Hopf Bifurcation for a Model of HIV Infection of CD 4 T Cells with Virus Released Delay

A viral model of HIV infection of CD4 T-cells with virus released period is formulated, and the effect of this released period on the stability of the equilibria is investigated. It is shown that the introduction of the viral released period can destabilize the system, and the period solution may arise. The direction and stability of the Hopf bifurcation are also discussed. Numerical simulations are presented to illustrate the results.


Introduction and Model Formulation
In the last decade, many mathematical models have been developed to describe the immunological response to infection with human immunodeficiency virus HIV see 1-11 .Simple HIV models have played a significant role in the development of a better understanding of the disease and the various drug therapy strategies used against it.Perelson et al. in 1 proposed a basic mathematical model to describe spread of HIV.Many other models 12-14 which take the model proposed in 1 as their inspiration have been formulated.Zhou  1.1 In 1.1 , T t , I t represent, respectively, the concentration of healthy CD4 T cells and infected CD4 T cells at time t, and V t represents the concentration of free HIV at time t.These parameters are defined as follows: s is the source of CD4 T cells precursors, d is the natural death rate of CD4 T cells, a is their growth rate thus, a > d in general , and T max is their carrying capacity, β is the contact rate between uninfected CD4 T cells and virus particles, δ is a blanket death term for infected cells, c is the clearance rate constant of virus, q is the lytic death rate for infected cells, and ρ is cure rate from infected cells to healthy cells.Parameters d, a, δ, β, ρ, c, T max , q, and s are positive values.They obtained the conditions for which system 1.1 exists an orbitally asymptotically stable periodic solution.Time delays of one type or another have been introduced to describe the time between viral entry into a target cell and the production of new virus particles by many authors see 6-9 .Culshaw and Ruan in 6 introduced a discrete time delay to the model to describe the time between infection of a CD4 T-cell and the emission of viral particles on a cellular level.They discussed locally asymptotically stable and obtained existence of the Hopf bifurcation under some conditions.Herz et al. in 3 used a discrete delay to model the intracellular delay that would substantially shorten the estimate for the half-life of free virus.
However, almost all of CD4 models were discussed and the discrete delay had denoted the time between infection of a CD4 T-cell and the emission of viral particles on a cellular level.According to reports of CDC government, they believed that there exists a "window period" when the infected cell released virus.The time interval between point of infection and detection of a seroconversion using US FDA-licensed third-generation antibody assays averages 22 days.In primary HIV infection, a localized viral replication eclipse takes place first, and lasts for 1-4 weeks 15 .If the amount of virus released does not attend a certain level, a patient infected by infected CD4 can not be examined.Therefore, we introduce a discrete delay denoted the "window period" in their model.Efficacy of the inhibition of viral replication is imposed upon the virus-host system by nucleoside analogue therapy.Based on the work of Zhou et al. see 5 , the HIV infection model with delay can be written as follows: All the biological meanings of parameters are same as system 1.1 .The time delay is introduced in the system describing the dynamics of "window period," that is, the released virus term of infected CD4 T cells is changed from ρI to qI t − τ .The initial conditions of system 1.2 are With a standard argument given in 16 , it easy to show that the solution T t , I t , V t with initial conditions 1.2 exists and is unique for all t ≥ 0.
Theorem 1.1.The solution of system 1.2 is positive and boundary.
Proof.First we proof the positivity of the solution of 1.2 .From the first equation of 1.2 , we obtain Solving it, we obtain From the second and the third equation, we obtain Therefore, the solution of 1.2 is positive.
Second, we will prove the boundary of the solution of 1. 1.9 It leads to L ≤ 2 max{s/ d, T max } .M 2 .Hence, for ε > 0 sufficiently small, there is a T 1 such Again, for ε > 0 sufficiently small, we drive from the third equation of system 1.2 that for t > T 1 τ Since this is true for arbitrary ε > 0, it follows that V ≤ q M 2 /c.If we set M max{M 2 , q M 2 /c}, such that S ≤ M, I ≤ M, V ≤ M. Hence, the solution of system 1.2 is boundary.

Equilibria Stability and Hopf Bifurcation
First we find all biologically feasible equilibria admitted by the system 1.2 and then study the dynamics of the system around each equilibrium.We introduce the reproduction number of differential-delay model 1.2 , which is given by a similar expression The R 0 stands if one virus is introduced in a population of uninfected cells which infect the total number of secondary infectious during their infectious period 1/c δ ρ .The equilibria of the system 1.2 are as follows: i uninfected equilibrium E 0 T 0 , 0, 0 , where T 0 T max /2a a − d a − d 2 4as/T max ; ii an infected equilibrium E T, I, V , which exists if R 0 > 1, where Following the analysis in 5 , we can see that, if R 0 > 1, then the infection-free equilibrium E 0 is unstable, and incorporation of a delay will not change the instability.Next, we focus on investigating the local stability and Hopf bifurcation of the positive equilibrium of 1.2 .To study the local stability of the positive equilibrium E T, I, V , we consider the linearization of system 1.2 at the point E. Let us define T t x T, I t y I, V t z V .

2.3
The following transcendental characteristic equation is obtained: where the coefficients in this equation are expressed as follows:

2.5
For τ 0, the characteristic equation 2.4 reduces to the following By the Routh-Hurwitz Criterion, it follows that all eigenvalues of 2.6 have negative real parts if and only if If R 0 > 1, and d − a 2aT/T max > 0,

2.8
We know that all eigenvalues of 2.6 have negative real parts.
Clearly, if λ iw with w > 0 is a root of 2.4 .This is the case if and only if w satisfies the following equation: Separating the real and imaginary parts, we have the following system that must be satisfied:

2.10
We eliminate the trigonometric functions by squaring both sides of each equation above and adding the resulting equations.We obtain the following sixth degree equation for w: Letting z w 2 , then 2.11 becomes a third order equation in z: where we have used the following notation for the coefficients of 2.12 :

2.13
In order to show that the endemic equilibrium E is locally stable we have to show that 2.12 does not have a positive real solution which might give the square of w, that is, that 2.4 cannot have purely imaginary solutions.The lemma below establishes conditions leading to that result.
Lemma 2.1 see 10 .For the polynomial equation 2.12 i if m 3 < 0, then 2.12 has at least one positive root;

has positive root if and only if z
Lemma 2.2 see 10 .For the polynomial equation 2.12 i if m 3 ≥ 0 and Δ m 2 1 − 3m 2 ≤ 0, then all roots with positive real parts of 2.4 have the same sum as those of the polynomial equation 2.12 for all τ; and h z * 1 ≤ 0, then all roots with positive real parts of 2.4 have the same sum as those of the polynomial equation 2.12 for τ ∈ 0, τ 0 .

Summarizing the above analysis and noting that
we have the following theorem.
Then the endemic equilibrium E of 2.4 is absolutely stable, that is, E is asymptotically stable for all values of the delay τ ≥ 0. Now, we turn to the bifurcation analysis.We use the delay τ as bifurcation parameter.We view the solutions of 2.4 as functions of the bifurcation parameter τ.Let λ τ η τ iw τ be the eigenvalue of 2.10 such that for some initial value of the bifurcation parameter τ 0 we have η τ 0 0, and w τ 0 w 0 without loss of generality we may assume w 0 > 0 .From 2.10 we have w 0 , j 0, 1, . . . .

2.15
Also, we can verify that the following transversal condition holds.By continuity, the real part of λ τ becomes positive when τ > τ 0 and the steady state becomes unstable.Moreover, a Hopf bifurcation occurs when τ passes through the critical value τ 0 see 16 .
To establish the Hopf bifurcation at τ τ 0 , we need to show that d Re λ τ /dτ| τ τ 0 > 0. Differentiating 2.4 from both sides with respect to τ, it follows that 2.17 This gives

2.21
The roots of 2.21 can be expressed as The above analysis can be summarized into the following theorem.
Theorem 2.4.Suppose that R 0 > 1.If both m 3 ≥ 0 and m 2 1 − 3m 2 ≥ 0 are satisfied, then the endemic equilibrium E of the delay model 1.2 is asymptotically stable when τ < τ 0 and unstable when τ > τ 0 , where when τ τ 0 , a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcates from E as τ passes through the critical value τ 0 .
In this way, using time delay as a bifurcation parameter, Theorem 2.4 indicates that the delay model could exhibit Hopf bifurcation at a certain value τ 0 of the delay if the parameters satisfy conditions.They show that a time delay in the infected-to-viral cells transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation.

Direction and Stability of the Hopf Bifurcation
In this section, we will study the direction, stability, and the period of the bifurcating periodic solutions.The approach we used here is based on the normal form approach, the center manifold theory, and delay differential equation theory see [16][17][18][19] .Throughout this section, we always assume that system 1.2 undergoes Hopf bifurcation at the positive equilibrium E S, I, V , for τ τ k , and then iw is corresponding purely imaginary roots of the characteristic equation at the positive equilibrium E S, I, V .Letting and f : R×C → R are given, respectively, by By the Riesz representation theorem, there exists a function η θ, μ of bounded variation for θ ∈ −1, 0 , such that In fact, we can choose where δ is the Dirac delta function.For φ ∈ C −1, 0 , R 3 , define

3.7
Then system 3.2 is equivalent to and a bilinear inner product where η θ η θ, 0 .Then A 0 and A * are adjoint operators.By the discussion in Section 2, we know that ±iwτ k are eigenvalues of A 0 .Thus, they are also eigenvalues of A * .We first need to compute the eigenvectors of A 0 and A * corresponding to iwτ k and −iwτ k , respectively.

3.11
It follows from the definition of A 0 and 3.3 , 3.5 , and 3.6 that That is, Thus, we can easily obtain

3.15
Suppose that q * s D 1, α * , β * 1 e iwτ k s , where 3.17 In order to assume q * θ , q θ 1, we need to determine the value of D. From 3.10 , we have

3.18
Thus, we can choose D as In the remainder of this section, we use the same notations as in 19 ; we first compute the coordinates to describe the center manifold C 0 at μ 0. Let x t be the solution of 3.8 when μ 0. Define z t q * , x t , W t, θ x t θ − 2 Re z t q θ .

3.20
On the center manifold C 0 , we have where z and z are local coordinates for center manifold C 0 in the direction of q and q * .Note that W is real if x is real.We only consider real solutions.For solution x t ∈ C 0 of 3.8 , since μ 0, we have ż t iw 0 τ k z q * 0 f 0, W z, z, 0 2 Re zq 0 .iw 0 τ k z q * 0 f 0 z, z .

3.23
We rewrite this equation as where It follows from 3.20 and 3.23 that

3.26
It follows together with 3.4 that

3.27
Hence one can obtain

3.28
Comparing the coefficients with 3.25 , we have  Since there are W 20 θ and W 11 θ in g 21 , we still need to compute them

3.30
where From 3.30 and 3.31 , we have In view of 3.32 , one can obtain

3.33
It follows from 3.32 and 3.33 that

3.34
Substituting the corresponding series into 3.30 and comparing the coefficients, we obtain

3.38
From 3.35 and 3.37 and the definition of A 0 , we have ẇ20 θ 2iw 20 g 20 q θ g 20 q θ .

3.40
Similarly, from 3.35 and 3.38 and the definition of A 0 , we have

3.41
In what follows, we shall seek appropriate E 1 and E 2 .From the definition of A and 3.35 , we obtain where g θ g 0, θ .By 3.37 and 3.38 , we have 44 Substituting 3.42 into 3.44 and noticing that

3.49
Similarly, substituting 3.43 into 3.45 , we can get It follows from 3.29 that g 21 can be expressed explicitly.Thus, we can compute the following values: Theorem 3.1.In 3.53 , the sign of σ 2 determined the direction of Hopf bifurcation: if σ 2 > 0, < 0 , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for τ k > τ k j < τ k j .β 2 determines the stability of the bifurcating periodic solution: the bifurcating periodic solution is stable (unstable) if β 2 < 0 > 0 , and T 2 determines the period of the bifurcating periodic solution: the period increases (decreases) if T 2 > 0 < 0 .

Simulation
In this section, we use numerical simulations to illustrate the theoretical results obtained in previous sections.As an example, we take the parameter values as follows: s 5, a 0.97, d 0.0002, T max 1200, δ 0.26, q 120, c 2.4, β 0.00024, τ 0.75, and ρ 0.01.By using the classical implicit format solving the delay differential equations and the method of steps for differential equations, we can solve the numerical solutions of 2.4 via the software package DEDiscover.
Simulation of the model in this situation produces stable dynamics as is presented in Figure 1.Plots a -c of Figure 1 show that uninfected cells, infected, cells and virus converge to their equilibrium with the parametric values.They show that the equilibrium E under some conditions see Theorem 2.3 is asymptotically stable.Next, we use a same set of parameter values as those in Figure 1, but we vary the value of τ 2.01.Thus the conditions of Theorem 2.4 are satisfied.Then the system 2.4 has an asymptotically stable periodic orbit see Figure 2 .Plots a -c of Figure 2 are the oscillations of uninfected cells, infected cells, and virus if τ attend a certain level see Theorem 2.4 .Figure 2 shows that there is a periodic solution.
We also find that the infection would always keep stability when the cure rate ρ is larger.This can be analyzed from the expression of R 0 and the conditions of Theorem 2.3.For example, we know that the oscillations of uninfected cells, infected cells and virus in Figure 3.And if we select ρ 1 0.01, ρ 2 0.61, and τ 2.01 and the other parameter values are same in Figure 1, then the infection would be stale see Figure 3 .Thus we can claim that the cure rate ρ is a very important parameter.The results show that if we improve the cure rate, we may control the disease.

Conclusion
An epidemic model of HIV infection of CD4 T cells with virus released period is studied.Mathematical analyses of the model equations with regard to dynamic behaviors of equilibria, Hopf bifurcation are analyzed.The basic reproduction number is obtained.In 5 , if R 0 < 1, the disease-free equilibrium is globally stable and the disease dies out.If R 0 > 1, a unique endemic equilibrium exists, and it is globally asymptotically stable.In our model, we determine criteria for Hopf bifurcation using the time delay as the bifurcation parameter based on the differential-delay model.We show that positive equilibrium is locally asymptotically stable when time delay is suitably small, while a Hopf bifurcation can occur as the delay increases.Hopf bifurcation has helped us in finding the existence of a region of instability in the neighborhood of a nonzero endemic equilibrium where the population will survive undergoing regular fluctuations.We should discuss the length of the delay which impact on the stability of our model.
et al. in 5 discussed the following ODE model:

Figure 1 :
Figure1: a -c show that uninfected cells, infected cells, and virus converge to their equilibrium with parametric values as stated in the text with τ 0.75.They show that the equilibrium is asymptotically stable.

Figure 2 :
Figure 2: a -c are the oscillations of uninfected cells, infected cells, and virus.

Figure 3 :
Figure 3: a -c show the uninfected cells, infected cells and virus with ρ 1 0.01 and ρ 2 0.61.They show the cure rate is a important parameter.
Adding the first equation and the second equation of 1.2 , it is easy to see 2 .Define d min{d, δ} and L T I. if L ≥ T max , 1.7 become the following: dL dt ≤ s − dL.1.8 It is easy to get L ≥ s/ d .M 1 .If L ≤ T max , 1.7 become the following: