Global dynamics of a latent HIV infection model with general incidence function and multiple delays

Yu Yang; Yueping Dong; Yasuhiro Takeuchi

doi:10.3934/dcdsb.2018207

Article Contents

2019, Volume 24, Issue 2: 783-800. Doi: 10.3934/dcdsb.2018207

This issue Previous Article Valuation of American strangle option: Variational inequality approach Next Article A two-species weak competition system of reaction-diffusion-advection with double free boundaries

Global dynamics of a latent HIV infection model with general incidence function and multiple delays

1.
School of Science and Technology, Zhejiang International Studies University, Hangzhou 310023, China
2.
Faculty of Engineering, Kyushu University, Fukuoka 819-0395, Japan
3.
College of Science and Engineering, Aoyama Gakuin University, Kanagawa 252-5258, Japan

* Corresponding author: yangyu@zisu.edu.cn
* Corresponding author: yangyu@zisu.edu.cn

Received: July 31, 2017

Revised: December 31, 2017

Early access: June 2018

Published: January 2019

The first author was partially supported by National Natural Science Foundation of China (No. 11501519) and the third author was partially supported by the Japan Society Promotion of Science) through the "Grant-in-Aid 26400211"

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Abstract

In this paper, we propose a latent HIV infection model with general incidence function and multiple delays. We derive the positivity and boundedness of solutions, as well as the existence and local stability of the infection-free and infected equilibria. By constructing Lyapunov functionals, we establish the global stability of the equilibria based on the basic reproduction number. We further study the global dynamics of this model with Holling type-Ⅱ incidence function through numerical simulations. Our results improve and generalize some existing ones. The results show that the prolonged time delay period of the maturation of the newly produced viruses may lead to the elimination of the viruses.

Keywords:

Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 92D30.

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Figure 1. The solution converges to the infection-free equilibrium $E_0(10^6, 0, 0, 0)$ when $\mathcal{R}_0 = 0.1781<1$. The parameter values are taken from Data1 in Table 1

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Figure 2. The solution converges to the infected equilibrium $E^*(9.0262\times10^5, 68.5219,944.7647, 8.1662\times10^4)$ when $\mathcal{R}_0 = 2.0126>1$. The parameter values are taken from Data2 in Table 1

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Figure 3. The function $\mathcal{R}_0(\tau_3)$ with respect to $\tau_3$

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Figure 4. Effect of different values of $\tau_3$ on the dynamics of system (12)

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Figure 5. The solution converges to $E_0(10^6, 0, 0, 0)$ when $\tau_3$ becomes large. Here we choose $\tau_3 = 80$. The parameter values are taken from Data3 in Table 1

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Table 1. List of parameters

Parameters	Data1	Data2	Data3	Source
$s$ (cells ml$^{-1}$ day$^{-1}$)	10$^{4}$	10$^{4}$	10$^{4}$	[4]
$d_T$ (day$^{-1}$)	0.01	0.01	0.01	[30]
$\beta$ (ml virion$^{-1}$ day$^{-1}$)	2.4$\times10^{-8}$	2.4$\times10^{-8}$	2.4$\times10^{-8}$	[43]
$\alpha_1$	0.00001	0.00001	0.00001	Assumed
$k$	1.5$\times10^{-6}$	0.001	0.001	[4,2]
$\delta_1$ (day$^{-1}$)	0.05	0.05	0.05	[2]
$\delta_L$ (day$^{-1}$)	0.004	0.004	0.004	[4]
$\delta_2$ (day$^{-1}$)	0.01	0.01	0.01	[28]
$\alpha$ (day$^{-1}$)	0.01	0.01	0.01	[47]
$\delta$ (day$^{-1}$)	0.7	1	1	[4,22]
$N$ (virions cell$^{-1}$)	100	2000	2000	[4,47]
$c$ (day$^{-1}$)	13	23	23	[4,45]
$\tau_1$	0.3	0.3	0.3	[2]
$\tau_2$	0.6	0.6	0.6	[2]
$\tau_3$	0.6	0.6	Assumed	[41]

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