![A schematic diagram of leukemogenesis and haematopoietic stem cell lineages regarding immune response and time delays. The black arrows above S, A and L represent their proliferation process, the black arrows from S to A, A to D, and L to T denote the division process. The red dot-head arrows indicate the feedback inhibitions to the proliferation and division from the cell (D) to S and A. τ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1$$\end{document} and τ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _2$$\end{document} denote the time delays during the inhibition processes. τ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _3$$\end{document} is the delay during the immune response of L and αγ+L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\alpha }{\gamma +L}$$\end{document} is the immune response](https://www.researchgate.net/publication/362084660/figure/fig1/AS:11431281090917456@1666231580556/A-schematic-diagram-of-leukemogenesis-and-haematopoietic-stem-cell-lineages-regarding_Q320.jpg)
![The number of purely leukemia steady state El\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_l$$\end{document} is determined by the values of p30\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{30}$$\end{document} and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. The black and blue lines represent Δ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta =0$$\end{document} and α=p30v30(2-1/p30)γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =p_{30}v_{30}(2-1/p_{30})\gamma $$\end{document}, respectively. The symbol II (I, III) means that the system (4) has 2(1, 0) steady states El\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_l$$\end{document}](https://www.researchgate.net/publication/362084660/figure/fig2/AS:11431281090917457@1666231580597/The-number-of-purely-leukemia-steady-state-Eldocumentclass12ptminimal_Q320.jpg)
![The existence of coexisting steady state (Ec\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_c$$\end{document}). The number of intersections of the curves G(L,D)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(L,D)=0$$\end{document} (red) and H(L,D)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(L,D)=0$$\end{document} (black) corresponds to the number of coexisting steady state (Ec\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_c$$\end{document}). If p10=0.55,p30=0.9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{10}=0.55, p_{30}=0.9$$\end{document} (discontinuous line), p10=0.6,p30=0.95\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{10}=0.6, p_{30}=0.95$$\end{document} (continuous line) and p10=0.65,p30=0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{10}=0.65, p_{30}=0.7$$\end{document} (dot line), the system (4) will have two or one or zero coexisting steady state](https://www.researchgate.net/publication/362084660/figure/fig3/AS:11431281090932939@1666231580677/The-existence-of-coexisting-steady-state-Ecdocumentclass12ptminimal_Q320.jpg)
![The stability of El\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_l$$\end{document} versus α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} with p10=0.45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{10}=0.45$$\end{document},p20=0.68\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{20}=0.68$$\end{document}, p30=0.8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{30}=0.8$$\end{document}. Black (green) stars denote the stable (unstable) equilibria El\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_l$$\end{document}](https://www.researchgate.net/publication/362084660/figure/fig4/AS:11431281090909427@1666231580725/The-stability-of-Eldocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![The stability of three types of equilibria against α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} with p10=0.51\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{10}=0.51$$\end{document}, p20=0.85\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{20}=0.85$$\end{document}, p30=0.98\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{30}=0.98$$\end{document}. Black (green) points represent stable (unstable) equilibria. Square, round dots, and star signs denote the states Ec\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_c$$\end{document},Eh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_h$$\end{document} and El\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_l$$\end{document}, respectively](https://www.researchgate.net/publication/362084660/figure/fig5/AS:11431281090897164@1666231580756/The-stability-of-three-types-of-equilibria-against-adocumentclass12ptminimal_Q320.jpg)
July 2022
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1 Citation
Nonlinear Dynamics
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.
![A few of the 1024 automatically generated bifurcation diagrams near configuration #576 (circled in red dash) studied in the text. Each configuration is a different definition of the asymmetry parameter bcommon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{common}$$\end{document}. Configurations all present a high level of similarity in the presence of saddle-node lobes. But, because discovered branches depend on the success of randomly generated starting point for the bifurcation calculations, not all diagrams show a branch with the main Hopf bifurcation in the system (red dot). (Color figure online)](https://www.researchgate.net/publication/360382367/figure/fig1/AS:1182639709011980@1658974687409/A-few-of-the-1024-automatically-generated-bifurcation-diagrams-near-configuration-576_Q320.jpg)
![a The subset of edges included in definition of bcommon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{common}$$\end{document} for configuration #576 studied in the text. Note that each pair of nodes has two mutual suppression edges, but, on the diagram, for clarity, we only show that edge that is part of the definition of bcommon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{common}$$\end{document}. Also, each node has positive auto-regulation, also not indicated on the diagram, for clarity. b All edges of the five gene system. Each gene pair presents mutual repression “i” onto “j” and “j” onto “i” of different strengths; but for clarity we draw only one connecting line. Blunt termination indicates suppression; pointed arrow indicates activation. Each gene is auto activating](https://www.researchgate.net/publication/360382367/figure/fig2/AS:1182639708999704@1658974687487/a-The-subset-of-edges-included-in-definition-of-bcommondocumentclass12ptminimal_Q320.jpg)
![The bifurcation diagram of configuration #576 studied in the text. The x-axis shows the parameter asymmetry bcommon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{common}$$\end{document} and the y-axis shows the state variables X1,X2,X3,X4andX5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1,X_2,X_3,X_4 and X_5$$\end{document} color coded as black, red, green, blue and magenta on the diagram. Filled circles indicate stable and dots indicate unstable dynamics. The limit cycles terminate short of the saddle-node bifurcation because of limits in the numerical calculations. H denotes the Hopf bifurcation](https://www.researchgate.net/publication/360382367/figure/fig3/AS:1182639709007907@1658974687521/The-bifurcation-diagram-of-configuration-576-studied-in-the-text-The-x-axis-shows-the_Q320.jpg)
![a: At asymmetry = 0.5535, the impact of several saddles can be seen as trajectory all feed into the main limit cycle. The view is in X1X2X4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1X_2X_4$$\end{document}. b: As discussed in the text, we focused on the latter part of the trajectories where cycling occurs. The colors indicate the local speed of the dynamics, in arbitrary units. Distinctive slowdowns are identified by circles and correspond to the impact of the nearby, but not local saddles. See text for more details](https://www.researchgate.net/publication/360382367/figure/fig4/AS:1182639708995628@1658974687570/a-At-asymmetry-05535-the-impact-of-several-saddles-can-be-seen-as-trajectory-all_Q320.jpg)
![Search governed by Metropolis algorithm maximizing the integration time as a surrogate for complexity of trajectories. The asymmetry configuration is #576\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\#576$$\end{document}, and the search was performed at asymmetry of 0.5535. The search is accomplished by randomly sampling coupling parameters in the network. As complexity increases the number of calculations necessary to achieve the required accuracy causes the integration time to increase. The probability that the search increases is p=0.48 indicating a mild cooling trend for the process. More details in the text](https://www.researchgate.net/publication/360382367/figure/fig5/AS:1182639709003828@1658974687666/Search-governed-by-Metropolis-algorithm-maximizing-the-integration-time-as-a-surrogate_Q320.jpg)
