Article Contents
Article Contents

# Hopf bifurcation of an age-structured virus infection model

• * Corresponding author
• 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.

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

 Citation:

• 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 $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$

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.

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 $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$

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.

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.

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.

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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