# Stability and Hopf Bifurcation in an HIV-1 Infection Model with Latently Infected Cells and Delayed Immune Response

Authors: Haibin Wang; Rui Xu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

An HIV-1 infection model with latently infected cells and delayed immune response is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria is established and the existence of Hopf bifurcations at the CTL-activated infection equilibrium is also studied. By means of suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infectionR0≤1; if the basic reproduction ratio for viral infection R0>1 and the basic reproduction ratio for CTL immune response R1≤1, the CTL-inactivated infection equilibrium is globally asymptotically stable. If the basic reproduction ratio for CTL immune response R1>1, the global stability of the CTL-activated infection equilibrium is also derived when the time delay τ=0. Numerical simulations are carried out to illustrate the main results.


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## Body

## 1. Introduction

Mathematical and computational models of the human immune response under HIV-1 infection have received great attention in recent years [1–9]. It is a useful tool of better understanding disease dynamics and making prediction of disease outbreak and evaluations of prevention strategies and drug therapy strategies used against HIV-1 infection.It is well known that when HIV-1 enters the body, it targets cells with CD4 receptors, including the CD4+ T-cells, the main driver of the immune response. Recent studies have shown that a significant proportion of CD4+ T-cells are infected by the virus, and that this specific population of T-cells might be preferentially infected [10]. In human’s immune system, cytotoxic T lymphocytes (CTLs) play an important role in antiviral defense by attacking infected cells. Therefore, it is important and yet has been a hot topic to formulate models to explain the exhaustion of the CD4+ T-cells and CTLs. Such models involve the concentrations of uninfected CD4+ T-cells, x, infected CD4+ T-cells that are producing virus, y, free virus, v, and CTLs, z. A basic mathematical model describing HIV-1 infection dynamics that has been studied in [1, 2] is of the form
(1)x˙(t)=λ-dx(t)-βx(t)v(t),y˙(t)=βx(t)v(t)-ay(t)-py(t)z(t),v˙(t)=ky(t)-uv(t),z˙(t)=cy(t)-bz(t),
where uninfected, susceptible CD4+ T-cells are created from sources within the body at a rate, λ, uninfected CD4+ T-cells die at rate dx(t) and become infected at rate βx(t)v(t), where β is the rate constant describing the infection process; infected cells die at rate ay(t) and are lysed by CTLs at a rate py(t)z(t); free virus is produced from infected cells at rate ky(t) and is removed at rate uy(t). The CTLs expand at a rate cy(t) and decay at a rate bz(t).Time delays cannot be ignored in models for immune response, especially the delay between viral appearance and the production of new immune particles [6–9]. Wang et al. [6] studied the effects of the time delay for immune response on a three-dimensional system with z˙=cy(t-τ)-bz. By assuming that the production of CTLs also depends on the population of CTL cells, Canabarro et al. [7] investigated the effects of a time delay on a four-dimensional system with z˙=cy(t-τ)z(t-τ)-bz.However, in all the previous works mentioned above, they all neglected the fact that once in the cells not all viruses initiate active virus production. A large proportion of CD4+ T-cells are latently infected following the integration of proviral DNA into the host cell genome, some of which can remain quiescent for long periods of time before becoming activated [11]. In [12], such cells are defined as latently infected cells. The capability of the HIV-1 to persist latent inside CD4+ T-cells is currently regarded as a barrier to recovery from infection. But till now, as far as we know, only a few works (see, [13, 14]) concern the effects that latently infected cells are expected to have on HIV-1 infection process. In this paper, motivated by the works of [1, 7, 13], we modify the basic virus infection model and add a further state variable, w, which represents the population of latently infected cells and propose a four-dimensional delayed HIV-1 infection model with latently infected cells. The model is given by
(2)x˙(t)=λ-dx(t)-βx(t)y(t)1+αy(t),w˙(t)=(1-q)βx(t)y(t)1+αy(t)-ew(t)-δw(t),y˙(t)=qβx(t)y(t)1+αy(t)-ay(t)+δw(t)-py(t)z(t),z˙(t)=cy(t-τ)z(t-τ)-bz(t),
where the parameters λ, d, β, a, p, c, and b are the same as those defined in model (1). In our model, we assume that the free virus interacts with the uninfected cells to produce actively infected cells at rate qβx(t)y(t)/[1+αy(t)] and latently infected cells at rate (1-q)βx(t)y(t)/[1+αy(t)] due to the saturation response of the infection rate, where 0<q<1 and α>0. Latently infected cells containing proviral DNA die at rate ew(t) and become activated at rate δw(t). The CTL response is activated at a rate proportional to the number of infected cells at a previous time, cy(t-τ)z(t-τ), where τ is the time delay of CTL response.The initial conditions for system (2) take the form(3)x(θ)=ψ1(θ),w(θ)=ψ2(θ),y(θ)=ψ3(θ),z(θ)=ψ4(θ),ψ1(θ)≥0,ψ2(θ)≥0,ψ3(θ)≥0,ψ4(θ)≥0,θ∈[-τ,0),ψ1(0)>0,ψ2(0)>0,ψ3(0)>0,ψ4(0)>0,
where (ψ1(θ),ψ2(θ),ψ3(θ),ψ4(θ))∈C([-τ,0],R+04) and the Banach space of continuous functions maps the interval [-τ,0] into R+04, where R+04={(x1,x2,x3,x4):xi≥0,i=1,2,3,4}.Our primary goal is to carry out a complete mathematical analysis of system (2) and establish its global dynamics. The organization of this paper is as follows. In Section 2, by analyzing the corresponding characteristic equations, we study the local stability of an infection-free equilibrium and a CTL-inactivated infection equilibrium of model (2). In Section 3, we discuss the local stability and the existence of Hopf bifurcations at the CTL-activated infection equilibrium. In Section 4, by means of suitable Lyapunov functionals and LaSalle’s invariance principle, we study the global stability of the infection-free equilibrium and the CTL-inactivated infection equilibrium with any τ≥0 and the CTL-activated infection equilibrium with τ=0, respectively. Numerical simulations are carried out in the last section to illustrate the main results. A brief remark is also given in this section to conclude our work.


## 2. Equilibria and Their Local Stability

In this section, we firstly establish the nonnegativity and boundedness of solutions of system (2) and then give sufficient conditions for the existence of each of feasible equilibria of system (2) and discuss the local stability of the infection-free equilibrium and the CTL-inactivated infection equilibrium, respectively.Theorem 1.
Let(x(t),w(t),y(t),v(t)) be any solution of system (2). Then, under the initial conditions (3), all solutions (x(t),w(t),y(t),v(t)) are nonnegative on [0,+∞) and ultimately bounded.Proof.
Since the right hand side of system (2) is completely continuous and locally Lipschitzian on C, the solution (x(t),w(t),y(t),v(t)) of (2) with initial conditions (3) exists and is unique on [0,T), where 0<T<+∞ [15, Chapter 2].
Let us show thatx(t)>0 for all t∈[0,+∞). If x(t) were to lose its nonnegativity on some local existence interval [0,T) for some constant t>0, there would be a time at t1>0. By the first equation of (2), we have x˙(t1)=λ>0. That means x(t)<0 for t∈(t1-ε,t1), where ε is an arbitrarily small positive constant. This leads to a contradiction. It follows that x(t) is always positive. Further, from the second and the third equation in (2), we have, respectively,
(4)w(t)=w(0)exp{∫0t‍[(1-q)βx(σ)y(σ)w(σ)(1+αy(σ))-(e+δ)]dσ},y(t)=y(0)exp{∫0t‍[qβx(σ)1+αy(σ)-affffffffffffffffffffff+δw(σ)y(σ)-pz(σ)qβx(σ)1+αy(σ)]dσ},
which show that w(t), y(t)>0 for all t≥0. Then by the last equation in (2)
(5)z(t)>z(0)e-bt,
confirming that z(t)>0 for all t≥0.
Next, we prove the ultimate boundedness of (2). Denote
(6)N(t)=x(t)+w(t)+y(t)+pcz(t+τ).
By nonnegativity of the solution, it follows that
(7)N˙(t)=λ-dx-ew-ay-b·pcz(t+τ)≤λ-min{a,b,d,e}N(t).
Then the comparison theorem implies that there exists a t2>0 such that N(t)≤λ/min{a,b,d,e}, for t≥t2. Let M1=max{λ/a,λ/b,λ/d,λ/e}; then N(t)≤M1, for t≥t2. This implies that N(t) is ultimately bounded, and so are x(t), w(t), y(t), and v(t). Thus the solutions of (2) are ultimately bounded. This completes the proof.Clearly, system (2) always has an infection-free equilibrium E0(x0,0,0,0), where x0=λ/d.Denote(8)R0=βλ(eq+δ)ad(e+δ),R1=cβλ(eq+δ)a(e+δ)[d(c+αb)+βb].
Here R0 and R1 are called the basic reproduction ratios for viral infection and CTL immune response of system (2), respectively. It is easy to show that R0>R1 always holds. If R0>1, system (2) admits a CTL-inactivated infection equilibrium E1(x1,w1,y1,0) besides E0, where
(9)x1=α(eq+δ)+a(e+δ)(eq+δ)(dα+β),w1=ad(1-q)(R0-1)(eq+δ)(dα+β),y1=d(R0-1)dα+β.Furthermore, ifR1>1, system (2) also admits a CTL-activated infection equilibrium E*(x*,w*,y*,z*), where
(10)x*=λ(c+αb)d(c+αb)+βb,w*=βλb(1-q)(e+δ)[d(c+αb)+βb],y*=bc,z*=ap(R1-1).Then we give the stability result ofE0 by the following theorem.Theorem 2.
The infection-free equilibriumE0(x0,0,0,0) of system (2) is locally asymptotically stable if R0<1 and is unstable if R0>1.Proof.
The characteristic equation of system (2) at the infection-free equilibrium E0(x0,0,0,0) is of the form
(11)(s+b)(s+d)(s2+g1s+g0)=0,
where
(12)g0=a(e+δ)-(eq+δ)βλd,g1=a+e+δ-qβλd.
Clearly, (11) always has two negative real roots s1=-b and s2=-d. Noting that, if R0<1, (eq+δ)βλ/d<a(e+δ), then g0>0. Since 0<q<1, (eq+δq)βλ/d<a(e+δ), that is, qβλ/d<a, then we get that g1>0. It is easy to see that (11) has another two negative real roots. Therefore, if R0<1, the infection-free equilibrium E0 is locally asymptotically stable. If R0>1, g0<0, (11) has at least one positive real root. Hence, E0 is unstable in this case. The proof of Theorem 2 is completed.As to the stability ofE1, we have the following result.Theorem 3.
IfR1<1, the CTL-inactivated infection equilibrium E1(x1,w1,y1,0) is locally asymptotically stable. If R1>1, E1(x1,w1,y1,0) is unstable.Proof.
The characteristic equation of system (2) at the CTL-inactivated infection equilibrium E1(x1,w1,y1,0) is of the form
(13)(s+b-cy1e-sτ)(s3+As2+Bs+C)=0,
where
(14)A=a+e+δ+d+βy11+αy1+qβx1(1+αy1)2,B=a(e+δ)-δβx1(1+αy1)2+d(a+e+δ)+(a+e+δ)βy11+αy1+2βy11+αy1qβx1(1+αy1)2+(eq+2δq+dq)βx1(1+αy1)2,C=ad(e+δ)-dδβx1(1+αy1)2+a(e+δ)βy11+αy1+2(e+δ)qβy11+αy1βx1(1+αy1)2+(eq+2δq)dβx1(1+αy1)2.
We first consider the following equation
(15)s3+As2+Bs+C=0.
Obviously, A>0. Since
(16)-δβx1(1+αy1)2>-δβx11+αy1=-δy111-q(1-q)βx1y11+αy1=-δy111-q(e+δ)w1=-a(e+δ)δeq+δ>-a(e+δ),
it is easy to see that B>0, C>0. By direct calculation,
(17)AB-C=d(a+e+δ)(a+e+δ+d)+d(a+e+δ)×(βy11+αy1+qβx1(1+αy1)2)+a2βy11+αy1+(βy11+αy1+qβx1(1+αy1)2+(e+δ+d))×(a+e+δ)βy11+αy1+d2qβx1(1+αy1)2+(βy11+αy1+qβx1(1+αy1)2+(a+e+δ))×(eq+2δq+dq)βx1(1+αy1)2+(a+d+βy11+αy1+qβx1(1+αy1)2)×2βy11+αy1qβx1(1+αy1)2+(βy11+αy1+qβx1(1+αy1)2+(a+e+δ))a(e+δ)-(βy11+αy1+qβx1(1+αy1)2+(a+e+δ))×δβx1(1+αy1)2>0.
Therefore, by Routh-Hurwitz criterion, all roots of (15) have negative real parts.
Then we discuss the location of the roots of the following equation(18)s+b-cy1e-sτ=0.
Denote f(s)=s+b-cy1e-sτ. If R1>1, it is easy to verify
(19)f(0)=d(c+αb)+βbdα+β(1-R1)<0,lims→+∞f(s)=+∞.
Hence, f(s)=0 has one positive real root. E1(x1,w1,y1,0) is unstable.
IfR1<1, then we prove that all roots of f(s)=0 have negative real parts. Assume that Res≥0, then from f(s)=0, we derive that
(20)Res=b[d(c+αb)+βbb(dα+β)(R1-1)e-τRescos(τIms)hhhh+e-τRescos(τIms)-1d(c+αb)+βbb(dα+β)]<0,
which comes to a contradiction. Hence, Res<0. From the discussion above, we can see that, if R1<1, the CTL-inactivated infection equilibrium E1(x1,w1,y1,0) is locally asymptotically stable. This completes the proof of Theorem 3.


## 3. Stability and Hopf Bifurcation at  the CTL-Activated Infection Equilibrium

From the results above, whenR1>1, the CTL-inactivated infection equilibrium E1 is unstable, and at the same time a CTL-activated infection equilibrium E* emerges. Now we regard τ as a parameter to study the stability of E* and the existence of Hopf bifurcations.The characteristic equation of system (2) at the CTL-activated infection equilibrium E*(x*,w*,y*,z*) is of the form
(21)s4+p3s3+p2s2+p1s+p0-(q3s3+q2s2+q1s+q0)e-sτ=0,
where
(22)p3=e+δ+a+pz*+b+d+βy*1+αy*-qβx*(1+αy*)2,p2=(b+e+δ)(d+a+pz*)+d(a+pz*)+b(e+δ)+(b+e+δ+a+pz*)βy*1+αy*-(bq+eq+dq+δ)βx*(1+αy*)2,p1=(e+δ)(a+pz*)(b+d)+bd(e+δ+a+pz*)-βx*(1+αy*)2(b+d)(eq+δ)+βy*1+αy*×[b(a+pz*)+(e+δ)(a+pz*)+b(e+δ)]-bdqβx*(1+αy*)2,p0=bd(e+δ)(a+pz*)+βy*1+αy*b(e+δ)(a+pz*)-bd(eq+δ)βx*(1+αy*)2,q3=b,q2=(e+δ+a+d)b+bβy*1+αy*-bqβx*(1+αy*)2,q1=bd(a+e+δ)+ab(e+δ)+b(e+δ+a)βy*1+αy*-b(eq+dq+δ)βx*(1+αy*)2,q0=abd(e+δ)+ab(e+δ)βy*1+αy*-bd(eq+δ)βx*(1+αy*)2.
When τ=0, (21) becomes
(23)s4+(p3-q3)s3+(p2-q2)s2+(p1-q1)s+p0-q0=0.
It is easy to verify that
(24)p0-q0=bpz*(e+δ)(d+βy*1+αy*)>0.
Since
(25)qβx*(1+αy*)2<qβx*1+αy*=qβx*y*1+αy*1y*=a+pz*-δw*y*<a+pz*,
by direct calculation, we can easily derive that
(26)p3-q3=e+δ+a+pz*+d+βy*1+αy*-qβx*(1+αy*)2>0.
Then for τ=0, according to Routh-Hurwitz criterion, all the roots of (21) have negative real parts if and only if the following conditions hold:(H1)(27)(p3-q3)(p2-q2)-(p1-q1)>0,(p3-q3)(p2-q2)(p1-q1)-(p3-q3)2(p0-q0)-(p1-q1)2>0.From what has been discussed above, we have the following result.Theorem 4.
Letτ=0. Then the CTL-activated infection equilibrium E*(x*,w*,y*,z*) of system (2) is locally asymptotically stable if R1>1 and the condition (H1) holds.Forτ>0, it is not easy to find rigorously local stability of E*. In the following, we will firstly investigate the existence of purely imaginary roots to (21) following the framework of that in [9].Ifiω(ω>0) is a root of (21), separating real and imaginary parts yields
(28)(q0-q2ω2)cosωτ+(q1ω-q3ω3)sinωτ=ω4-p2ω2+p0,(q2ω2-q0)sinωτ+(q1ω-q3ω3)cosωτ=-p3ω3+p1ω.
Squaring and adding the two equations of (28), we obtain that
(29)ω8+G1ω6+G2ω4+G3ω2+G4=0,
where
(30)G1=p32-q32-2p2,G2=p22+2p0-2p1p3-q22+2q1q3,G3=p12+2q0q2-2p0p2-q12,G4=p02-q02.
Let μ=ω2. Then, from (29), we have that
(31)μ4+G1μ3+G2μ2+G3μ+G4=0.
Denote h(μ)≡μ4+G1μ3+G2μ2+G3μ+G4. Then we get
(32)h′(μ)=4μ3+3G1μ2+2G2μ+G3.Suppose that (31) has positive real roots. Without loss of generality, we assume that it has n(1≤n≤4) positive real roots, defined by μ1<μ2<⋯<μn, respectively. Then (29) has n positive real roots
(33)ω1=μ1,ω2=μ2,…,ωn=μn.
From (28), we have(34)cosωτ=((-q2ω02+q0)(ω04-p2ω02+p0)+(-q3ω03+q1ω0)(-p3ω03+p1ω0))×((-q2ω02+q0)2+(-q3ω03+q1ω0)2)-1.
Therefore, if we denote
(35)τk(j)=1ωkarccos[((-q2ω02+q0)2+(-q3ω03+q1ω0)2)-1((-q2ω02+q0)(ω04-p2ω02+p0)ddddddddd+(-q3ω03+q1ω0)(-p3ω03+p1ω0))aaaaaaaaaa×((-q2ω02+q0)2+(-q3ω03+q1ω0)2)-1]+2jπωk,
where k=1,2,…,n; j=0,1,2,…, then ±iωk are a pair of purely imaginary roots of (21) with τk(j).Lets(τ)=α(τ)+iω(τ) be the root of (21) near τ=τk(j) satisfying α(τk(j))=0, ω(τk(j))=ωk, then we have the following result.Theorem 5.
Suppose thatμk=ωk2, h′(μk)≠0, where h′(μ) is defined by (32). Then
(36)d(α(τk(j)))dτ≠0
and d(α(τk(j)))/dτ and -h′(μk) have the same sign.Proof.
Calculating the derivative with respective toτ, we obtain
(37)(4s3+3p3s2+2p2s+p1)dsdτ+[τ(q3s3+q2s2+q1s+q0)-(3q3s2+2q2s+q1)]e-sτdsdτ=s(q3s3+q2s2+q1s+q0)e-sτ.
Then we derive from (37) that
(38)(dsdτ)-1=-3s4-2p3s3-p2s2+p0-s2(s4+p3s3+p2s2+p1s+p0)+2q3s3+q2s2-q0-s2(q3s3+q2s2+q1s+q0)-τs.
Therefore, we have
(39)[d(Res(τ))dτ]τ=τk(j)-1=Re[-3ωk4+2p3iωk3+p2ωk2+p0ωk2(ωk4-p3iωk3-p2ωk2+p1iωk+p0)]+Re[-2q3iωk3-q2ωk2-q0ωk2(-q3iωk3-q2ωk2+q1iωk+q0)]=((-3μk2+p2μk+p0)(μk2-p2μk+p0)+2p3μk2(p1-p3μk))×(μk(μk2-p2μk+p0)2+μk2(p1-p3μk)2)-1+-2q3μk2(q1-q3μk)+(q22μk2-q02)μk(-q2μk+q0)2+μk2(q1-q3μk)2,
where μk=ωk2. From (29) and (31), we get
(40)(μk2-p2μk+p0)2+μk(p1-p3μk)2=(-q2μk+q0)2+μk(q1-q3μk)2,p02-q02=G4=-(μk4+G1μk3+G2μk2+G3μk).
Then we get that
(41)[d(Res(τ))dτ]τ=τk(j)-1=((-3μk2+p2μk+p0)(μk2-p2μk+p0)+2p3μk2×(p1-p3μk)-2q3μk2(q1-q3μk)+(q22μk2-q02))×(μk[(-q2μk+q0)2+μk(q1-q3μk)2])-1=-4μk4-3G1μk3-2G2μk2-G3μkμk[(-q2μk+q0)2+μk(q1-q3μk)2]=-h′(μk)(-q2μk+q0)2+μk(q1-q3μk)2.
Furthermore, it follows that
(42)sign[d(α(τ))dτ]τ=τk(j)=sign[d(Res(τ))dτ]τ=τk(j)=sign[-h′(μk)(-q2μk+q0)2+μk(q1-q3μk)2].
Then, this completes the proof of Theorem 5.Applying the Hopf bifurcation theorem for functional differential equation [15], we derive the existence of a Hopf bifurcation at E* as stated in the following theorem.Theorem 6.
Suppose that (31) has at least one simple positive root and μ0 is the last such root. Then there is a Hopf bifurcation for system (2) as τ passes through τ0 leading to a periodic solution that bifurcates from E*, where
(43)τ0=1μ0arccos[((-q2μ0+q0)2+μ0(-q3μ0+q1)2)-1((-q2μ0+q0)(μ02-p2μ0+p0)+μ0(-q3μ0+q1)(-p3μ0+p1)(-q2μ0+q0)(μ02-p2μ0+p0))×((-q2μ0+q0)2+μ0(-q3μ0+q1)2)-1]+2jπμ0.Remark 7.
As an example, we suppose that (31) has three simple positive roots, denoted by μ1<μ2<μ3, respectively. Then we have h′(μ1)>0, h′(μ2)<0, and h′(μ3)>0. By Theorem 5, it is easy to see that
(44)d(α(τ1(j)))dτ>0,d(α(τ2(j)))dτ<0,d(α(τ3(j)))dτ>0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhj=0,1,2,…,
from which we can conclude that the stability switches may appear as τ increases.


## 4. Global Stability

In this part, we study the global stability of the infection-free equilibrium and the CTL-inactivated infection equilibrium and then discuss the global stability of the CTL-activated infection equilibrium whenτ=0. The strategy of proofs is to use suitable Lyapunov functionals and LaSalle’s invariance principle in [16].Define(45)F(x)=x-1-lnx.
Clearly, for x∈(0,+∞), F(x) is nonnegative and has the global minimum at x=1 and F(1)=0.Theorem 8.
The infection-free equilibriumE0(x0,0,0,0) of system (2) is globally asymptotically stable if R0≤1.Proof.
Let(x(t),w(t),y(t),z(t)) be any positive solution of system (2) with initial conditions (3).
Define(46)W0(t)=12[x(t)-x0]2+δeq+δw(t)+e+δeq+δy(t)+p(e+δ)c(eq+δ)z(t)+p(e+δ)eq+δx0∫t-τt‍y(θ)z(θ)dθ.
The time derivative of W0(t) along positive solutions of system (2) is given by
(47)W˙0(t)=[x(t)-x0](λ-dx(t)-βx(t)y(t)1+αy(t))+δeq+δx0[(1-q)βx(t)y(t)1+αy(t)-ew(t)-δw(t)]+e+δeq+δx0[qβx(t)y(t)1+αy(t)-ay(t)hhhhhhhhhh+δw(t)-py(t)z(t)qβx(t)y(t)1+αy(t)]+p(e+δ)c(eq+δ)x0[cy(t-τ)z(t-τ)-bz(t)]+p(e+δ)eq+δx0[y(t)z(t)-y(t-τ)z(t-τ)].
On substituting λ=dx0 into (47) we derive that
(48)W˙0(t)=-d[x(t)-x0]2-[x(t)-x0]×[βy(t)1+αy(t)(x(t)-x0)+βx0y(t)1+αy(t)]+x0[βy(t)1+αy(t)(x(t)-x0)+βx0y(t)1+αy(t)dddd-a(e+δ)eq+δy(t)-p(e+δ)eq+δy(t)z(t)βx0y(t)1+αy(t)]+p(e+δ)eq+δx0y(t-τ)z(t-τ)-p(e+δ)c(eq+δ)x0bz(t)+p(e+δ)eq+δx0y(t)z(t)-p(e+δ)eq+δx0y(t-τ)z(t-τ)=-(d+βy(t)1+αy(t))[x(t)-x0]2-x0[-βx0y(t)1+αy(t)+a(e+δ)eq+δy(t)]-p(e+δ)c(eq+δ)x0bz(t)=-(d+βy(t)1+αy(t))[x(t)-x0]2-x0a(e+δ)eq+δy(t)1+αy(t)[1-R0+αy(t)]-p(e+δ)c(eq+δ)x0bz(t).
Noting that R0≤1, by (48), we have that W˙0(t)≤0. By Theorem  5.3.1 in [15], the solutions of system (2) limit to M, the largest invariant subset of {W˙0(t)=0}. Clearly, it follows from (48) that W˙0(t)=0 if and only if x=x0 and y=z=0. Noting that M is invariant, for each element in M, we have y=0, and y˙(t)=0. We therefore derive from the third equation of system (2) that
(49)0=y˙(t)=δw,
which yields w=0. Hence, W˙0(t)=0 if and only if (x,w,y,z)=(x0,0,0,0). Accordingly, the global asymptotic stability of E0 follows LaSalle’s invariance principle. This completes the proof.We are now in a position to establish the global stability of the CTL-inactivated infection equilibriumE1 of system (2).Theorem 9.
The CTL-inactivated infection equilibriumE1(x1,w1,y1,0) of system (2) is globally asymptotically stable if R0>1 and R1≤1.Proof.
Let(x(t),w(t),y(t),z(t)) be any positive solution of system (2) with initial conditions (3).
Define(50)W11(t)=x1F[x(t)x1]+δeq+δw1F[w(t)w1]+e+δeq+δy1F[y(t)y1]+p(e+δ)c(eq+δ)z(t),
where the function F(x) is defined in (45).
The derivative ofW11(t) along positive solutions of system (2) is given by
(51)W˙11(t)=(1-x1x(t))(λ-dx(t)-βx(t)y(t)1+αy(t))+δeq+δ(1-w1w(t))×[(1-q)βx(t)y(t)1+αy(t)-ew(t)-δw(t)]+e+δeq+δ(1-y1y(t))×(qβx(t)y(t)1+αy(t)-ay(t)+δw(t)-py(t)z(t))+p(e+δ)c(eq+δ)[cy(t-τ)z(t-τ)-bz(t)].
On substituting λ=dx1+βx1y1/(1+αy1) into (51) we derive that
(52)W˙11(t)=(1-x1x(t))[-d(x(t)-x1)]+βx1y11+αy1(1-x1x(t))+βx1y(t)1+αy(t)+δeq+δ[-(1-q)βx(t)y(t)1+αy(t)w1w(t)+(e+δ)w1]-e+δeq+δay(t)-e+δeq+δpy(t)z(t)+e+δeq+δ(-qβx(t)y(t)1+αy(t)y1y(t)+ay1-δw(t)y1y(t)dddd+py1z(t)qβx(t)y(t)1+αy(t))+e+δeq+δpy(t-τ)z(t-τ)-pb(e+δ)c(eq+δ)z(t).
Noting that
(53)ay1=qβx1y11+αy1+δw1,(1-q)βx1y11+αy1=(e+δ)w1,y1=βλ(eq+δ)-ad(e+δ)a(e+δ)(dα+β),
we derive from (52) that
(54)W˙11(t)=-d[x(t)-x1]2x(t)+βx1y11+αy1(1-x1x(t))+βx1y(t)1+αy(t)-e+δeq+δay(t)-e+δeq+δpy(t)z(t)-δeq+δ(1-q)βx(t)y(t)1+αy(t)w1w(t)+δeq+δ(1-q)βx1y11+αy1-e+δeq+δqβx(t)y(t)1+αy(t)y1y(t)+e+δeq+δ(qβx1y11+αy1+δw1)-e+δeq+δδw1w(t)w1y1y(t)+e+δeq+δpy1z(t)-pb(e+δ)c(eq+δ)z(t)+e+δeq+δpy(t-τ)z(t-τ)=-d[x(t)-x1]2x-e+δeq+δpy(t)z(t)+e+δeq+δpy(t-τ)z(t-τ)+δeq+δ(1-q)βx1y11+αy1×(4-x1x(t)-w1w(t)x(t)y(t)x1y11+αy11+αy(t)sssss-y1w(t)w1y(t)-1+αy(t)1+αy1)+e+δeq+δqβx1y11+αy1×(3-x1x(t)-x(t)x11+αy11+αy(t)-1+αy(t)1+αy1)-αβx1[y(t)-y1]2(1+αy1)2[1+αy(t)]+p(e+δ)[d(c+αb)+βb]c(eq+δ)(dα+β)(R1-1)z(t).
Define
(55)W1(t)=W11(t)+p(e+δ)eq+δ∫t-τt‍y(θ)z(θ)dθ.
Then we have
(56)W˙1(t)=W˙11(t)+p(e+δ)eq+δ[y(t)z(t)-y(t-τ)z(t-τ)]=-d(x(t)-x1)2x(t)+δeq+δ(1-q)βx1y11+αy1×(4-x1x(t)-w1w(t)x(t)y(t)x1y11+αy11+αy(t)-y1w(t)w1y(t)fffff-1+αy(t)1+αy1)+e+δeq+δqβx1y11+αy1×(3-x1x(t)-x(t)x11+αy11+αy(t)-1+αy(t)1+αy1)-αβx1[y(t)-y1]2(1+αy1)2[1+αy(t)]+p(e+δ)[d(c+αb)+βb]c(eq+δ)(dα+β)(R1-1)z(t).
Noting that x1,w1,y1>0, R0>1, and R1≤1, by (56), we have that W˙1(t)≤0. By Theorem  5.3.1 in [15], the solutions of system (2) limit to M, the largest invariant subset of {W˙1(t)=0}. It is readily seen from (56) that W˙1(t)=0 holds if and only if (x,w,y,z)=E1(x1,w1,y1,0). Using a similar argument as that in the proof of Theorem 8 and by LaSalle’s invariance principle, the global asymptotic stability of E1 follows. This completes the proof.Finally, we give the global stability result ofE* when there is no time delay, that is, τ=0.Theorem 10.
Letτ=0. The CTL-activated infection equilibrium E*(x*,w*,y*,z*) of system (2) is globally asymptotically stable if R1>1.Proof.
Define(57)W*(t)=x*F(xx*)+δeq+δw*F(ww*)+e+δeq+δy*F(yy*)+p(e+δ)c(eq+δ)z*F(zz*),
where the function F(x) is defined in (45).
Calculating the derivative ofW*(t) along positive solutions of system (2), it follows that
(58)W˙*(t)=(1-x*x)(λ-dx-βxy1+αy)+δeq+δ(1-w*w)[(1-q)βxy1+αy-ew-αw]+e+δeq+δ(1-y*y)(qβxy1+αy-ay+δw-pyz)+p(e+δ)c(eq+δ)(1-z*z)(cyz-bz).
On substituting λ=dx*+βx*y*/(1+αy*) into (58) we derive that
(59)W˙*(t)=(1-x*x)[-d(x-x*)+βx*y*1+αy*]+βx*y1+αy+δeq+δ[-(1-q)βxy1+αyw*w+(e+α)w*]-e+δeq+δay+e+δeq+δ×[-qβxy1+αyy*y+ay*-δwy*y+py*z]+p(e+δ)c(eq+δ)(-bz-cyz*+bz*).
At the steady state E*,
(60)(e+δ)w*=(1-q)βx*y*1+αy*,ay*+py*z*=qβx*y*1+αy*+δw*,cy*=b.
Then it follows from (59) that
(61)W˙*(t)=-d(x-x*)2x+βx*y*1+αy*(1-x*x)+βx*y1+αy-e+δeq+δay-δeq+δ(1-q)βxy1+αyw*w+δeq+δ(1-q)βx*y*1+αy*-e+δeq+δqβxy1+αyy*y+e+δeq+δ(qβx*y*1+αy*+δw*-py*z*)-e+δeq+δδw*ww*y*y+e+δeq+δpy*z+p(e+δ)c(eq+δ)(-bz-cyz*+bz*)=-d(x-x*)2x-αβx*(y-y*)2(1+αy*)2(1+αy)+δeq+δ(1-q)βx*y*1+αy*×(4-x*x-w*w1+αy*1+αyxyx*y*-ww*y*yssss-1+αy1+αy*)+e+δeq+δqβx*y*1+αy*×(3-x*x-1+αy*1+αyxx*-1+αy1+αy*).
Noting that x*,w*,y*,z*>0, by (61), we have that W˙*(t)≤0. Also, it is easy to verify that W˙*(t)=0 if and only if x=x*, w=w*, and y=y*. We therefore derive from the third equation of system (2) that
(62)0=y˙(t)=qβx*y*1+αy*-ay*+δw*-py*z,
which yields z=z*. The maximal compact invariant set in {(x,ω,y,v):W˙*(t)=0} is the singleton {E*} when R1>1. Accordingly, the global asymptotic stability of E* follows. This completes the proof.


## 5. Numerical Simulations and Conclusions

In the following, we give examples to illustrate the main theoretical results above. In system (2), we choose a set of parameters as follows: λ=270, d=0.02, β=0.001, α=0.01, q=0.8, e=0.3, δ=0.5, a=0.8, p=0.04, c=0.025, and b=0.3. The initial values are x0=8000, w0=20, y0=10, and z0=150. Then R1≈9.0752>1, and the CTL-activated infection equilibrium E* exists. The values of these parameters satisfy the condition of Theorem 6. By calculation, we derive that μ0≈0.2860 and τ0≈0.3746.Whenτ=0.2<τ0, the trajectory converges to the CTL-activated infection equilibrium E* (see Figure 1). While, when τ=1.2>τ0, the CTL-activated infection equilibrium E* of (2) becomes unstable (see Figure 2), and the Hopf bifurcation occurs.Whenτ=0.2<τ0, the CTL-activated infection equilibrium E*(8790.7081,23.5465,12.0000,161.5044) of system (2) is asymptotically stable.
(a)
(b)
(c)
(d)Whenτ=1.2>τ0, the CTL-activated infection equilibrium E*(8790.7081,23.5465,12.0000,161.5044) of system (2) becomes unstable.
(a)
(b)
(c)
(d)In this paper, we have investigated the global dynamics of an HIV-1 infection model with latently infected cells and delayed immune response. By giving the explicit expressions of two basic reproduction ratiosR0 and R1, we first gave sufficient conditions for the existence of each of feasible equilibria of system (2). Then, a detailed analysis on the local asymptotic stability of the equilibria was carried out. It was shown that, if R0<1 (hence, E1 and E* are not feasible), the infection-free equilibrium E0 is locally asymptotically stable. If R1<1, the CTL-inactivated infection equilibrium E1 is locally asymptotically stable. If R1>1, the CTL-activated infection equilibrium E* exists. By giving Theorem 6, we studied the existence of Hopf bifurcations at E*. By constructing suitable Lyapunov functionals and using LaSalle’s invariance principle, we proved that, for any time delay τ≥0, if R0<1, the infection-free equilibrium E0 is globally asymptotically stable. If R0>1 and R1<1, the CTL-inactivated infection equilibrium E1 is globally asymptotically stable. From this we can see that time delay has no effect on the stability of the infection-free equilibrium and the CTL-inactivated infection equilibrium. When τ=0, we also proved the global asymptotic stability of the CTL-activated infection equilibrium E* of system (2) whenever it exists.



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*Source: 169427-2013-12-31.xml*