Bautin bifurcation in a minimal model of immunoediting

Joaquín Delgado; Eymard Hernández–López; Lucía Ivonne Hernández–Martínez

doi:10.3934/dcdsb.2019233

Article Contents

2020, Volume 25, Issue 4: 1397-1414. Doi: 10.3934/dcdsb.2019233

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Bautin bifurcation in a minimal model of immunoediting

1.
Departamento de Matemáticas, UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México City, C.P. 09340, México
2.
Posgrado en Ciencias Naturales e Ingeniería, UAM-Cuajimalpa, Av. Vasco de Quiroga 4871, Col. Santa Fé Cuajimalpa, México City, 05348, México
3.
Universidad Autónoma de la Ciudad de México, Plantel San Lorenzo Tezonco, Calle Prolongación San Isidro 151, Col. San Lorenzo Tezonco, México City, 09790, México

^* Corresponding author: Joaquín Delgado
^* Corresponding author: Joaquín Delgado

Received: June 30, 2018

Revised: April 30, 2019

Early access: November 2019

Published: April 2020

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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Abstract

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.

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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 $

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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)

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Figure 2. Local diagram of Bautin bifurcacion

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

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

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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 $

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

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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)

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Figure 9. Ilustration of Proposition 3

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

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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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References

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