# Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Analysis of the HIV Model

#### 2.1. Existence and Local Stability

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.2. Global Stability Results

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 3. Numerical Results and Simulations

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Existence and Local Stability Results Proof

## Appendix B. Global Stability Result Proof

## References

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**Figure 2.**The behavior of the infection dynamics for $\lambda =10$, ${d}_{1}=0.0139$, ${k}_{1}=0.04$, ${d}_{2}=0.0495$, ${k}_{2}=1.1$, ${d}_{3}=0.5776$, $a=2$, ${d}_{4}=0.6$, $p=0.0024$, $c=0.15$, $b=0.5$. For the parameters used in this figure, the basic reproduction number is ${R}_{0}=0.2209<1.$ Solutions of the system converge to the disease-free equilibrium point ${E}_{f}=(827.22,0,0,0,0)$.

**Figure 3.**Selective phase portraits to illustrate the solution behavior of the infection dynamics when $\lambda =10$, ${d}_{1}=0.0139$, ${k}_{1}=0.04$, ${d}_{2}=0.0495$, ${k}_{2}=1.1$, ${d}_{3}=0.5776$, $a=2$, ${d}_{4}=0.6$, $p=0.0024$, $c=0.15$, $b=0.5$.

**Figure 4.**The behavior of the infection dynamics for $\lambda =1$, ${d}_{1}=0.0139$, ${k}_{1}=0.04$, ${d}_{2}=0.0495$, ${k}_{2}=1.1$, ${d}_{3}=0.5776$, $a=100$, ${d}_{4}=0.6$, $p=0.0024$, $c=0.15$, $b=0.5$. For the parameters used in this figure, the basic reproduction number is $11.049>1.$ The immune response reproduction number ${R}_{CTL}=3.596\times {10}^{-1}<1.$ In this case, all the solutions converge towards the endemic equilibrium ${E}_{1}=(19.96,5.98\times {10}^{-1},1.14,199.78,0).$

**Figure 5.**Selective phase portraits to illustrate the solution behavior of the infection dynamics when $\lambda =1$, ${d}_{1}=0.0139$, ${k}_{1}=0.04$, ${d}_{2}=0.0495$, ${k}_{2}=1.1$, ${d}_{3}=0.5776$, $a=100$, ${d}_{4}=0.6$, $p=0.0024$, $c=0.15$, $b=0.5$.

**Figure 6.**The behavior of the infection dynamics for $\lambda =10$, ${d}_{1}=0.0139$, ${k}_{1}=0.04$, ${d}_{2}=0.0495$, ${k}_{2}=1.1$, ${d}_{3}=0.5776$, $a=100$, ${d}_{4}=0.6$, $p=0.0024$, $c=0.15$, $b=0.5$. For the chosen parameters in this figure, we have ${R}_{0}=11.049>1$ and ${R}_{CTL}=4.13>1$. Solutions tending to the infection steady state ${E}_{3}=(285.12,6.55,3.33,555.55,660.86)$ are observed.

**Figure 7.**Selective phase portraits to illustrate the solution behavior of the infection dynamics when $\lambda =10$, ${d}_{1}=0.0139$, ${k}_{1}=0.04$, ${d}_{2}=0.0495$, ${k}_{2}=1.1$, ${d}_{3}=0.5776$, $a=100$, ${d}_{4}=0.6$, $p=0.0024$, $c=0.15$, $b=0.5$.

Parameters | Meaning | Value | References |
---|---|---|---|

$\lambda $ | Source rate of CD4${}^{+}$ T cells | $[0,10]$ | [16] |

${k}_{1}$ | Average of infection | $[2.5\times {10}^{-4},0.5]$ | [14] |

${d}_{1}$ | Decay rate of healthy cells | $0.0139$ | [14] |

${d}_{2}$ | Death rate of exposed CD4${}^{+}$ T cells | $0.0495$ | [14] |

${k}_{2}$ | The rate that exposed become infected CD4${}^{+}$ T cells | $1.1$ | [14] |

${d}_{3}$ | Death rate of infected CD4${}^{+}$ T cells, not by CTL killing | $0.5776$ | [14] |

a | The rate of production the virus by infected CD4${}^{+}$ T cells | $[2,1250]$ | [14] |

${d}_{4}$ | Clearance rate of virus | $[0.3466,2.4]$ | [14] |

p | Clearance rate of infection | $0.0024$ | [17] |

c | Activation rate CTL cells | $0.15$ | [17] |

b | Death rate of CTL cells | $0.5$ | [17] |

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**MDPI and ACS Style**

Allali, K.; Danane, J.; Kuang, Y. Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase. *Appl. Sci.* **2017**, *7*, 861.
https://doi.org/10.3390/app7080861

**AMA Style**

Allali K, Danane J, Kuang Y. Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase. *Applied Sciences*. 2017; 7(8):861.
https://doi.org/10.3390/app7080861

**Chicago/Turabian Style**

Allali, Karam, Jaouad Danane, and Yang Kuang. 2017. "Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase" *Applied Sciences* 7, no. 8: 861.
https://doi.org/10.3390/app7080861