Figure 23.
With dde23 solver, when
$ \tau_1=7, \tau_2\in(0,2.5) $ $ \tau_2 $ $ \tau_2 $ -
Figure 24.
With dde23 solver, when
$ \tau_1=7, \tau_2\in(0,2.5) $ $ \tau_2 $ $ \tau_2 $ -
Figure 25.
With dde23 solver, when
$ \tau_2=2, \tau_1\in(0,7) $ $ \tau_1 $ $ \tau_1 $ -
Figure 26.
With dde23 solver, when
$ \tau_2=2, \tau_1\in(0,7) $ $ \tau_1 $ $ \tau_1 $ -
Figure 27.
With ode45 solver, in absence of delay, (a) the influence of production rate of CD4+ T cells
$ a_2\in(0.8,1) $ $ a_3\in(4.5,6.5) $ -
Figure 28.
With ode45 solver, in absence of delay, (a) the influence of natural death rate of CD4+ T cells
$ c_1\in(0.01,0.12) $ $ c_2\in(25,40) $ -
Figure 29.
With ode45 solver, in absence of delay, (a) the influence of tumor-killing rate by cytokines
$ a_1\in(0.1,0.5) $ $ r_2\in(0.1,0.5) $ -
Figure 30.
With ode45 solver, in absence of delay, (a) the influence of intrinsic growth rate of tumor cells
$ r_1\in(0.15,0.6) $ $ d_2\in(10^{-8},10^{-6}) $ -
Figure 31.
With ode45 solver, in absence of delays, without injection treatment, (a) the influence of initial values
$ (8.5\times 10^3, 10^6, 1.5\times10^5, 9.5\times10^8) $ $ (8.5\times 10^6, 10^6, 1.5\times10^5, 9.5\times10^8) $ -
Figure 32.
With ode45 solver, in absence of delays, (a) the influence of injection treatment of only CD4+ T cells
$ I_1=10^5, I_2=0 $ $ (8.5\times 10^3, 10^6, 1.5\times10^5, 9.5\times10^8) $ $ I_1=10^5, I_2=0 $ $ (8.5\times 10^6, 10^6, 1.5\times10^5, 9.5\times10^8) $ $ I_1=1.6\times 10^8, I_2=0 $ $ (8.5\times 10^6, 10^6, 1.5\times10^5, 9.5\times10^8) $ -
Figure 33.
With ode45 solver, in absence of delays, (a) the influence of injection treatment of only cytokines
$ I_1=0, I_2=10^4 $ $ (8.5\times 10^3, 10^6, 1.5\times10^5, 9.5\times10^8) $ $ I_1=0, I_2=10^4 $ $ (8.5\times 10^6, 10^6, 1.5\times10^5, 9.5\times10^8) $ $ I_1=0, I_2=10^9 $ $ (8.5\times 10^6, 10^6, 1.5\times10^5, 9.5\times10^8) $ -
Figure 34.
With ode45 solver, in absence of delays, with the same initial condition
$ (8.5\times 10^6, 10^6, 1.5\times10^5, 9.5\times10^8) $ $ I_1=10^4, I_2=10^5 $ $ I_1=10^7, I_2=10^8 $ $ I_1=10^7, I_2=6\times 10^8 $