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Bautin bifurcation in a minimal model of immunoediting

  • * Corresponding author: Joaquín Delgado

    * Corresponding author: Joaquín Delgado 

Authors were supported by Conacyt grant A1-S-41007, "Bifurcaciones en el estudio de la existencia y estabilidad de EDP". EHL was also supported by a Ph.D. Conacyt grant

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  • One of the simplest model of immune surveillance and neoplasia was proposed by Delisi and Resigno [6]. Later Liu et al [10] proved the existence of non-degenerate Takens-Bogdanov (BT) bifurcations defining a surface in the whole set of five positive parameters. In this paper we prove the existence of Bautin bifurcations completing the scenario of possible codimension two bifurcations that occur in this model. We give an interpretation of our results in terms of the Immuno Edition Theory (IET) of three phases: elimination, equilibrium and escape.

    Mathematics Subject Classification: Primary: 34C23, 34C60; Secondary: 37G15.


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  • Figure 1.  The catastrophe surface in coordinates $ (\psi, x_c, x_0) $, where $ x_0 $ is the abscissa of the critical point. For a given value of $ (\psi,x_c) $ there are up to two critical points with $ x_0>0 $ and the trivial critical point corresponding to $ x_0 = 0 $. Notice that there are critical points with $ x_0<0 $ that are not considered. The folding of the surface projects into the saddle–node curve given by (10) in the plane $ \psi $–$ x_c $

    Figure 3.  Schema of the bifurcation diagram computed numerically in Figure 10-b. The codes of the lines are as follows: SS (blue) symmetric–saddle; H (green) Hopf; SN (black dotted) saddle–node; LPC (red) limit point of cycles; Hom (magenta) homoclinic. Special points are BT(Takens–Bogdanov) and GH(Bautin)

    Figure 2.  Local diagram of Bautin bifurcacion

    Figure 10.  Numerical continuation of bifurcation diagram with MatCont. Saddle-node: black; Hopf: green; limit point of cycles: red; symmetric saddles: blue; homoclinic: violet

    Figure 4.  Qualitative phase portrait along the line $ CKDPAT $ of the bifurcation scheme in Figure 3

    Figure 5.  Qualitative phase portrait along the line $ C'K'D'P'A'T' $ of the bifurcation scheme in Figure 3. The phase portrait along the segment $ C'K'D'P'A' $ is the same as $ CKPDA $

    Figure 6.  Qualitative phase portrait along the line $ C''K''D''P''A''T'' $ of the bifurcation scheme in Figure 3

    Figure 7.  Coexistence of two limit cycles along the line $ C'T' $: (a), (c) and (b). Along the line $ C''T'' $: (b), (d) and (f)

    Figure 9.  Ilustration of Proposition 3

    Figure 8.  Graphs of coexisting limit cycles of Figure 7. Stable in blue, unstable in red

    Table 1.  Parameters value of system (1), [6], [10]

    Parameter-Definition Dimension Value Scaled
    $ \lambda_{1} $: Lymphocyte growth rate $ \frac{1}{day} $ $ 0.01 $ $ 0.01 $
    $ \lambda_{2} $: Death rate of cancer cells $ \frac{1}{day} $ $ 0.020016 $ $ 0.006672 $
    $ \alpha_{1} $: Rate of interaction $ \frac{1}{day} $ $ 1.5 \times 10^{-7} $ $ 0.297312 $
    $ \alpha_{2} $: Rate of interaction $ \frac{1}{day} $ $ 4.6135 \times 10^{-9} $ $ 0.00318 $
    $ x_{c} $: Saturation level $ volume $ $ 2.5 \times 10^{11} $ $ 2500 $
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  • [1] J. A. Adam, Effects of vascularization on lymphocyte/tumor cell dynamics: Qualitative features, Math. Comput. Modelling, 23 (1996), 1-10. 
    [2] B. Al-HdaibatW. GovaertsY. A. Kuznetsov and H. G. E. Meijer, Initialization of homoclinic solutions near Bogdanov-Takens points: Lindstedt-Poincaré compared with regular perturbation method, SIAM J. Applied Dynamical Systems, 15 (2016), 952-980.  doi: 10.1137/15M1017491.
    [3] G. I. Bell, Predator-prey simulating an immune response, Mathematical Biosciences, 16 (1973), 291-314. 
    [4] W.-J. Beyn, Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point, IMA Journal of Numerical Analysis, 14 (1994), 381-410.  doi: 10.1093/imanum/14.3.381.
    [5] A. R. Champneys and Y. A. Kuznetzov, Numerical detection and continuation of codimension-two homoclinic bifurcation, International Journal of Bifurcation and Chaos, 4 (1994), 785-822.  doi: 10.1142/S0218127494000587.
    [6] C. DeLisi and A. Rescigno, Immune surveillance and neoplasia-I. A minimal mathematical model, Bull. Math. Bio., 39 (1977), 201-221.  doi: 10.1007/bf02462859.
    [7] G. P. DunnL. J. Old and R. D. Schreiber, The three ES of Cancer Immunoediting, Annual Review of Immunology, 22 (2004), 329-360. 
    [8] R. KimM. Emi and K. Tanabe, Cancer immunoediting from immune surveillance to immune escape, Immunology, 21 (2007), 11-14. 
    [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.
    [10] D. LiuS. G. Ruan and D. M. Zhu, Bifurcation analysis in models of tumor and immune system interactions, Discrete and Continuous Dynamical Systems series B, 12 (2009), 151-168.  doi: 10.3934/dcdsb.2009.12.151.
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