# Eradication Conditions of Infected Cell Populations in the 7-Order HIV Model with Viral Mutations and Related Results

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## Abstract

**:**

## 1. Introduction

## 2. On the Localization Problem of Compact Invariant Sets

## 3. Preliminary Remarks

- (1)
- (2)
- This system possesses the invariant plane ${x}_{1}={x}_{3}={x}_{4}=0$; this is the case when there are no wild-type HIV particles and no $Th$ cells infected by them. The subsystem defined on this plane is explored in Section 7.
- (3)
- Suppose that ${a}_{5}$ is a source term for uninfected CD4${}^{+}$ T cells that is zero, the concentration of uninfected Th cells (${x}_{2}$) is zero as well, and between the death rate and growth rate of the uninfected CD4${}^{+}$ T cell population, the following inequalities are fulfilled: ${a}_{6}>{a}_{7}$ and ${u}_{3}<{a}_{6}{a}_{7}^{-1}-1$. Then the system (4) becomes a linear asymptotically stable system for which free, wild-type HIV particles and all other cell populations vanish after a sufficiently long observation time.

## 4. Equilibrium Points

**Example**

**1.**

## 5. Ultimate Upper Bounds

**Remark**

**1.**

**Theorem**

**1.**

**Remark**

**2.**

## 6. On the Location of $\mathbf{\omega}$-Limit Sets

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

- 1.
- Conditions of Theorem 2 do not depend on controls ${u}_{3}$ and ${u}_{4}$.
- 2.
- If the condition ${d}_{2}<1$ holds, then Theorem 2 is true. Indeed, taking into account (3), we get that this condition implies the condition ${d}_{1}<1$ and, consequently, conditions (9). Note that the condition ${d}_{2}<1$ does not depend on controls, that is, it is satisfied with zero values of controls.

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 7. Concluding Remarks

- Calculate equilibrium points;
- Present local stability conditions;
- Find ultimate upper bounds for all variables of this model that define the polytope containing all $\omega $-limit sets.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Starkov, K.E.; Kanatnikov, A.N. Eradication Conditions of Infected Cell Populations in the 7-Order HIV Model with Viral Mutations and Related Results. *Mathematics* **2021**, *9*, 1862.
https://doi.org/10.3390/math9161862

**AMA Style**

Starkov KE, Kanatnikov AN. Eradication Conditions of Infected Cell Populations in the 7-Order HIV Model with Viral Mutations and Related Results. *Mathematics*. 2021; 9(16):1862.
https://doi.org/10.3390/math9161862

**Chicago/Turabian Style**

Starkov, Konstantin E., and Anatoly N. Kanatnikov. 2021. "Eradication Conditions of Infected Cell Populations in the 7-Order HIV Model with Viral Mutations and Related Results" *Mathematics* 9, no. 16: 1862.
https://doi.org/10.3390/math9161862