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The Effect of a Linear Tuning between the Antigenic Stimulations of CD4^{+} T Cells and CD4^{+} Tregs

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## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Equilibria of the Model

**Theorem**

**1.**

**Proof.**

## 4. Stability Analysis

## 5. Time Evolutions

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Equilibria manifold obtained from Theorem 1. (

**a**) balance between the concentration of T cells $x=T+{T}^{*}$ and that of Tregs $y=R+{R}^{*}$. The shading color indicates the real part of the largest eigenvalue $Re\left(\lambda \right)$, increasing from black to blue for stable equilibria, and unstable equilibria from green to yellow. The red and the magenta lines show the bifurcations, when $Re\left(\lambda \right)=0$. (

**b**) cross-section for $m=0.2765$. The line type indicates stable (solid) or unstable (dashes) equilibria.

**Figure 2.**Equilibria manifold obtained from Theorem 1. (

**a**,

**b**) relationship between the antigenic stimulation b of T cells and the concentration of T cells $x=T+{T}^{*}$. (

**c**,

**d**) relationship between the antigenic stimulation b of T cells and the concentration of Tregs $y=R+{R}^{*}$. (

**a**,

**c**) the axis pointing upwards to the right is the slope parameter m. The shading color indicates the real part of the largest eigenvalue $Re\left(\lambda \right)$, increasing from black to blue for stable equilibria, and unstable equilibria from green to yellow. The red and the magenta lines show the bifurcations, when $Re\left(\lambda \right)=0$. (

**b**,

**d**) cross-sections for $m=0.2765$. The line type indicates stable (solid) or unstalbe (dashes) equilibria.

**Figure 3.**Equilibria manifold obtained from Theorem 1. Relationship between the antigenic stimulation b of T cells and (

**a**) the concentration of non-secreting T cells T, (

**b**) the concentration of secreting T cells ${T}^{*}$, (

**c**) the concentration of inactive Tregs R, and (

**d**) the concentration of active Tregs ${R}^{*}$. The axis pointing upwards to the right is the slope parameter m. The shading color indicates the real part of the largest eigenvalue $Re\left(\lambda \right)$, increasing from black to blue for stable equilibria, and from green to yellow for unstable equilibria. The red and the magenta lines show the bifurcations, when $Re\left(\lambda \right)=0$.

**Figure 4.**Relation between the eigenvalues ($\lambda $) with the largest real part (blue line) and the second largest real part (green dashes) with the antigenic stimulation b of T cells, for $m=0.2765$. (

**a**) the largest real part of the eigenvalues can be positive for b between ${b}_{L}\approx 2.8\times {10}^{-1}$ and ${b}_{H}\approx 6.4\times {10}^{2}$; and that the second largest real part of the eigenvalues is negative. (

**b**) the two shown eigenvalues can be complex conjugate for b between ∼$2.1\times {10}^{-2}$ and ∼$2.5\times {10}^{-2}$, and for b between ∼$1.9$ and ∼$6.1\times {10}^{2}$.

**Figure 5.**Time evolutions for two sets of values of the parameters and four initial conditions (see Table 2). (

**a**,

**b**) $b={10}^{-1}$ and $m=0.2765$. Here, the only stable steady state is the controlled state. (

**c**,

**d**) $b=30$ and $m=0.2765$. In this case, there are two stable steady states. (

**a**,

**c**) black solid lines—total concentration of T cells x; blue dots—concentration of secreting T cells ${T}^{*}$; green dashes—concentration of non-secreting T cells T. (

**b**,

**d**) black solid lines—total concentration of Tregs y; blue dots—concentration of active Tregs ${R}^{*}$; green dashes—concentration of inactive Tregs R.Time evolutions

Parameter | Symbol | Range | Value |
---|---|---|---|

T cellT, ${T}^{*}$ | |||

T cell Maximum growth rate ${}^{1}$ | $\rho /\alpha $ | $<6$day${}^{-1}$ | 4 day${}^{-1}$ |

Death rate of inactive T cells (day${}^{-1}$) | ${d}_{T}$ | 0.1–0.01 [27] | $0.1$ |

Death rate ratio of active: inactive T cells | ${d}_{{T}^{*}}/{d}_{T}$ | 0.01–100 | $0.1$ |

Capacity of T cells ${}^{2}$ | $\rho /\left(\alpha \beta \right)$ | ${10}^{6}$–${10}^{7}$ cells/ml [28] | ${10}^{7}$ cells/ml |

Input rate of inactive T cells (cells/ml/day) | ${T}_{in}$ | 0–${10}^{4}$ | 100 |

Secretion reversion (constant) ${}^{3}$ | k | hrs-days | 0.1 h${}^{-1}$ |

Antigen stimulation level | $bk$ | ${10}^{-4}$–${10}^{5}\times \widehat{k}$ | Bifurcation parameter |

TregsR, ${R}^{*}$ | |||

Growth rate ratio T${}_{reg}$:T | $\u03f5$ | $<1$ | 0.6 |

Relaxation rate | $\widehat{k}$ | hrs-days | 0.1 h${}^{-1}$ |

Death rate ratio of inactive Tregs: inactive T cells | ${d}_{R}/{d}_{T}$ | $0.01\u2013100$ | 1 |

Death rate relative ratio of Tregs: T cells | $\frac{{d}_{{R}^{*}}}{{d}_{R}}/\frac{{d}_{{T}^{*}}}{{d}_{T}}$ | 0.01–100 | 1 |

Input rate ratio of inactive Tregs: inactive T cells | ${R}_{in}/{T}_{in}$ | $0\u2013{10}^{2}$ | 1 |

Homeostatic capacity ${}^{4}$ | ${R}_{hom}$ | 10–${10}^{5}$ cells/ml | ${10}^{4}$ cells/ml |

Tregs basal antigen stimulation level (for $b=0$) | $a\widehat{k}$ | 0–10 per day | 1 per day |

Homeostatic capacity ${}^{4}$ | ${R}_{hom}$ | 10–${10}^{5}$ cells/ml | ${10}^{4}$ cells/ml |

Secretion inhibition | $\gamma $ | 0.1–100 $\times {R}_{hom}^{-1}$ | 10 ${R}_{hom}^{-1}$ |

Slope of the tuning | m | 0–1 | Bifurcation parameter |

Cytokines | |||

Max. cytokine concentration ${}^{5}$ | $1/\alpha $ | 100–500 pM | 200 pM |

IL2 secretion rate | $\sigma $ | 0.07, 2 fgrms h${}^{-1}$ [29] ${}^{6}$ | 10${}^{6}$ molecs s${}^{-1}$ cell${}^{-1}$ |

Cytokine decay rate | $\sigma \delta $ | hrs-days | 1.5 h${}^{-1}$ [30] |

**Table 2.**Initial conditions for the time evolutions. Note that the initial condition 2 has higher T cell concentrations and lower Tregs concentrations than the initial condition 3.

Initial Condition | $\mathit{R}$ | ${\mathit{R}}^{\mathbf{*}}$ | $\mathit{T}$ | ${\mathit{T}}^{\mathbf{*}}$ | $\mathit{I}$ |
---|---|---|---|---|---|

1: immune response | 30 | 30 | 0 | ${10}^{7}$ | 200 |

2: intermediate ^{+} | $4.0\times {10}^{4}$ | $4.0\times {10}^{4}$ | $1.3\times {10}^{5}$ | $1.3\times {10}^{5}$ | 6 |

3: intermediate ^{-} | $4.5\times {10}^{4}$ | $4.5\times {10}^{4}$ | $1.2\times {10}^{5}$ | $1.2\times {10}^{5}$ | 5 |

4: controlled | 500 | 500 | ${10}^{3}$ | 0 | 0 |

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**MDPI and ACS Style**

Yusuf, A.A.; Figueiredo, I.P.; Afsar, A.; Burroughs, N.J.; Pinto, A.A.; Oliveira, B.M.P.M.
The Effect of a Linear Tuning between the Antigenic Stimulations of CD4^{+} T Cells and CD4^{+} Tregs. *Mathematics* **2020**, *8*, 293.
https://doi.org/10.3390/math8020293

**AMA Style**

Yusuf AA, Figueiredo IP, Afsar A, Burroughs NJ, Pinto AA, Oliveira BMPM.
The Effect of a Linear Tuning between the Antigenic Stimulations of CD4^{+} T Cells and CD4^{+} Tregs. *Mathematics*. 2020; 8(2):293.
https://doi.org/10.3390/math8020293

**Chicago/Turabian Style**

Yusuf, Aliyu A., Isabel P. Figueiredo, Atefeh Afsar, Nigel J. Burroughs, Alberto A. Pinto, and Bruno M. P. M. Oliveira.
2020. "The Effect of a Linear Tuning between the Antigenic Stimulations of CD4^{+} T Cells and CD4^{+} Tregs" *Mathematics* 8, no. 2: 293.
https://doi.org/10.3390/math8020293