Global dynamics of a delay virus model with recruitment and saturation effects of immune responses

Cuicui Jiang; Kaifa Wang; Lijuan Song

doi:10.3934/mbe.2017063

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2017, Volume 14, Issue 5&6: 1233-1246. Doi: 10.3934/mbe.2017063

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Global dynamics of a delay virus model with recruitment and saturation effects of immune responses

Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China

^* Corresponding authorr: Kaifa Wang
^* Corresponding authorr: Kaifa Wang

Received: July 26, 2016

Revised: October 19, 2016

Published: November 2017

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Abstract

In this paper, we formulate a virus dynamics model with the recruitment of immune responses, saturation effects and an intracellular time delay. With the help of uniform persistence theory and Lyapunov method, we show that the global stability of the model is totally determined by the basic reproductive number $R_0$. Furthermore, we analyze the effects of the recruitment of immune responses on virus infection by numerical simulation. The results show ignoring the recruitment of immune responses will result in overestimation of the basic reproductive number and the severity of viral infection.

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

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Figure 1. Illustration of the proportion of infected cells ($I_1$) and virus load ($V_1$) at the endemic equilibrium $E_{1}$. Here parameters are $\lambda=50, \beta=5\times 10^{-7}, d_1=0.008, d_2=0.8,$$ d_3=3, d_4=0.05, d_5=0.1,$$ p=0.05, r=2500, q=0.2, k_1=0.12, h_1=1200, k_2=1.5, s=0.001, \tau=1.5$

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Table 1. Parameter definitions and values used in numerical simulations

Par.	Value	Description	Ref.
$\lambda$	0-50 cells ml-day$^{-1}$	Recruitment rate of healthy cells	[33,38]
$d_1$	$0.007-0.1$day $^{-1}$	Death rate of healthy cells	[38]
$\beta$	$5\times10^{-7}-0.5$ ml virion-day$^{-1}$	Infection rate of target cells by virus	[33,38]
$d_2$	$0.2-0.8$ day$^{-1}$	Death rate of infected cells	[41,46]
$r$	$10-2500$ virions/cell	Burst size of virus	[38]
$d_3$	$2.4-3$ day$^{-1}$	Clearance rate of free virus	[38]
$p$	$0.05-1$ day$^{-1}$	Killing rate of CTL cells	[41,40]
$q$	$0.1-1$ day$^{-1}$	Neutralizing rate of antibody	[41]
$k_1$	$0.1-0.12$ day$^{-1}$	Proliferation rate of CTL response	[2,41]
$k_2$	$1.5$ day$^{-1}$	Production rate of antibody response	[41]
$d_4$	$0.05-2$ day$^{-1}$	Mortality rate of CTL response	[2,40]
$d_5$	$0.1$ day$^{-1}$	Clearance rate of antibody	[41]
$s$	$0.001-1.4$	1/s is the average time	[32,47]
$\tau$	$0-2$ days	Virus replication time	[38]
$h_1$	1200	Saturation constant	Assumed
$h_2$	1500	Saturation constant	Assumed
$\lambda_1$	Varied	Rate of CTL export from thymus	[9]
$\lambda_2$	Varied	Recruitment rate of antibody	[9]

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