# Spatiotemporal Dynamics of an HIV Infection Model with Delay in Immune Response Activation

Authors: Mehdi Maziane; Khalid Hattaf; Noura Yousfi
Journal: International Journal of Differential Equations
 (2018)

---

## Abstract

We propose and analyse an human immunodeficiency virus (HIV) infection model with spatial diffusion and delay in the immune response activation. In the proposed model, the immune response is presented by the cytotoxic T lymphocytes (CTL) cells. We first prove that the model is well-posed by showing the global existence, positivity, and boundedness of solutions. The model has three equilibria, namely, the free-infection equilibrium, the immune-free infection equilibrium, and the chronic infection equilibrium. The global stability of the first two equilibria is fully characterized by two threshold parameters that are the basic reproduction numberR0 and the CTL immune response reproduction number R1. The stability of the last equilibrium depends on R0 and R1 as well as time delay τ in the CTL activation. We prove that the chronic infection equilibrium is locally asymptotically stable when the time delay is sufficiently small, while it loses its stability and a Hopf bifurcation occurs when τ passes through a certain critical value.


---

## Body

## 1. Introduction

HIV is a virus that attacks the CD4+ T cells and reduces their number in the body. It is known that when the number of these cells is less than 200 cells per μl, the patient enters the phase of acquired immunodeficiency syndrome (AIDS). This phase is characterized by the appearance of opportunistic infections caused by bacteria, viruses, or fungi or by the appearance of certain types of cancer. From the world health organization (WHO) [1], HIV continues to be a major global public health issue, having claimed more than 35 million lives so far. In 2016, 1 million people died from HIV-related causes globally. Also, there were approximately 36.7 million people living with HIV at the end of 2016 with 1.8 million people becoming newly infected in 2016 globally. Therefore, many mathematical models have been developed to better understand the dynamics of HIV infection. One of the earliest of these models was presented by Nowak and Bangham [2] that considers three populations: uninfected target cells, productive infected cells, and free viral particles. Rong et al. [3] extended the model of [2] by including the infected cells in eclipse stage (unproductive infected cells) and considered that a portion of these cells returns to the uninfected state. In 2014, Hu et al. [4] replaced the bilinear incidence rate in [3] by a saturated infection rate and they studied the global stability of equilibria. In 2015, Maziane et al. [5] improved the model of [4] by considering the Hattaf’s incidence rate [6] that includes the common types such as the bilinear incidence rate, the saturated incidence rate, the Beddington-DeAngelis functional response [7, 8] and the Crowley-Martin functional response [9].Cytotoxic T lymphocytes (CTL) cells are responsible for cellular immunity and they play an important role in antiviral defense by killing the productive infected cells. For this, Lv et al. [10] proposed an HIV model with Beddington-DeAngelis functional response and CTL immune response. In 2016, Maziane et al. [11] generalized and extended the model of Lv et al. [10] by considering the mobility of cells and virus. They assumed that the motion of virus follows the Fickian diffusion and proposed the following model: (1)∂T∂t=λ-μTTx,t-fTx,t,Vx,tVx,t+ρEx,t,∂E∂t=fTx,t,Vx,tVx,t-μE+ρ+γEx,t,∂I∂t=γEx,t-μIIx,t-pIx,tCx,t,∂V∂t=dΔVx,t+kIx,t-μVVx,t,∂C∂t=aIx,tCx,t-μCCx,t,where T(x,t), E(x,t), I(x,t), V(x,t), and C(x,t) represent the densities of uninfected CD4+ T cells, unproductive infected cells, productive infected cells, and free virus particles and CTL cells at location x and time t, respectively. The positive parameters λ, γ, k, and a are the production rate of uninfected cells, the rate at which infected cells in the eclipse stage become productive infected cells, the production rate of virions by infected cells, and the proliferation rate of CTL cells, respectively. The positive constants μT, μE, μI, μV, and μC are, respectively, the death rates of uninfected CD4+ T cells, unproductive infected cells, productive infected cells, free virus, and CTL cells. The unproductive infected cells return to the uninfected cells at rate ρ while the productive infected cells are killed by CTL at rate p. In model (1), the infection transmission process is modeled by Hattaf’s incidence rate [6] of the form f(T,V)=βT/1+α1T+α2V+α3TV, where α1, α2, and α3≥0 are the saturation factors measuring the psychological or inhibitory effect and β>0 is the infection coefficient. Here Δ=∑i=1n∂2/∂xi2 is the Laplacian operator and d is the diffusion coefficient of virus.In the reality, the activation of the immune response is not instantaneous. When the virus invades the body, the immune system takes time to recognize and react to the virus. Therefore, system (1) becomes (2)∂T∂t=λ-μTTx,t-fTx,t,Vx,tVx,t+ρEx,t,∂E∂t=fTx,t,Vx,tVx,t-μE+ρ+γEx,t,∂I∂t=γEx,t-μIIx,t-pIx,tCx,t,∂V∂t=dΔVx,t+kIx,t-μVVx,t,∂C∂t=aIx,t-τCx,t-τ-μCCx,t,where τ denotes the time needed for the activation of the CTL immune response, namely, the immunological delay. The other parameters have the same biological meaning as system (1). In addition, we consider our model (2) with homogenous Neumann boundary condition (3)∂V∂ν=0on∂Ω×0,+∞,and initial conditions (4)Tx,θ=ϕ1x,θ≥0,Ex,θ=ϕ2x,θ≥0,Ix,θ=ϕ3x,θ≥0,Vx,θ=ϕ4x,θ≥0,Cx,θ=ϕ5x,θ≥0,x∈Ω-,θ∈-τ,0,where Ω is a bounded domain in Rn with smooth boundary ∂Ω, ϕi(x,θ)(i=1,2,3,4,5) is Hölder continuous in Ω-×-τ,0, and ∂V/∂ν is the outward normal derivative on ∂Ω.The rest of the paper is outlined as follows. In the next section we investigate the well-posedness and equilibria for system (2)–(4). The stability analysis and the existence of Hopf bifurcation are studied in Section 3. Finally, a brief conclusion is given in Section 4.


## 2. Well-Posedness and Equilibria

In this section, we establish the existence, positivity, and boundedness of solutions of problem (2)–(4) because this model describes the evolution of a cell population. Hence the densities of cells should remain nonnegative and bounded. In addition, we determine the basic reproduction number, the CTL immune response reproduction number, and equilibria of the model (2)–(4).Before proceeding, we shall set some notations and terminology.X will denote a Banach space over a real or complex field. C=C-τ,0,X will denote the Banach space of X-valued functions on [-τ,0], with supremum norm, where τ>0. Here, X=C(Ω-,R5). If u is a continuous function from [-τ,b] to X and t∈[0,b], then ut denotes the element of C given by ut(θ)=u(t+θ), -τ≤θ≤0.Proposition 1.
For any initial conditions satisfying (4), there exists a unique solution of problem (2)–(4) defined on 0,+∞ and this solution remains nonnegative and bounded for all t≥0.Proof.
Letϕ=(ϕ1,ϕ2,ϕ3,ϕ4,ϕ5)T∈C and x∈Ω-. We define F=(F1,F2,F3,F4,F5) by (5)F1ϕx=λ-μTϕ1x,0-fϕ1x,0,ϕ4x,0ϕ4x,0+ρϕ2x,0,F2ϕx=fϕ1x,0,ϕ4x,0ϕ4x,0-ρ+μE+γϕ2x,0,F3ϕx=γϕ2x,0-μIϕ3x,0-pϕ3x,0ϕ5x,0,F4ϕx=kϕ3x,0-μVϕ4x,0,F5ϕx=aϕ3x,-τϕ5x,-τ-μCϕ5x,0.Hence, system (2)–(4) can be written of the form (6)u′t=Aut+Futt,t>0u0=ϕ∈X,where u=(T,E,I,V,C)T and Au(t)=(0,0,0,dΔV,0)T. Obviously, F is locally Lipschitz in X. By [12–16], we deduce that system (2) admits a unique local solution on 0,Tmax, where Tmax is the maximal existence time for solution of system (2). In addition, (0,0,0,0,0) is a lower solution of each solution of system (2); then we deduce that T(x,t)≥0, E(x,t)≥0, I(x,t)≥0, V(x,t)≥0 and C(x,t)≥0.
Next, we prove the boundedness of solutions by considering the following function:(7)Sx,t=Tx,t+Ex,t+Ix,t+paCx,t+τ.From system (2), we obtain (8)∂Sx,t∂t=λ-μTTx,t-μEEx,t-μIIx,t-μCpaCx,t+τ≤λ-μSx,t,where μ=min⁡μT,μE,μI,μC. Thus, (9)Sx,t≤max⁡λμ,maxx∈Ω¯ϕ1x,0+ϕ2x,0+ϕ3x,0+paϕ5x,-τ.Then, T, E, I, and C are bounded.
To prove the boundedness ofV, from system (2), we get (10)∂Vx,t∂t-dVΔV≤kδ-μVV,∂Vx,t∂ν=0,Vx,0=maxx∈Ω-⁡ϕ4x,0,where δ=max⁡λ/μ,maxx∈Ω¯⁡ϕ1x,0+ϕ2x,0+ϕ3x,0+p/aϕ5x,-τ.
Using the comparison principle [17], we have V(x,t)≤V-(t), where V-(t)=ϕ4(x)e-μVt+kδ/μV(1-e-μVt) is the solution of the problem (11)dV-dt=kδ-μVV-V-0=maxx∈Ω-⁡ϕ4x,0.Since V-t≤max⁡kδ/μV,maxx∈Ω-⁡ϕ4x,0, ∀x,t∈Ω-×0,Tmax, we have that V is bounded.
Therefore, we have proved thatT(x,t), E(x,t), I(x,t), V(x,t), and C(x,t) are bounded on Ω-×0,Tmax. Hence, it follows from the standard theory for semilinear parabolic systems [18] that Tmax=+∞.As in [11], the basic reproduction number of virus in the absence of spatial dependence is given by(12)R0=λβkγμIμVλα1+μTμE+γ.In addition to R0, we define the CTL immune response reproduction number R1 of our model by (13)R1=aI1μC,which represents the threshold level to activate the CTL cells response.Theorem 2.
(i)
IfR0≤1, system (2) has always an infection-free equilibrium of the form Q0(λ/μT,0,0,0,0). (ii)
IfR0>1, system (2) has an immune-free equilibrium of the form Q1(T1,E1,I1,V1,0) with T1∈(0,λ/μT), E1≥0, I1≥0, and V1≥0. (iii)
IfR1>1, system (2) has a chronic infection equilibrium of the form Q2(T2,E2,I2,V2,C2) with T2∈(0,λ/μT-μIμC(μE+γ)/aγμT), E2≥0, I2≥0, V2≥0, and C2≥0.


## 3. Stability Analysis and Hopf Bifurcation

First, we discuss the global stability of the infection-free equilibriumQ0 and the immune-free equilibrium Q1.Theorem 3.
(i)
The infection-free equilibriumQ0 is globally asymptotically stable if R0≤1. (ii)
The immune-free equilibriumQ1 is globally asymptotically stable if R1≤1<R0 and(14)R0≤1+μTμIμVμE+γ+α2μTλkγμE+ρ+γ+ρα3kγλ2ρμIμVμE+ρ+γμT+α1λ.Proof.
By using the method proposed by Hattaf and Yousfi [19], we propose the following Lyapunov functional for system (2)–(4) at Q0:(15)W0=∫ΩTx,t-T0-∫T0Tx,tfT0,0fS,0dS+ρTx,t-T0+Ex,t221+α1T0μT+μE+γT0+ρ+μE+γγIx,t+Ex,t+μIρ+μE+γkγVx,t+pρ+μE+γaγCx,t+pρ+μE+γγ∫t-τtIx,θCx,θdθdx,where T0=λ/μT. For convenience, we let ψ(x,t)=ψ and ψ(x,t-τ)=ψτ, for any ψ∈T,E,I,V,C. By calculation, we have(16)dW0dt=∫Ω1-fT0,0fT,0∂T∂t+ρT-T0+E∂T/∂t+∂E/∂t1+α1T0μT+μE+γT0+ρ+μE+γγ∂I∂t+∂E∂t+μIρ+μE+γkγ∂V∂t+pρ+μE+γaγ∂C∂t+pρ+μE+γγ∂∂t∫t-τtIθCθdθdx.Noting that λ=μTT0, the time derivative of W0 along the positive solutions of system (2) satisfies (17)dW0dt=∫Ω1-fT0,0fT,0μTT0-T+fT0,0fT,VfT,0V+ρ1-fT0,0fT,0E-ρμTT-T021+α1T0μT+μE+γT0-ρμE+γE21+α1T0μT+μE+γT0+ρE1+α1T0T0T0-T-pρ+μE+γγIC-μIμVρ+μE+γkγV+pρ+μE+γγIτCτ-μCpρ+μE+γaγC+pρ+μE+γγIC-IτCτ+dμIρ+μE+γkγΔVdx=-∫Ω1T+ρμT+μE+γT0μTT-T021+α1T0+ρμE+γE21+α1T0μT+μE+γT0+ρT-T02E1+α1T0TT0-μIμVρ+μE+γkγR0-1V+α2+α3TV21+α1T+α2V+α3TVfT0,0+μCpρ+μE+γaγCdx.Therefore, dW0/dt≤0 if R0≤1. In addition, it is not hard to verify that the largest compact invariant set in T,E,I,V,C∣dW0/dt=0 is just the singleton Q0. From LaSalle invariance principle [20], we deduce that Q0 is globally asymptotically stable.
Next, we construct the Lyapunov functional for system (2)–(4) at Q1: (18)W1=∫ΩT-T1-∫T1TfT1,V1fS,V1dS+ρ1+α2V1T-T1+E-E1221+α1T1+α2V1+α3T1V1μT+μE+γT1+fT1,V1V1γE1I1ΦII1+E1ΦEE1+μIfT1,V1V1kγE1V1ΦVV1+pfT1,V1V1aγE1C+pfT1,V1V1γE1∫t-τtIθCθdθdx,where Φx=x-1-ln⁡x. Obviously, the function Φ has a global minimum at 1 and satisfies Φ(1)=0. Calculating the time derivative of W1 along the positive solutions of system (2) and applying λ=μTT1+f(T1,V1)V1-ρE1, we obtain (19)dW1dt=∫Ω1-fT1,V1fT,V1μTT1-T+fT1,V1fT,VfT,V1V+ρ1-fT1,V1fT,V1E-μTρ1+α2V11+α1T1+α2V1+α3T1V1μT+μE+γT1T-T12-ρ1+α2V1μE+γ1+α1T1+α2V1+α3T1V1μT+μE+γT1E-E12-ρ1+α2V1E-E1T-T121+α1T1+α2V1+α3T1V1T1T-fT1,V12fT,V1V1+4fT1,V1V1-fT1,V1V-fT1,V1V1I1EIE1-fT,VVE1E-fT1,V1V1V1II1V+pfT1,V1V1γE1I1-μCaCdx+dμIfT1,V1V1kγE1∫Ω1-V1VΔVdx=-∫Ω1+α2V1T-T12TT11+α1T1+α2V1+α3T1V1μTT1-ρE1+ρμTTμT+μE+γ+ρE+ρE-E121+α2V1μE+γT11+α1T1+α2V1+α3T1V1μT+μE+γ-fT1,V1V15-fT1,V1fT,V1-I1EE1I-fT,VfT1,V1VE1V1E-V1IVI1-fT,V1fT,V+fT1,V11+α1Tα2+α3TV-V121+α1T+α2V1+α3TV11+α1T+α2V+α3TV-pμCfT1,V1V1aγE1R1-1Cdx-dμIfT1,V1V12kγE1∫Ω∇V2V2dx.Using the arithmetic-geometric inequality, we get (20)5-fT1,V1fT,V1-I1EE1I-fT,VfT1,V1VE1V1E-V1IVI1-fT,V1fT,V≤0.Therefore, dW1/dt≤0 if R1≤1 and ρE1⩽μTT1.
Obviously, the conditionρE1⩽μTT1 is equivalent to (21)R0≤1+μTμIμVμE+γ+α2μTλkγμE+ρ+γ+ρα3kγλ2ρμIμVμE+ρ+γμT+α1λ.In addition, dW1/dt=0 if and only if T=T1, E=E1, I=I1, V=V1, and C=0. Hence, the largest compact invariant set in T,E,I,V,C∣dW1/dt=0 is the singleton Q1. This proves the global stability of Q1 by using LaSalle’s invariance principle [20].From the above theorem, we deduce that the time delay in the activation of CTL immune response has no effect on the stability ofQ0 and Q1. Next we investigate the stability and existence of Hopf bifurcation at the chronic infection equilibrium Q2.Whenτ=0, system (2) becomes system (1). By Theorem 3 (iii) [11], we deduce the following result.Theorem 4.
Whenτ=0, the chronic infection equilibrium with immune response Q2 is globally asymptotically stable if R1>1 and (22)kβμCρ≤α1λρaμV+μTρ+μE+γα2kμC+aμV+α3ρλkμC.Now, we study the existence of Hopf bifurcation by regarding time delayτ as the bifurcation parameter.Let0=μ0<μ1<⋯<μn<⋯ be the eigenvalues of -Δ on Ω with homogeneous Neumann boundary conditions, and for =0,1,2,…, let E(μi) be the space of eigenfunctions corresponding to μi in C1(Ω). Let ϕij:j=1,2,…,dim⁡Eμi be an orthonormal basis of E(μi), X=[C1(Ω)]5 and Xij=cϕij:c∈R5. Then, X=⨁i=1∞Xi, Xi=⨁j=1dim⁡[S(μi)]Xij.The linearization of system (2) at the constant solution Q2(T2,E2,I2,V2,C2) can be expressed by (23)∂Z∂t=DΔZ+JZ+J∗Zτ,where Z=(T,E,I,V,C), Zτ=(Tτ,Eτ,Iτ,Vτ,Cτ), and D=(0,0,0,d,0),(24)J=-∂fT2,V2∂TV2-μTρ-∂fT2,V2∂VV2-fT2,V200∂fT2,V2∂VV2-μE+ρ+γ0∂fT2,V2∂VV2+fT2,V200γμI-pC20-pI200k-μV00000-μC,J∗=0000000000000000000000aC20aI2.LetLZ=DΔZ+JZ(x,t)+J∗Z(x,t-τ). For each i=0,1,2,…, Xi is invariant under the operator L, and ξ is an eigenvalue of L if and only if it is satisfying the characteristic equation(25)det⁡-ξI-μiD+J+e-ξτJ∗=0.Then, at Q2, the associated characteristic equation of system (2) is given by (26)ξ5+a1ξ4+a2ξ3+a3ξ2+a4ξ+a5+e-ξτb1ξ4+b2ξ3+b3ξ2+b4ξ+b5=0,where(27)a1=∂fT2,V2∂TV2+μT+μI+μV+μC+pC2+μE+ρ+γ+μid,a2=∂fT2,V2∂TV2+μTμE+ρ+γ+μV+μid+μI+μC+pC2+μE+ρ+γμV+μid+μI+μC+pC2+μI+pC2μV+μid+μC+μV+μidμC+μTρ,a3=∂fT2,V2∂TV2+μTμV+μid+μCμI+pC2+μV+μidμC+∂fT2,V2∂TV2+μTμE+γμI+μV+μid+μC+pC2+μCμV+μidμI+pC2+μE+ρ+γ-kγ∂fT2,V2∂VV2+μTρμV+μid+μI+μC+pC2,a4=∂fT2,V2∂TV2+μTμV+μidμCμI+pC2+μE+γμV+μid+μI+pC2μC-kγ∂fT2,V2∂VV2-kγμC∂fT2,V2∂VV2+ρμTμI+pC2μC+μV+μid+μV+μidμC+∂fT2,V2∂TV2kγ∂fT2,V2∂VV2+μV+μidμE+γμI+pC2,a5=-∂fT2,V2∂TV2+μTkγ∂fT2,V2∂VV2μC-kγμTμV+μid∂fT2,V2∂VV2+∂fT2,V2∂TV2μC∂fT2,V2∂VV2+fT2,V2,b1=-μC,b2=-μC∂fT2,V2∂TV2+μT+μI+μV+μid+μE+ρ+γ,b3=-μC∂fT2,V2∂TV2+μTμI+μV+μid+μE+γ+μIμV+μid+μI+μV+μidμE+ρ+γ+ρμT,b4=-μC∂fT2,V2∂TV2+μTμIμV+μid+μE+γμI+μV+μid-kγ∂fT2,V2∂VV2+fT2,V2+μTρμI+μV+μid,b5=-μC∂fT2,V2∂TV2+μTμIμV+μidμE+γ-kγμT∂fT2,V2∂VV2+fT2,V2+ρμTμIμV+μid.Forτ≠0, we suppose that (26) has a purely imaginary root ξ=iω with ω>0. Substituting ξ=iω in (26) and separating the real and the imaginary parts, we get (28)w5-a2w3+a4w=b1w4-b3w2+b5sin⁡ωτ+b2w3-b4wcos⁡ωτ,a1w4-a3w2+a5=-b1w4-b3w2+b5cos⁡ωτ+b2w3-b4wsin⁡ωτ.Squaring and adding the two equations of (28), we have (29)ω10+c1ω8+c2ω6+c3ω4+c4ω2+c5=0,where (30)c1=a12-2a2-b12,c2=a22+2a4-2a1a3+2b1b3-b22,c3=a32-2a2a4-b32+2b2b4+2a1a5-2b1b5,c4=a42-b42-2a3a5+2b3b5,c5=a52-b52.Letting z=ω2 we obtain (31)hz=z5+c1z4+c2z3+c3z2+c4z+c5=0.Denote(32)p1=-625c12+35c2,q1=8125c13+625c1c2+25c3,r1=-3625c14+3125c12c2-225c1c3+15c4,p2=-13p12-4r1,q2=-227p13+83p1r2-q12,Δ1=127p23+14q22,s∗=-q22+Δ13+-q22-Δ13+13p1,Δ2=-s∗-p1+2q1s∗-p1,Δ3=-s∗-p1-2q1s∗-p1,z-=q12p1-s∗-15p.Suppose that (31) has positive roots zk, k=1,2,3,4,5, where wk=zk. From (28), we have (33)cos⁡ωkτ=w5-a2w3+a4wb2w3-b4w-a1w4-a3w2+a5b1w4-b3w2+b5b2w3-b4w2+b1w4-b3w2+b52=Lωk.Therefore (34)τjk=1ωkarccos⁡Lωk+2jπ,k=1,2,3,4,5,j=0,1,2,….Then ±iwk are a pair of purely imaginary roots of (26) with τ=τjk.Define(35)τ0=τj0k0=min1≤k≤5,j≥1⁡τjk,w0=wk0.From Theorem4 and by a similar argument as that in [21, 22], we have the following results.Lemma 5.
Suppose thatR1>1 and (22) hold. (i)
If one of the following holds: (a)c5<0; (b) c5≥0, q1=0, Δ0≥0, and p1<0 or r1≤0 and there exist z∗∈z1,z2,z3,z4 such that z∗>0 and h(z∗)≤0; (c) c5≥0, q1≠0, s∗>p1, Δ2≥0, or Δ3≥0 and there exist z∗∈z1,z2,z3,z4 such that z∗>0 and h(z∗)≤0; (d) c5≥0, q1≠0, s∗<p1, q12/4p1-s∗2+1/2s∗=0, z->0, and h(z-)≤0, then all the roots of (26) have negative real parts when τ∈0,τ0.(ii)
If all the conditions (a)–(d) of (i) are not satisfied, then all roots of (26) have negative real parts for all τ≥0.We considerξ(τ)=ξ(τ)+iω(τ) to be a root of (26) satisfying ξ(τ)=0 and ω(τ)=ω0. Differentiating the two sides of (26) with respect to τ and noticing that ξ is a function of τ, then (36)dξdτ-1=-5ξ4+4a1ξ3+3a2ξ2+2a3ξ+a4ξξ5+a1ξ4+a2ξ3+a3ξ2+a4ξ+a5+4b1ξ3+3b2ξ2+2b3ξ1+b4ξb1ξ4+b2ξ3+b3ξ2+b4ξ+b5-τξ.From (28) we obtain (37)dRe⁡ξτdττ=τkj-1=-5ωk4-3a2ωk2+a4-ωk6+a2ωk4-a4ωk2-ωk6+a2ωk4-a4ωk22+a1ωk5-a3ωk3+a5ωk2+4a1ωk3-2a3ωka1ωk5-a3ωk3+a5ωk-ωk6+a2ωk4-a4ωk22+a1ωk5-a3ωk3+a5ωk2+-3b2ωk2+b4b2ωk4-b4ωk2b2ωk4-b4ωk22+b1ωk5-b3ωk3+b5ωk2+-4b1ωk3+2b3ωkb1ωk5-b3ωk3+b5ωkb2ωk4-b4ωk22+b1ωk5-b3ωk3+b5ωk2.By (26) we get (38)ω5-a2ω3+a4ω2+a1ω4-a3ω2+a52=b2ω3-b4ω2+b1ω4-b3ω2+b5.Then (39)dRe⁡ξτdττ=τkj-1=5zk4+4c1zk3+3c2zk2+2c3zk+a4b1ωk4-b3ωk2+b52+b2ωk2-b42=h′zkb1ωk4-b3ωk2+b52+b2ωk2-b42.Therefore, it follows that (40)signdRe⁡ξτdττ=τkj=signdRe⁡ξτdττ=τkj-1=signh′zk.Sincezk>0, then Re⁡[dξk(τ)/dτ]τ=τkj and h′zk have the same sign.From the above analysis and the Hopf bifurcation theorem for functional differential equation [20], we have the following result.Theorem 6.
Suppose thatR1>1 and (22) hold. (i)
If the conditions (a)–(d) of Lemma5 are all not satisfied, then the chronic infection equilibrium Q2 is locally asymptotically stable for all time delay τ≥0.(ii)
If one of the conditions (a)–(d) of Lemma5 is satisfied, then the chronic infection equilibrium Q2 is locally asymptotically stable for τ∈0,τ0.(iii)
If the condition of (ii) is satisfied andh′(zk)≠0, then system (2) undergoes a Hopf bifurcation at Q2 when τ=τ0.


## 4. Conclusion

In this paper, we have studied an HIV infection model including infected cells in eclipse stage and delay in the activation of CTL immune response. The model is governed by reaction diffusion equations and the transmission process is modeled by a specific nonlinear incidence rate that includes many types of special incidence functions as special cases. First, we discussed the nonnegativity and boundedness of solutions and the existence of equilibria of system (2). The global stability of the infection-free equilibrium Q0 has been given by the Lyapunov’s direct method and LaSalle’s invariance principal when the basic reproductive number R0≤1, which means that the infection is cleared and the virus dies out. We also obtained the global asymptotic stability of the immune-free infection equilibrium when R0>1 and condition (14) is satisfied, which means that the infection will become chronic without persistent CTL immune response. If R0>1 and R1>1, there exists a chronic infection equilibrium with CTL immune response Q2. We have shown that the chronic infection equilibrium Q2 is locally asymptotically stable when the delay is sufficiently small, but with the increase of the time delay, the stability of Q2 may destabilize and lead to Hopf bifurcation.The results of this paper reflect the fact that the immunological delay in (2) do not affect the positivity and boundedness of solutions and the global stability of the infection-free equilibrium and immune-free equilibrium. For the chronic infection equilibrium, in the absence of delay, the globally stability is obtained, a small delay does not affect the local stability, and Hopf bifurcation may occur when the time delay is large enough.



---

*Source: 3294268-2018-03-01.xml*