Hopf bifurcation of an age-structured virus infection model

Hossein Mohebbi; Azim Aminataei; Cameron J. Browne; Mohammad Reza Razvan

doi:10.3934/dcdsb.2018046

Article Contents

2018, Volume 23, Issue 2: 861-885. Doi: 10.3934/dcdsb.2018046

This issue Previous Article Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method Next Article Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis

Hopf bifurcation of an age-structured virus infection model

1.
Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box:16315-1618, Tehran, Iran
2.
Department of Mathematics, University of Louisiana, Lafayette, LA, USA
3.
Department of Mathematical Sciences, Sharif University of Technology, P. O. Box: 11155-9415, Tehran, Iran

* Corresponding author
* Corresponding author

Received: November 30, 2016

Revised: August 31, 2017

Early access: November 2017

Published: June 2018

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Abstract

In this paper, we introduce and analyze a mathematical model of a viral infection with explicit age-since infection structure for infected cells. We extend previous age-structured within-host virus models by including logistic growth of target cells and allowing for absorption of multiple virus particles by infected cells. The persistence of the virus is shown to depend on the basic reproduction number $R_{0}$. In particular, when $R_{0}≤1$, the infection free equilibrium is globally asymptotically stable, and conversely if $R_{0}> 1$, then the infection free equilibrium is unstable, the system is uniformly persistent and there exists a unique positive equilibrium. We show that our system undergoes a Hopf bifurcation through which the infection equilibrium loses the stability and periodic solutions appear.

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Mathematics Subject Classification: Primary: 37N25, 37B25; Secondary: 92B05.

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Figure 1. A numerical solution of system (40) tends to the infection-free equilibrium $E_0$, as time tends to infinity, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] = [0.038, 0.0045, 4.6137, 1, 0.093, 0.4, 0.028]$. In this case $R_0 = 0.6518$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space. Eigenvalues of linearized matrix about $\overline{E}$ are $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$

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Figure 2. A numerical solution of system (40) approaches to $\overline{E}$, as time tends to infinity, and $\bar{E}$ is stable, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] = $ $[0.03285, 0.01, 4.6137, 1.3, 0.045, 0.1, 0.0351]$. In this case $R_0 = 16.0999$, $\bar{E} = (\bar{T},\bar{V},\bar{I}) =[0.2212, 3.1275, 0.1971]$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space.

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Figure 3. A numerical solution of system (40) tends to the limit cycle, as time tends to infinity, and $\bar{E}$ is unstable, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] = $ $[0.03285, 0.01, 4.6137, 1.3, 0.03, 0.1, 0.0351]$. In this case $R_0 = 22.4437$, $(\bar{T},\bar{V},\bar{I}) =[ 0.1115, 3.2056, 0.1018]$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space. Eigenvalues of linearized matrix about $\overline{E}$ are $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$

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Figure 4. A numerical solution of system (41)-(44) tends to the DFE, as time tends to infinity, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] = $ $[.1,100000,0.0000005,200,13,0.000003,2,0.05,0.7]$. In this case $R_0 =0.9905$ and $(\bar{T},\bar{V},\bar{I}) =[ 10^5, 0 , 0]$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.

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Figure 5. A numerical solution of system (41)-(44) tends to the $\bar{E}$, as time tends to infinity, and $\bar{E}$ is stable, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] = $ $[.1,100000,0.0000008,200,13,0.000003,2,0.05,0.7]$. In this case $R_0 =1.5812$, $(\bar{T},\bar{V},\bar{J}+\bar{I}) =[ 6.3209\times 10^4, 4.5989\times 10^4, 7.4321\times 10^3]$. The probability of re-infection of infected cells during eclipse phase (during age $0\leq a \leq \tau$) calculated at $\bar{E}$ is $\pi(\tau) = 0.23$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.

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Figure 6. A numerical solution of system (41)-(44) tends to the limit cycle, as time tends to infinity, and $\bar{E}$ is unstable, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] = $ $[1,100000, 0.000005,200, 13, 0.000001, 2, 0.05, 0.7]$. In this case $R_0 =9.5750$, $(\bar{T},\bar{V},\bar{J}+\bar{I}) =[ 1.0119\times 10^4, 1.7976\times 10^5, 2.9066\times 10^4]$. The probability of re-infection of infected cells during eclipse phase calculated at $\bar{E}$ is $\pi(\tau) = 0.2882$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.

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Table 1. Parameter definition and values from literatures.

Parameter	Value	Description	Reference
$e$	day$^{-1}$	Maximum proliferation rate	See text
$g$	0.008 day$^{-1}$	Death rate of uninfected cells	[27]
$T_{\text{max}}$	mm$^{-3}$	Density of $T$ cell at which proliferation shouts off	See text
$k$	$5 \times 10^{-7}$ ml virion day$^{-1}$	Infection rate of target cells by virus	[27]
$\delta$	0.8 day$^{-1}$	Death rate of infected cells	[32]
p	Varied	Virion production rate of an infected cell	See text
$d$	3 day$^{-1}$	clearance rate of free virus	[22]
$\gamma$	day$^{-1}$	Reinfection rate of infected cells by virus	See text

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