# Spatiotemporal Dynamics of a Generalized Viral Infection Model with Distributed Delays and CTL Immune Response

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

**(H**_{0})- g(0,I) = 0, for all $I\ge 0$; $\frac{\partial g}{\partial T}(T,I)\ge 0$ (or $g(T,I)$ is a strictly monotone increasing function with respect to T when $f\equiv 0$) and $\frac{\partial g}{\partial I}(T,I)\le 0$, for all $T\ge 0$ and $I\ge 0$.
**(H**_{1})- $f(0,I,V)=0$, for all $I\ge 0$ and $V\ge 0$,
**(H**_{2})- $f(T,I,V)$ is a strictly monotone increasing function with respect to T (or $\frac{\partial f}{\partial T}(T,I,V)\ge 0$ when $g(T,I)$ is a strictly monotone increasing function with respect to T), for any fixed $I\ge 0$ and $V\ge 0$,
**(H**_{3})- $f(T,I,V)$ is a monotone decreasing function with respect to I and V.

## 2. Model Formulation and Preliminaries

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

- (i)
- If${\mathcal{R}}_{0}\le 1$, then system (2) has a unique infection-free equilibrium${E}_{0}({T}_{0},0,0,0)$, where$T}_{0}=\frac{\lambda}{d$.
- (ii)
- If${\mathcal{R}}_{0}>1$, then system (2) has a unique infection equilibrium without cellular immunity${E}_{1}({T}_{1},{I}_{1},{V}_{1},0)$besides${E}_{0}$, where${T}_{1}\in (0,\frac{\lambda}{d})$, $I}_{1}=\frac{{\eta}_{1}(\lambda -d{T}_{1})}{a$and$V}_{1}=\frac{k{\eta}_{1}{\eta}_{2}(\lambda -d{T}_{1})}{a\mu$.
- (iii)
- If${\mathcal{R}}_{1}^{Z}>1$, then system (2) has a unique infection equilibrium with cellular immunity${E}_{2}({T}_{2},{I}_{2},{V}_{2},{Z}_{2})$besides${E}_{0}$and${E}_{1}$, where${T}_{2}\in (0,\frac{\lambda}{d}-\frac{ab}{dc{\eta}_{1}})$, $I}_{2}=\frac{b}{c$, $V}_{2}=\frac{kb{\eta}_{2}}{\mu c$and$Z}_{2}=\frac{c{\eta}_{1}(\lambda -d{T}_{2})-ab}{pb$.

## 3. Global Stability

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**(i)**, we construct the Lyapunov functional as follows

_{4}) that

**(ii)**, we construct the Lyapunov functional as follows

**Remark**

**1.**

_{4}) comes from (17) and (18). This hypothesis is a sufficient condition for that the time derivatives of the Lyapunov functionals${L}_{1}$and${L}_{2}$to be non-negative. When cell-to-cell mode is ignored (i.e.,$g\equiv 0$), the assumption (H

_{4}) can be reduced to

## 4. Application and Numerical Simulations

_{4}) is verified. By applying Theorems 3 and 4, we obtain the following result.

**Corollary**

**1.**

- If${\mathcal{R}}_{0}\le 1$, then the infection-free equilibrium${E}_{0}$of system (21) is globally asymptotically stable.
**(i)**- the infection equilibrium without cellular immunity${E}_{1}$that is globally asymptotically stable if${\mathcal{R}}_{1}^{Z}\le 1$;
**(ii)**- the infection equilibrium with cellular immunity${E}_{2}$that is globally asymptotically stable if${\mathcal{R}}_{1}^{Z}>1$.

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Spatiotemporal dynamics of the model (21) when ${\mathcal{R}}_{0}=2.0830>1$ and ${\mathcal{R}}_{1}^{Z}=0.8441\le 1$.

**Figure 3.**Spatiotemporal dynamics of the model (21) when ${\mathcal{R}}_{0}=2.0830>1$ and ${\mathcal{R}}_{1}^{Z}=1.2662>1$.

Parameter | Value | Parameter | Value |
---|---|---|---|

$\lambda $ | 10 | ${\alpha}_{2}$ | $0.01$ |

d | $0.0139$ | ${\gamma}_{2}$ | $0.1$ |

${\beta}_{1}$ | $2.4\times {10}^{-5}$ | k | 50 |

a | $0.29$ | b | $0.1$ |

${\u03f5}_{1}$ | $0.05$ | p | $0.01$ |

${\u03f5}_{2}$ | $0.07$ | $\mu $ | 3 |

${\gamma}_{1}$ | $0.1$ | ${\beta}_{2}$ | Varied |

${\alpha}_{1}$ | $0.01$ | c | Varied |

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Hattaf, K.
Spatiotemporal Dynamics of a Generalized Viral Infection Model with Distributed Delays and CTL Immune Response. *Computation* **2019**, *7*, 21.
https://doi.org/10.3390/computation7020021

**AMA Style**

Hattaf K.
Spatiotemporal Dynamics of a Generalized Viral Infection Model with Distributed Delays and CTL Immune Response. *Computation*. 2019; 7(2):21.
https://doi.org/10.3390/computation7020021

**Chicago/Turabian Style**

Hattaf, Khalid.
2019. "Spatiotemporal Dynamics of a Generalized Viral Infection Model with Distributed Delays and CTL Immune Response" *Computation* 7, no. 2: 21.
https://doi.org/10.3390/computation7020021