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Effects of immune response and time delays in models of acute myeloid leukemia

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Abstract

In this paper, we propose a general acute myeloid leukemia (AML) model and introduce an immune response and time delays into this model to investigate their effects on the dynamics. Based on the existence, stability and local bifurcation of three types of equilibria, we show that the immune response is a best strategy for the control of the AML on the condition that the rates of proliferation and differentiation of the hematopoietic lineage exceed a threshold. In particular, a powerful immune response leads to a bistability feature meaning there exist the leukemia cells and healthy cells in the bone marrow or only the healthy cells. In addition, we further reveal that the time delays in the feedback regulation and immune response process induce a series of oscillations around the steady state, which shows that the leukemia cells are hardly eliminated. Our work in this paper aims to investigate the complex dynamics of this AML model with the immune response and time delays on the basis of mathematical models and numerical simulations, which may provide a theoretical guidance for the treatments of the AML.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

This work was supported by the National Natural Science Foundation of China (No. 11902106).

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Authors and Affiliations

  1. School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, China

    Weigang Sun, Lei Yang & Min Luo

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  1. Weigang Sun
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Correspondence to Weigang Sun.

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Appendix

Appendix

1.1 A. Numerical simulations

The non-delayed bifurcation diagrams are performed by Oscill8 software and the delayed bifurcation diagrams are performed by DDE-BIFTOOL, a Matlab toolbox for bifurcation analysis of time-delayed systems, and others are finished by the Matlab.

1.2 B. Denotations of involved parameters

Table 1 The definitions, basal values and references of all the parameters
Full size table

1.3 C. Representations of the parameters

$$\begin{aligned} a_0&= 8 v_{10}v_{20} (p_{20} - 1)^2 (2 p_{10} - 1)(p_{10} -1)\\ a_1&=2(p_{20}-1)v_{20}\big (p_{10}(2p_{20}-1)\mu _D\\&\quad -4v_{10}(3g_1p_{10}p_{20}\\&\quad +4g_2p_{10}^2-3g_1p_{10}-2g_1p_{20}\\&\quad -6g_2p_{10}+2g_1+2g_2)\big )\\ a_2&=\mu _D^2p_{10}p_{20}+2p_{10}v_{20}(2p_{20}^2(g_1+h_1)\\&\quad -3p_{20}(g_1+g_2+h_1)+g_1+h_1+2g_2)\mu _D\\&\quad +8v_{10}v_{20}(g_1^2(p_{20}^2-2p_{20}+1)\\&\quad +g_1g_2(6p_{10}p_{20}-6p_{10}-4p_{20}+4)\\&\quad +g_2^2(2p_{10}^2-3p_{10}+1))\\ a_3&=p_{10}p_{20}(g_1+g_2+h_1+h_2)\mu _D^2\\&\quad -8g_1g_2v_{10}v_{20}(2g_1p_{20}+3g_2p_{10}-2g_1-2g_2)\\&\quad -2p_{10}v_{20}(3g_1g_2p_{20}-2g_1h_1p_{20}^2\\&\quad +3g_1h_1p_{20}+3g_2h_1p_{20}\\&\quad -2g_1g_2-g_1h_1-g_2^2-2g_2h_1)\mu _D\\ a_4&=p_{10}p_{20}((h_1+h_2)(g_1+g_2)+g_1g_2\\&\quad +h_1h_2)\mu _D^2+2g_2p_{10}v_{20}(g_1g_2-3g_1h_1p_{20}\\&\quad +2g_1h_1+g_2h_1)\mu _D+8g_1^2g_2^2v_{10}v_{20}\\ a_5&=\mu _D^2p_{10}p_{20}(g_1g_2(h_1+h_2)+h_1h_2(g_1+g_2))\\&\quad +2\mu _D g_1g_2^2h_1p_{10}v_{20}\\ a_6&=p_{10}p_{20}g_1g_2h_1h_2\mu _D^2\\ A_0&=f_{3A}f_{2D}f_{4L}f_{1S}f_{5T}+f_{4A}f_{3D}f_{2L}f_{1S}f_{5T}\\&\quad -f_{2A}f_{3D}f_{4L}f_{1S}f_{5T}-f_{3A}f_{1D}f_{4L}f_{2S}f_{5T}\\ A_1&=f_{2A}f_{3D}f_{4L}f_{1S}+f_{3A}f_{1D}f_{4L}f_{2S}\\&\quad -f_{3A}f_{2D}f_{4L}f_{1S}-f_{4A}f_{3D}f_{2L}f_{1S}\\&\quad +f_{2A}f_{3D}f_{4L}f_{5T}-f_{3A}f_{2D}f_{4L}f_{5T}\\&\quad -f_{4A}f_{3D}f_{2L}f_{5T}+f_{2A}f_{3D}f_{1S}f_{5T}\\&\quad +f_{3A}f_{1D}f_{2S}f_{5T}-f_{3A}f_{2D}f_{1S}f_{5T}\\&\quad -f_{3A}f_{4L}f_{1S}f_{5T}-f_{4A}f_{2L}f_{1S}f_{5T}\\&\quad +f_{3D}f_{4L}f_{1S}f_{5T}\\ A_2&=f_{3A}f_{2D}f_{4L}-f_{2A}f_{3D}f_{4L}+f_{4A}f_{3D}f_{2L}\\&\quad -f_{2A}f_{3D}f_{1S}-f_{3A}f_{1D}f_{2S}+f_{3A}f_{2D}f_{1S}\\&\quad -f_{2A}f_{3D}f_{5T}+f_{3A}f_{2D}f_{5T}-f_{2A}f_{4L}f_{1S}\\&\quad +f_{4A}f_{2L}f_{1S}-f_{2A}f_{4L}f_{5T}+f_{4A}f_{2L}f_{5T}\\&\quad -f_{3D}f_{4L}f_{1S}-f_{3D}f_{4L}f_{5T}-f_{2A}f_{1S}f_{5T}\\&\quad -f_{3D}f_{1S}f_{5T}-f_{4L}f_{1S}f_{5T}\\ A_3&=f_{2A}f_{3D}-f_{3A}f_{2D}+f_{2A}f_{4L}-f_{4A}f_{2L}\\&\quad +f_{3D}f_{4L}+f_{1S}f_{2A}+f_{2A}f_{5T}+f_{3D}f_{1S}\\&\quad +f_{3D}f_{5T}+f_{4L}f_{1S}+f_{4L}f_{5T}+f_{1S}f_{5T}\\ A_4&=-f_{1S}-f_{2A}-f_{3D}-f_{4L}-f_{5T}\\ m_0&=a_{44}\mu _Df_{1S}f_{2A}f_{5T}-\mu _Df_{1S}f_{2L}f_{4A}f_{5T}\\ m_1&=\mu _D\big (f_{1S}f_{2L}f_{4A}+f_{2L}f_{4A}f_{5T}-f_{1S}f_{2A}f_{5T}\\&\quad -a_{44}f_{1S}f_{5T}-a_{44}f_{2A}f_{5T}-a_{44}f_{1S}f_{2A}\big )\\&\quad +a_{44}f_{1S}f_{2A}f_{5T}-f_{1S}f_{2L}f_{4A}f_{5T}\\ m_2&=-a_{44}\big (f_{1S}f_{2A}+f_{2A}f_{5T}\\&\quad +f_{1S}f_{5T}-f_{2A}\mu _D-f_{1S}\mu _D-f_{5T}\mu _D\big )\\&\quad -\mu _D\big (f_{2L}f_{4A}-f_{1S}f_{2A}-f_{2A}f_{5T}-f_{1S}f_{5T}\big )\\&\quad +f_{1S}f_{2L}f_{4A}+f_{2L}f_{4A}f_{5T}-f_{1S}f_{2A}f_{5T}\\ m_3&=a_{44}(f_{1S}+f_{2A}+f_{5T})-\mu _D(a_{44}+f_{1S}\\&\quad +f_{2A}+f_{5T}) -f_{4A}f_{2L}+f_{1S}f_{2A}\\&\quad +f_{2A}f_{5T}+f_{1S}f_{5T}\\ m_4&=\mu _D-a_{44}-f_{1S}-f_{2A}-f_{5T}\\ n_0&=a_{44}b_{23}f_{1S}f_{3A}f_{5T}-a_{44}f_{1D}f_{2S}f_{3A}f_{5T}\\&\quad +c_{33}f_{1S}f_{2L}f_{4A}f_{5T}+a_{44}c_{23}f_{1S}f_{3A}f_{5T}\\&\quad -a_{44}c_{33}f_{1S}f_{2A}f_{5T}\\ n_1&=-a_{44}b_{23}f_{1S}f_{3A}-a_{44}b_{23}f_{3A}f_{5T}\\&\quad +a_{44}f_{1D}f_{2S}f_{3A}-b_{23}f_{1S}f_{3A}f_{5T}\\&\quad +f_{1D}f_{2S}f_{3A}f_{5T} -a_{44}c_{23}f_{1S}f_{3A}\\&\quad +a_{44}c_{33}f_{1S}f_{2A}-a_{44}c_{23}f_{3A}f_{5T}\\&\quad +a_{44}c_{33}f_{2A}f_{5T}+a_{44}c_{33}f_{1S}f_{5T}\\&\quad -c_{33}f_{1S}f_{2L}f_{4A}-c_{33}f_{2L}f_{4A}f_{5T}\\&\quad -c_{23}f_{1S}f_{3A}f_{5T}+c_{33}f_{1S}f_{2A}f_{5T}\\ n_2&=a_{44}b_{23}f_{3A}+b_{23}f_{1S}f_{3A}+b_{23}f_{3A}f_{5T}\\&\quad -f_{1D}f_{2S}f_{3A}+a_{44}\big (c_{23}f_{3A}\\&\quad -c_{33}f_{2A}-c_{33}f_{1S}-c_{33}f_{5T}\big )\\&\quad +c_{33}\big (f_{2L}f_{4A}-f_{1S}f_{2A}\\&\quad -f_{2A}f_{5T}-f_{1S}f_{5T}\big )\\&\quad +c_{23}f_{3A}f_{1S}+c_{23}f_{3A}f_{5T} \\ n_3&=-(c_{23}+b_{23})f_{3A}+c_{33}\big (a_{44}+f_{2A}\\&\quad +f_{1S}+f_{5T}\big )\\ n_4&=-c_{33}\\ p_0&=d_{44}\mu _Df_{1S}f_{2A}f_{5T}\\ p_1&=d_{44}\big (f_{1S}f_{2A}f_{5T}-\mu _Df_{1S}f_{2A}\\&\quad -\mu _Df_{2A}f_{5T}-\mu _Df_{1S}f_{5T}\big )\\ p_2&=-d_{44}\big (f_{1S}f_{2A}+f_{2A}f_{5T}+f_{1S}f_{5T}\\&\quad -f_{2A}\mu _D-f_{1S}\mu _D-f_{5T}\mu _D\big )\\ p_3&=d_{44}\big (f_{1S}+f_{2A}+f_{5T}-\mu _D \big )\\ p_4&=-d_{44}\\ q_0&=d_{44}\big (b_{23}f_{1S}f_{3A}f_{5T}-f_{1D}f_{2S}f_{3A}f_{5T}\\&\quad +c_{23}f_{1S}f_{3A}f_{5T}-c_{33}f_{1S}f_{2A}f_{5T}\big )\\ q_1&=d_{44}\big (-b_{23}f_{1S}f_{3A}-b_{23}f_{3A}f_{5T}\\&\quad +f_{1D}f_{2S}f_{3A}+c_{33}f_{1S}f_{2A}\\&\quad -c_{23}f_{1S}f_{3A}-c_{23}f_{3A}f_{5T}\\&\quad +c_{33}f_{2A}f_{5T}+c_{33}f_{1S}f_{5T}\big )\\ q_2&=-d_{44}\big (c_{33}f_{1S}+c_{33}f_{2A}+c_{33}f_{5T}\\&\quad -c_{23}f_{3A}-b_{23}f_{3A}\big )\\ q_3&=c_{33}d_{44}\\ M_R&=m_4\omega ^4-m_2\omega ^2+m_0,\\ M_I&=\omega ^5-m_3\omega ^3+m_1\omega \qquad \qquad \qquad \\ N_R&=n_4\omega ^4-n_2\omega ^2+n_0,\\ N_I&=-n_3\omega ^3+n_1\omega \\ P_R&=p_4\omega ^4-p_2\omega ^2+p_0,\\ P_I&=-p_3\omega ^3+p_1\omega \\ Q_R&=-q_2\omega ^2+q_0,\\ Q_I&=-q_3\omega ^3+q_1\omega \\ X_1&=(M_IQ_I-M_RQ_R)(P_I-M_I)\\&\quad -(M_RQ_I+M_IQ_R)(M_R-P_R)\\&\quad +(Q_R^2+Q_I^2)(Q_I-N_I)\\ Y_1&=(M_IQ_I-M_RQ_R)(M_R+P_R)\\&\quad -(M_RQ_I+M_IQ_R)(M_I+P_I)\\&\quad +(Q_R^2+Q_I^2)(N_R+Q_R)\\ Z_1&=(M_IQ_I-M_RQ_R)N_R\\&\quad -(M_RQ_I+M_IQ_R)N_I+(Q_R^2+Q_I^2)P_R\\ X_2&=(M_RQ_R-M_IQ_I)(M_R-P_R)\\&\quad -(M_RQ_I+M_IQ_R)(P_I-M_I)\\&\quad +(Q_R^2+Q_I^2)(N_R-Q_R)\\ Y_2&=(M_RQ_R-M_IQ_I)(M_I+P_I)\\&\quad -(M_RQ_I+M_IQ_R)(M_R+P_R)\\&\quad +(Q_R^2+Q_I^2)(N_I+Q_I)\\ Z_2&=(M_RQ_R-M_IQ_I)N_I\\&\quad -(M_RQ_I+M_IQ_R)N_R+(Q_R^2+Q_I^2)P_I\\ M&=\lambda ^5+m_4\lambda ^4+m_3\lambda ^3+m_2\lambda ^2+m_1\lambda +m_0\\ N&=n_4\lambda ^4+n_3\lambda ^3+n_2\lambda ^2+n_1\lambda +n_0\\ P&=p_4\lambda ^4+p_3\lambda ^3+p_2\lambda ^2+p_1\lambda +p_0\\ Q&=q_3\lambda ^3+q_2\lambda ^2+q_1\lambda +q_0\\ M'&=5\lambda ^4+4m_4\lambda ^3+3m_3\lambda ^2+2m_2\lambda +m_1\\ N'&=4n_4\lambda ^3+3n_3\lambda ^2+2n_2\lambda +n_1\\ P'&=4p_4\lambda ^3+3p_3\lambda ^2+2p_2\lambda +p_1\\ Q'&=3q_3\lambda ^2+2q_2\lambda +q_1\\ F_R&=\text {Re}(M')+\text {Re}(N')\cos (\omega \tau )\\&\quad +\text {Im}(N')\sin (\omega \tau )+\text {Re}(P')\cos (2\omega \tau )\\&\quad +\text {Im}(P')\sin (2\omega \tau )\\&\quad +\text {Re}(Q')\cos (3\omega \tau )+\text {Im}(Q')\sin (3\omega \tau )\\ F_I&=\text {Im}(M')+\text {Im}(N')\cos (\omega \tau )\\&\quad -\text {Re}(N')\sin (\omega \tau )+\text {Im}(P')\cos (2\omega \tau )\\&\quad -\text {Re}(P')\sin (2\omega \tau )+\text {Im}(Q')\cos (3\omega \tau )\\&\quad -\text {Re}(Q')\sin (3\omega \tau )\\ T_R&=\text {Re}(N)\cos (\omega \tau )+\text {Im}(N)\sin (\omega \tau )\\&\quad +2\text {Re}(P)\cos (2\omega \tau )+2\text {Im}(P)\sin (2\omega \tau )\\&\quad +3\text {Re}(Q)\cos (3\omega \tau )+3\text {Im}(Q)\sin (3\omega \tau )\\ T_I&=\text {Im}(N)\cos (\omega \tau )\\&\quad -\text {Re}(N)\sin (\omega \tau )+2\text {Im}(P)\cos (2\omega \tau )\\&\quad -2\text {Re}(P)\sin (2\omega \tau )\\&\quad +3\text {Im}(Q)\cos (3\omega \tau )\\&\quad -3\text {Re}(Q)\sin (3\omega \tau )\\ \end{aligned}$$

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Sun, W., Yang, L. & Luo, M. Effects of immune response and time delays in models of acute myeloid leukemia. Nonlinear Dyn 110, 1789–1805 (2022). https://doi.org/10.1007/s11071-022-07697-9

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