Research article Special Issues

Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays

  • In this paper, we investigate a delayed HIV-1 infection model with immune response. Though a logistic growth is incorporated in the growth of the target cells, our focus is on the effect of delays on the infection dynamics. We first study the existence of steady states, which depends on the basic reproduction number R0. When R01, there is only the infection-free steady state, which is globally asymptotically stable if R0<1. When R0>1, besides the unstable infection-free steady state, there is a unique infected steady state. We then study the local stability of the infected steady state and local Hopf bifurcation at it. The theoretical analysis indicates that the dynamics scenario is complicated. For example, there can be three sequences of critical values, stability switches and double Hopf bifurcation can occur. Some of the theoretical results are also illustrated with numerical simulations.

    Citation: Juan Wang, Chunyang Qin, Yuming Chen, Xia Wang. Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2587-2612. doi: 10.3934/mbe.2019130

    Related Papers:

    [1] Xuejuan Lu, Lulu Hui, Shengqiang Liu, Jia Li . A mathematical model of HTLV-I infection with two time delays. Mathematical Biosciences and Engineering, 2015, 12(3): 431-449. doi: 10.3934/mbe.2015.12.431
    [2] Yu Yang, Gang Huang, Yueping Dong . Stability and Hopf bifurcation of an HIV infection model with two time delays. Mathematical Biosciences and Engineering, 2023, 20(2): 1938-1959. doi: 10.3934/mbe.2023089
    [3] Jiawei Deng, Ping Jiang, Hongying Shu . Viral infection dynamics with mitosis, intracellular delays and immune response. Mathematical Biosciences and Engineering, 2023, 20(2): 2937-2963. doi: 10.3934/mbe.2023139
    [4] Ning Bai, Rui Xu . Mathematical analysis of an HIV model with latent reservoir, delayed CTL immune response and immune impairment. Mathematical Biosciences and Engineering, 2021, 18(2): 1689-1707. doi: 10.3934/mbe.2021087
    [5] Cameron Browne . Immune response in virus model structured by cell infection-age. Mathematical Biosciences and Engineering, 2016, 13(5): 887-909. doi: 10.3934/mbe.2016022
    [6] Bing Li, Yuming Chen, Xuejuan Lu, Shengqiang Liu . A delayed HIV-1 model with virus waning term. Mathematical Biosciences and Engineering, 2016, 13(1): 135-157. doi: 10.3934/mbe.2016.13.135
    [7] Ting Guo, Zhipeng Qiu . The effects of CTL immune response on HIV infection model with potent therapy, latently infected cells and cell-to-cell viral transmission. Mathematical Biosciences and Engineering, 2019, 16(6): 6822-6841. doi: 10.3934/mbe.2019341
    [8] Yan Wang, Minmin Lu, Daqing Jiang . Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays. Mathematical Biosciences and Engineering, 2021, 18(1): 274-299. doi: 10.3934/mbe.2021014
    [9] Xiaohong Tian, Rui Xu, Jiazhe Lin . Mathematical analysis of an age-structured HIV-1 infection model with CTL immune response. Mathematical Biosciences and Engineering, 2019, 16(6): 7850-7882. doi: 10.3934/mbe.2019395
    [10] Jinhu Xu, Yicang Zhou . Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences and Engineering, 2016, 13(2): 343-367. doi: 10.3934/mbe.2015006
  • In this paper, we investigate a delayed HIV-1 infection model with immune response. Though a logistic growth is incorporated in the growth of the target cells, our focus is on the effect of delays on the infection dynamics. We first study the existence of steady states, which depends on the basic reproduction number R0. When R01, there is only the infection-free steady state, which is globally asymptotically stable if R0<1. When R0>1, besides the unstable infection-free steady state, there is a unique infected steady state. We then study the local stability of the infected steady state and local Hopf bifurcation at it. The theoretical analysis indicates that the dynamics scenario is complicated. For example, there can be three sequences of critical values, stability switches and double Hopf bifurcation can occur. Some of the theoretical results are also illustrated with numerical simulations.


    AIDS (acquired immunodeficiency syndrome) is a syndrome caused by HIV (human immunodeficiency virus). HIV is a lentivirus (a subetaoup of retrovirus). It infects vital cells in the human immune system, such as helper T cells (specifically CD4+ T cells), macrophages, and dendritic cells [1]. When the number of CD4+ T cells declines below a critical level, cell-mediated immunity is lost, and the body becomes progressively more susceptible to opportunistic infections, leading to the development of AIDS. Mathematical modeling has contributed a lot to the understanding of HIV infection (see, for example, the review by Perelson and Ribeiro [2] for within-host models).

    In the simplest and earliest models of HIV infection, only the key players were taken into account. These models include uninfected target cells (T), productively infected cells (T), and free viruses (V). One such model is described by the following system of ordinary differential equations,

    dTdt=λdTkVT,dTdt=kVTδT,dVdt=pTcV.

    For more detail, we refer the readers to Ribeiro and Perelson [3]. Inspired by this model, researchers have proposed many other HIV models by considering, for example, different uninfected target cell growth and incidence, latently infected CD4+ T cells, treatment, drug resistance, and immune response (to name a few, see [4,5,6,7,8,9,10,11,12,13,14,15,16]).

    Time delay is commonly observed in many biological processes. For HIV infection, on the one hand, it roughly takes about 1 day for a newly infected cell to become productive and then to be able to produce new virus particles [17]. On the other hand, during CTL response, effector CTLs need time to recognize infected cells and destroy them. Herz et al. [18] were the first to introduce an intracellular delay to describe the time between the initial viral entry into a target cell and subsequent viral production. They obtained the effect of the delay on viral load change. Since then, delayed HIV models have attracted the attention of many researchers. See, for example, [19,20,21,22,23,24,25,26] and the references therein.

    In this paper, motivated by the studies in [7,10,27], we propose and study the following delayed HIV model,

    dT(t)dt=sdT(t)+rT(t)(1T(t)Tmax)kV(t)T(t), (1.1a)
    dT(t)dt=k1V(tτ1)T(tτ1)δT(t)dxE(t)T(t), (1.1b)
    dV(t)dt=NδT(t)cV(t), (1.1c)
    dE(t)dt=λE+pT(tτ2)dEE(t). (1.1d)

    Here T(t), T(t), V(t), and E(t) represent the densities of uninfected CD4+ T-cells, productively infected CD4+ T-cells, free viruses, and immune effectors at time t, respectively. As in [7,28], k1=keατ1, where α[d,δ] is the death rate of infected cells before becoming productive. τ1 denotes the time delay between viral entry and viral production while τ2 stands for the time needed for the CTLs immune response to emerge to control viral replication. The interpretations of the parameters are summarized in Table 1, where their units and ranges will be given in Section 4. The logistic growth in target cells and natural growth of immune effectors combined is a new feature of Model (1.1). Our main focus is on the effects of delays, especially τ2, on the dynamics of (1.1).

    Table 1.  Descriptions of parameters in (1.1).
    ParameterDescription
    sProduction rate of uninfected CD+4 T-cells
    dDeath rate of uninfected CD+4 T-cells
    rProliferation rate of uninfected CD+4 T-cells
    TmaxCD+4 T-cells density where proliferation stops
    kInfection rate of CD+4 T-cells with virus particles
    αDeath rate of infected CD+4 T-cells before becoming productive
    δDeath rate of infected CD+4 T-cells
    dxCTL effectiveness
    NBursting term for viral production after lysis
    cClearance rate of virus
    λEProliferation rate of CTL from natural resources
    pProduction rate of CTL response
    dEDeath rate of CTL response

     | Show Table
    DownLoad: CSV

    The rest of the paper is organized as follows. In Section 2, we present some preliminary results of (1.1a), which include the positivity and boundedness of solution, the existence of steady states. Then we analyze the stability of steady states and possible Hopf bifurcation in Section 3. We conclude the paper with some numerical simulations to illustrate the main theoretical results.

    The suitable phase space for (1.1) is C=C1×C2×C1×R, where Ci=C([τi,0],R) is the Banach space of all continuous functions from [τi,0] to R equipped with the supremum norm, i=1, 2. The norm on C is the usual product norm. The nonnegative cone of Ci is C+i=C([τi,0],R+). To be biologically meaningful, in the sequel, the initial conditions of (1.1) will be always from C+=C+1×C+2×C+1×R+.

    For each Φ=(ϕ1,ϕ2,ϕ3,ϕ4)C+, by the standard theory of functional differential equations [29], Model (1.1) has a unique and global solution through it. For such a solution, we first claim that T(t)>0 for t>0. In fact, it is clear that there exists t0>0 such that T(t)>0 for t(0,t0). Suppose to the contrary that there exists t1>t0 such that T(t)>0 for t(0,t1) and T(t1)=0. Then by (1.1), dT(t1)dt=s>0 and hence there exists ε(0,t1) such that T(t)<0 for t(t1ε,t1), a contradiction. This proves the claim. Next, with step-by-step method we show that T(t)0 and V(t)0 for t0. Note that, for t0, by (1.1b) and (1.1c), we have

    T(t)=et0(δ+dxE(s))dsT(0)+t0etu(δ+dxE(s))dsk1V(uτ1)T(uτ1)du (2.1)

    and

    V(t)=ectV(0)+t0ec(ut)NδT(u)du, (2.2)

    respectively. It follows from (2.1) that T(t)0 for t[0,τ1]. This, combined with (2.2), gives V(t)0 for t[0,τ1], which together with (2.1) yields T(t)0 for t[0,2τ1]. In turn from (2.2) we have V(t)0 for t[0,2τ1]. Continuing this way gives the desired result. Finally, from (1.1d), we get

    E(t)=edEtE(0)+t0edE(ut)(λE+pT(uτ2))du

    for t0 and hence E(t)0 for t0. Therefore, the solution of (1.1a) with initial condition in C+ is nonnegative.

    Next, we consider the boundedness of solutions. Firstly, we obtain from (1.1a) that

    dT(t)dtsdT(t)+rT(t)(1T(t)Tmax)

    for t0. It follows that

    lim suptT(t)T0,

    where

    T0=Tmax2r[rd+(rd)2+4rsTmax ]

    is the unique positive zero of sdT+rT(1TTmax). Moreover, if T(0)T0 then T(t)T0 for t0. Secondly, consider the Lyapunov functional

    L1(t)=T(tτ1)+kk1T(t).

    The derivative of L1 along solutions of (1.1) is

    dL1(t)dt=sdT(tτ1)+rT(tτ1)(1T(tτ1)Tmax)kδk1T(t)kdxk1E(t)T(t)dT(tτ1)δkk1T(t)+rT(tτ1)rTmaxT2(tτ1)+sd1L1(t)+M0,

    where d1=min{δ,d} and M0=rTmax+4s4 (>0). Then lim suptL1(t)M0d1. In particular, lim suptT(t)k1M0kd1. Finally, this combined with (1.1c) and (1.1d) immediately gives

    lim suptV(t)Nδk1M0ckd1  and  lim suptE(t)λEkd1+pk1M0dEkd1,  respectively.

    Lastly, we study the lower boundedness of T. For any ε>0, there exists t2>0 such that V(t)Nδk1M0ckd1+ε for tt2. This, together with (1.1a), gives us

    dT(t)dtsdT+rT(1TTmax)kT(Nδk1M0ckd1+ε)for tt2.

    Hence, as ε is arbitrary, we get

    lim inftT(t)Tmax2r[rdNδk1M0cd1+(rdNδk1M0cd1)2+4rsTmax ].

    To summarize, we have shown the following result.

    Proposition 1. The solutions of (1.1) with initial conditions in C+ are nonnegative and bounded. Moreover, the region

    Γ={Φ=(ϕ1,ϕ2,ϕ3,ϕ4)C+|the solution (T(t),T(t),V(t),E(t)) of (1.1)through Φ satisfies ϕ1(0)T0ϕ1(0)+kk1T(τ1)M0d1,ϕ3(0)Nδk1M0ckd1,and ϕ4λEkd1+pk1M0dEkd1}

    is a positively invariant and attracting subset of (1.1) in C+.

    In the remaining of this section, we consider the steady states of (1.1). Note that a steady state is a solution of the following system of algebraic equations,

    sdT+rT(1TTmax)kVT=0, (2.3a)
    k1VTδTdxET=0, (2.3b)
    NδTcV=0, (2.3c)
    λE+pTdEE=0. (2.3d)

    It follows from (2.3c) that V=NδTc. Sustituting it into (2.3b) gives

    k1NδcTTδTdxET=0.

    Then T=0 or T=c(δ+dxE)Nδk1. When T=0, we get the infection-free steady state P0=(T0,0,0,E0), where E0=λEdE. Now assume T=c(δ+dxE)Nδk1. Combining it with E=λE+pTdE obtained from (2.3d), we can get after a little computation that

    T=[Nδk1Tc(δ+dxλEdE)]dEcdxp. (2.4)

    Then

    V=dENδc2dxp[Nδk1Tc(δ+dxλEdE)].

    Substituting it into (2.3a) yields

    G(T)=0,

    where

    G(T)=s+[rd+kdENδcdxp(δ+dxλEdE)]T[rTmax+(Nδc)2kk1dEdxp]T2.

    Note that G always has a positive zero and it only has one positive zero. However, for infected steady states, we have T>c(δ+dxλEdE)Nδk1 from (2.4), or equivalently, G(c(δ+dxλEdE)Nδk1)>0 or c(δ+dxλEdE)Nδk1<T0. Thus there is an infected steady state if and only if R0>1, where

    R0=Nδk1T0c(δ+dxλEdE).

    In summary we have obtained the following result.

    Theorem 2.1. (i) If R01 then (1.1) only has the infection-free steady state P0.

    (ii) If R0>1 then, besides P0, (1.1) also has a unique infected steady state P1=(T1,T1,V1,E1), where

    T1=b+b2+4as2a,a=rTmax+(Nδc)2kk1dEdxp,b=rd+(δdE+dxλE)Nδkcpdx,T1=dEdxp(Nδk1T1cδdxλEdE),V1=NδcT1,E1=λE+pT1dE.

    Note that, in epidemiology, R0 is called the basic reproduction number, whose expression can also be derived by the procedure in [30].

    We start with the local stability of the infection-free steady state P0.

    Theorem 3.1. (i) If R0<1, then the infection-free steady state P0 of (1.1) is locally asymptotically stable.

    (ii) If R0>1, then P0 is unstable.

    Proof. The characteristic equation at P0 is

    (λ+dE)(λ+dr+2rT0Tmax)[λ2+(c+δ+dxE0)λ+c(δ+dxE0)Nδk1T0eλτ1]=0.

    Clearly, dE and (dr+2rT0Tmax)=sT0rT0Tmax are eigenvalues and both are negative. The other eigenvalues are roots of the following transcendental equation,

    Δ0(λ)=λ2+(c+δ+dxE0)λ+c(δ+dxE0)Nδk1T0eλτ1=0. (3.1)

    Noting

    R0=Nδk1T0c(δ+dxE0),

    we can rewrite (3.1) as

    Δ0(λ)=λ2+(c+δ+dxE0)λ+c(δ+dxE0)(1R0eλτ1)=0 (3.2)

    or equivalently

    R0=(λc+1)(λδ+dxE0+1)eλτ1. (3.3)

    (ⅰ) Assume R0<1. We claim that all roots of (3.3) have negative real parts. Otherwise, (3.3) has a root λ=σ+ωi with σ0 and σ2+ω2>0 since 0 is not a root by R0>1. Taking moduli of both sides of (3.3) gives

    R0=eστ1[(σc+1)2+ω2c2][(σδ+dxE0+1)2+(ωδ+dxE0)2].

    This is impossible as the right side of the above is >1 and R0<1. This proves the claim and hence P0 is locally asymptotically stable if R0<1.

    (ⅱ) Assume R0>1. In this case, (3.2) has a positive root. In fact, this follows from the Intermediate Value Theorem and

    Δ0(0)=c(δ+dxE0)(1R0)<0andlimλΔ0(λ)=.

    Therefore, P0 is unstable if R0>1. This completes the proof.

    In fact, the local stability of P0 implies its global stability.

    Theorem 3.2. If R0<1, then the infection-free steady state P0 of (1.1) is globally asymptotically stable.

    Proof. Define the Lyapunov functional

    W0(t)=T(t)+k1T0cV(t)+k1ttτ1V(θ)T(θ)dθ.

    Then the time derivative of W0 along solutions of (1.1) is

    dW0(t)dt=dT(t)dt+k1T0cdV(t)dt+k1V(t)T(t)k1V(tτ1)T(tτ1)=k1V(tτ1)T(tτ1)δT(t)dxE(t)T(t)+Nδk1T0cT(t)k1T0V(t)+k1V(t)T(t)k1V(tτ1)T(tτ1)=(Nδk1T0c(δ+dxE(t)))T(t)+k1V(t)(T(t)T0)(Nδk1T0c(δ+dxE0))T(t)=(δ+dxE0)(R01)T(t)0.

    Moreover, dW0dt=0 if and only if T(t)=0 and V(t)(T(t)T0)=0. Then one can see that the largest invariant subset of {dW0dt=0} is {P0}. By the Lyapunov-LaSalle invariance principle (see [29,Theorem 5.3.1] or [31,Theorem 3.4.7]) and Theorem 3.1, we conclude that if R0<1 then P0 is globally asymptotically stable.

    Recall that P1 exists only when R0>1, which implies that necessarily Nδk1T0c(δ+dxλEdE)>1 as k1=keατ1. The purpose of this paper is to consider the effects of delays on the dynamics. As a result, in the sequel of this section, we always assume that Nδk1T0c(δ+dxλEdE)>1 and denote

    ˆτ1=1αlnNδkT0c(δ+dxλEdE).

    Then R0>1 is equivalent to τ1<ˆτ1.

    The characteristic equation at P1 is

    (λ+dr+2rT1Tmax+kV1)(λ+c)[(λ+δ+dxE1)(λ+dE)+pdxT1eλτ2]=(λ+dr+2rT1Tmax)(λ+dE)Nδk1T1eλτ1. (3.4)

    In the following, we follow the arguments in [23] to first show that P1 is locally stable for τ1[0,ˆτ1) and τ2=0. Then for given τ1[0,ˆτ1), we discuss the possible bifurcations.

    Theorem 3.3. Suppose τ1[0,ˆτ1), τ2=0, and 0r<d1T1Tmax. Then the infected steady state P1 is locally asymptotically stable.

    Proof. When τ2=0, the characteristic equation (3.4) reduces to

    (λ+dr+2rT1Tmax+kV1)(λ+c)[(λ+δ+dxE1)(λ+dE)+pdxT1]=(λ+dr+2rT1Tmax)(λ+dE)Nδk1T1eλτ1. (3.5)

    We will prove that all roots of (3.5) have negative real parts in three steps.

    Firstly, we show that (3.5) has no roots on the imaginary axis with contradictory arguments. Let λ=iω0 with ω00 be a root of (3.5). Then

    (iω0+dr+2rT1Tmax+kV1)(iω0+c)[(iω0+δ+dxE1)(iω0+dE)+pdxT1]=(iω0+dr+2rT1Tmax)(iω0+dE)Nδk1T1eiω0τ1. (3.6)

    Note that the modulus of the right hand side of (3.6) is

    Nδk1T1|iω0+dE||iω0+dr+2rT1Tmax|=c(δ+dxE1)|iω0+dE||iω0+dr+2rT1Tmax|.

    However, since

    |(iω0+δ+dxE1)(iω0+dE)+pdxT1|2(δ+dxE1)2|iω0+dE|2=ω20d2E+2pdEdxT1(δ+dxE1)+(ω20pdxT1)2>0

    and

    |(iω0+dr+2rT1Tmax+kV1)(iω0+c)|2|(iω0+dr+2rT1Tmax)|2c2>|(iω0+dr+2rT1Tmax)|2c2|(iω0+dr+2rT1Tmax)|2c2=0,

    it follows that the modulus of the left hand side of (3.6) is strictly larger than that of its right hand side, a contradiction. Thus we have verified that (3.5) has no roots on the imaginary axis.

    Secondly, we show that (3.5) has no nonnegative real roots. Again, by contradiction, assume that (3.5) has a root λ00 and we know that eλ0τ1(eλ0ˆτ1,1]. Noting

    (λ0+dr+2rT1Tmax)(λ0+dE)Nδk1T1eλ0τ1(λ0+dr+2rT1Tmax)(λ0+dE)Nδk1T1,

    we get from (3.5) that

    (λ0+dr+2rT1Tmax+kV1)(λ0+c)[(λ0+δ+dxE1)(λ0+dE)+pdxT1]Nδk1T1(λ0+dr+2rT1Tmax)(λ0+dE). (3.7)

    But

    (λ0+dr+2rT1Tmax+kV1)(λ0+c)[(λ0+δ+dxE1)(λ0+dE)+pdxT1]>(λ0+dr+2rT1Tmax)(λ0+c)(λ0+δ+dxE1)(λ0+dE)c(δ+dxE1)(λ0+dr+2rT1Tmax)(λ0+dE)=Nδk1T1(λ0+dr+2rT1Tmax)(λ0+dE)

    as Nδk1T1=c(δ+dxE1), which contradicts with (3.7). This proved that (3.5) has no nonnegative real root.

    Finally, we claim that there exists η0>0 such that all roots of (3.5) have negative real parts when 1<R0<1+η0. If this is not true, then there exists a sequence of values for the parameters where R0 (>1) 1 such that for each set of values for the parameters there exists a pair of conjugate roots for (3.5) with positive real parts (which follows from the results just proved above). Note that roots of (3.5) having nonnegative real parts are uniformly bounded. Without loss of generality, we suppose that the sequence of the conjugate roots converges to α0±iβ0, otherwise just consider a subsequence. Then α00 and α0±iβ0 are roots of the characteristic equation of the infection-free steady state P0 when R0=1. However, when R0=1, this characteristic equation has no roots with nonnegative real parts except the simple root 0. Then α0=β0=0, which implies that 0 is a root of at least multiplicity 2 of this characteristic equation, a contradiction. This proves the claim.

    Now the proof is done by noting the fact that the roots of (3.5) depend continuously on the parameters.

    Theorem 3.4. If τ1=τ2=0 and 0r<d1T1Tmax holds, then the infected steady state P1 is global asymptotically stable.

    Proof. We define a Lyapunov functional

    W1(t)=T1(TT11lnTT1)+kk1T1(TT11lnTT1)+kT1cV1(VV11lnVV1)+kdxE1k1pE1(EE11lnEE1).

    Then the time derivative of W1(t) along solutions of system (1.1) is

    dW1dt=(1T1T)dTdt+kk1(1T1T)dTdt+kT1c(1V1V)dVdt+kdxE1k1p(1E1E)dEdt=(1T1T)(sdT+rTrT2TmaxkVT)+kk1(1T1T)(k1VTδTdxET)+kT1c(1V1V)(NδTcV)+kdxE1k1p(1E1E)(λE+pTdEE)=2dT12rT1+rT21TmaxdT+rTrT2TmaxdT21T+rT21TrT31TTmax+rTT1Tmax+kV1T1kV1T21Tkk1δTkk1dxETkVTT1T+kk1δT1+kk1dxET1+kNδT1TckNδV1T1TcV+kT1V1+kdxE1k1pdEE1kk1dxE1T1+kk1dxE1TkdxE1k1pdEEkdxE1E1k1pEdEE1+kE1k1EdxE1T1kE1k1EdxE1T+kdxE1k1pdEE1=(rdrT1TmaxrTTmax)(TT1)2T+kV1T1(3T1TT1VTTV1T1V1TVT1)+kdxE1k1pdEE1(2EE1E1E)kk1dxE1T1(2EE1E1E)+kk1dxE1T(2EE1E1E)=(rdrT1TmaxrTTmax)(TT1)2T+kV1T1(3T1TT1VTTV1T1V1TVT1)+kk1dxE1(λEp+T)(2EE1E1E)0.

    It is clear that dW1dt=0 if and only if (T,T,V,E)=P1. By the Laypunov asymptotic stability theorem, we conclude that P1 of system (1.1) is globally asymptotically stable.

    In the following, we study the effects of delays by fixing τ1=τ01[0,ˆτ1) and changing τ2. Then the characteristic equation (3.4) can be rewritten as

    P(λ,τ01)+Q(λ,τ01)eλτ01+R(λ,τ01)eλτ2=0, (3.8)

    where

    P(λ,τ01)=λ4+A3λ3+A2λ2+A1λ+A0,Q(λ,τ01)=B2λ2+B1λ+B0,R(λ,τ01)=C2λ2+C1λ+C0,A3=c+δ+dxE1+dE+ST1+rT1Tmax,A2=c(δ+dxE1)+(c+δ+dxE1)(dE+ST1+rT1Tmax)+dE(ST1+rT1Tmax),A1=dE(c+δ+dxE1)(ST1+rT1Tmax)+c(δ+dxE1)(dE+ST1+rT1Tmax),A0=cdE(δ+dxE1)(ST1+rT1Tmax),B2=Nδk1T1=c(δ+dxE1),B1=c(δ+dxE1)(dE+dr+2rT1Tmax),B0=cdE(δ+dxE1)(dr+2rT1Tmax),C2=pdxT1,C1=pdxT1(c+ST1+rT1Tmax),C0=cpdxT1(ST1+rT1Tmax).

    Since

    P(0,τ01)+Q(0,τ01)+R(0,τ01)=A0+B0+C0=cdE(δ+dxE1)kV1+cpdxT1(ST1+rT1Tmax)>0,

    we know that λ=0 is not a root of (3.8). Therefore, for stability changes of P1 to occur, we first look for τ2 where (3.8) has a pair of conjugate roots λ=±iω(τ01,τ2) with ω(τ01,τ2)>0. Substitute λ=iω(τ01,τ2) into (3.8) and then separate the real and imaginary parts to obtain

    (C2ω2+C0)cosωτ2+C1ωsinωτ2=M1,C1ωcosωτ2(C2ω2+C0)sinωτ2=M2, (3.9)

    where

    M1=(ω4+A2ω2A0)(B2ω2+B0)cosωτ01B1ωsinωτ01,M2=(A3ω3A1ω)B1ωcosωτ01+(B2ω2+B0)sinωτ01.

    It follows that

    sinωτ2=M1C1ω(C2ω2+C0)M2(C2ω2+C0)2+C21ω2,cosωτ2=(C2ω2+C0)M1+C1ωM2(C2ω2+C0)2+C21ω2.

    Using sin2ωτ2+cos2ωτ2=1, we see that ω(τ01,τ2) satisfies F(ω,τ01)=0, where

    F(ω,τ01)=ω8+a6ω6+a5ω5+a4ω4+a3ω3+a2ω2+a1ω+a0 (3.10)

    with

    a6=A232A22B2cosωτ01,a5=2(B1A3B2)sinωτ01,a4=2A0+A222A1A3+B22+2(B0A3B1+A2B2)cosωτ01C22,a3=2(A3B0A2B1+A1B2)sinωτ01,a2=A212A0A2+B212B0B22(A2B0A1B1+A0B2)cosωτ01C21+2C0C2,a1=2(A1B0+A0B1)sinωτ01,a0=A20+B20+2A0B0cosωτ01C20.

    Therefore, ω(τ01,τ2) is independent of τ2. Denote

    Iτ01={ω:F(ω,τ01)=0},

    which is a finite set. If Iτ01= then P1 is stable for τ1=τ01 and τ20. Now, suppose Iτ01. For example, this is true if A0+B0<C0 since

    F(0,τ01)=(A0+B0)2C20,limωF(ω,τ01)=,

    and A0+B0+C0>0.

    Assume Iτ01={ω1(τ01),,ωj(τ01)(τ01)}. For jN(τ01)={1,,j(τ01)}, choose the unique angle θj(τ01)[0,2π) such that

    sinθj(τ01)=C1ωj(τ01)M1(C2ω2j(τ01)+C0)M2(C2ω2j(τ01)+C0)2+C21ω2j(τ01),cosθj(τ0j)=(C2ω2j(τ01)+C0)M1+C1ωj(τ01)M2(C2ω2j(τ01)+C0)2+C21ω2j(τ01). (3.11)

    Now, define

    τn2j(τ01)=θj(τ01)+2nπωfor nN={0,1,2,}.

    Then the characteristic equation (3.8) at τ2=τn2j has a pair of conjugate eigenvalues λ=±iωj(τ01) for jN(τ01) and nN. The following result comes from [32,Theorem 2.2].

    Theorem 3.5. Let τ01[0,ˆτ1). Then the following two statements are true.

    (i) If Iτ01=, then P1 is locallay asymptotically stable for τ1=τ01 and τ20.

    (ii) If Iτ01, then a pair of simple conjugate pure imaginary roots λ(τn2j(τ01))=±iωj(τ01) of (3.8) exists at τ2=τn2j(τ01) for jN(τ01) and nN, which crosses the imaginary axis from left to right if δ(τn2j(τ01))>0 and crosses the imaginary axis from right to left if δ(τn2j(τ01))<0, where

    δ(τn2j(τ01))=sign{dRe λdτ2|τ2=τn2j(τ01)}=sign{Fω(ωj(τ01),τ01)}.

    By Theorem 3.5, for any τ1[0,^τ1), there exists a τ2(τ1)(0,] such that P1 is locally asymptotically stable for τ<τ2(τ1).

    In general, it is hard to determine whether Iτ1 is empty or not. Moreover, if Iτ1 and has more than one element, then Theorem 3.5 indicates that there may be stability switches for P1. To get a clear picture of it, we consider the case where τ01=0. Moreover, F(ω,0) in (3.10) reduces to h(ω2), where

    h(z)=z4+b3z3+b2z2+b1z+b0 (3.12)

    with

    b3=A232(A2+B2),b2=2A0+A222A1A3+B22+2(B0A3B1+A2B2)C22,b1=A212A0A2+B212B0B22(A2B0A1B1+A0B2)C21+2C0C2,b0=(A0+B0)2C20.

    In this case, δ(τn2j(0)) in Theorem 3.5 is sign(h(ω2j(0))). As a result, for Hopf bifurcation to occur, we only focus on the situations where h(z) defined by (3.12) has simple positive real zeros.

    Though a simple calculation gives b3=d2E+(c+δ+dxE1)2+(sT1+rT1Tmax)2>0, we cannot easily get the signs of the other coefficients. By Descartes' rule of sign, the polynomial h(z) has at most three positive real zeros. In fact, Yan and Li [33] have obtained the conditions on the existence of at least one positive real zero for h(z). To cite the result, let

    p=8b23b2316,q=b334b3b2+8b132,Δ=q24+p327,z1=b34+3q2+Δ+3q2Δif Δ>0z2=max{b3423q2,b34+3q2}if Δ=0z3=max{b34+2Re{ρ},b34+2Re{ρϵ},b34+2Re{ρˉϵ}}if Δ<0

    where ρ is one of the cubic roots of the complex number q2+Δ and ϵ=1+3i2. Note that when Δ>0, z1 is the only real zero of h(z); when Δ=0, z1=b3423q2, z2=z3=b34+3q2 are the three real zeros of h(z); when Δ<0, b34+2Re{ρ}, b34+2Re{ρϵ}, and b34+2Re{ρˉϵ} are the three real zeros of h(z) and we arrange them as ˆz1<ˆz2<z3.

    Proposition 3.1 ([33], Lemma 2.1]).

    (i) If b0<0, then h(z) has at least one positive real zero. (ii) If b00, then h(z) has no positive real zero if one of the following conditions holds.

    (ii1) Δ>0 and z1<0;

    (ii2) Δ=0 and z2<0;

    (ii3) Δ<0 and z3<0.

    (iii) If b00, then h(z) has at least one positive real zero if one of the following conditions holds.

    (iii1) Δ>0, z1>0 and h(z1)<0;

    (iii2) Δ=0, z2>0 and h(z2)<0;

    (iii3) Δ<0, z3>0 and h(z3)<0.

    When τ1=0, by Proposition 3.1 we can have the following result.

    Theorem 3.6. Assume τ1=0 and one of the conditions in statement (ii) of Proposition 3.1 holds. Then the infected steady state P1 is locally asymptotically stable for all τ20.

    In the following result, we characterize the situations where h(z) has simple positive real zeros, which is not difficult to see by considering the possible graphs of h(z) and h(z). Recall that h(z) can have at most three positive real zeros.

    Proposition 3.2. For the polynomial h(z) defined by (3.12), the following results hold.

    (i) h(z) has one simple positive zero and no other positive zeros if and only if (H1): one of the following conditions hold.

    (i1) Δ0 and b0<0.

    (i2) Δ>0, b0=0, and z1>0.

    (i3) Δ=0, b0=0 and (z2=z1>0 or z2=z2>z1>0).

    (i4) Δ<0, b0<0 and ˆz20.

    (i5) Δ<0, b0<0, ˆz2>0 and h(ˆz2)<0.

    (i6) Δ<0, b0<0, ˆz2>0, h(ˆz2)>0 and h(z3)>0.

    (i7) Δ<0, b0=0 and ˆz2<0<z3.

    (i8) Δ<0, b0=0, ˆz1>0, h(ˆz2)>0 and h(z3)>0.

    (i9) Δ<0, b0=0, ˆz1>0 and h(ˆz2)<0.

    (ii) h(z) has two simple positive zeros and no other positive zeros if and only if (H2) : one of the following conditions holds.

    (ii1) Δ>0, b0>0, z1>0 and h(z1)<0.

    (ii2) Δ=0, b0>0, z2=z1>0 and h(z2)<0.

    (ii3) Δ=0, b0>0, z2=z2>z1>0 and h(z1)<0.

    (ii4) Δ<0, b0=0, ˆz10<ˆz2 and h(z3)<0.

    (ii5) Δ<0, b0>0, ˆz20<z3 and h(z3)<0.

    (ii6) Δ<0, b0>0, ˆz10<ˆz2 and h(z3)<0.

    (ii7) Δ<0, b0>0, ˆz1>0, h(ˆz1)>0 and h(z3)<0.

    (ii8) Δ<0, b0>0, ˆz1>0, h(ˆz1)<0 and h(ˆz2)<0.

    (ii9) Δ<0, b0>0, ˆz1>0, h(ˆz1)<0, h(ˆz2)>0 and h(z3)>0.

    (iii) h(z) has three simple positive zeros and no other positive zeros if and only if (H3): one of the following conditions holds.

    (iii1) Δ<0, b0<0, ˆz2>0, h(ˆz2)>0, and h(z3)<0.

    (iii2) Δ<0, b0=0, ˆz1>0, h(ˆz2)>0 and h(z3)<0.

    If (H1) holds, let ˜z>0 be the unique simple positive zero of h(z) and denote ˜ω=˜z. Solving (3.11) to obtain the unique ˜θ[0,2π). Define

    τn2=2nπ+˜θ˜ωfor nN.

    As h(˜z)>0, we have δ(τn2)=1 and hence the following result holds.

    Theorem 3.7. Assume that τ1=0 and assumption (H1) holds. Then there exists a sequence 0<τ02<τ12<τ22< such that P1 is locally asymptotically stable for τ2[0,τ02) and unstable for τ>τ02, and system (1.1) undergoes a Hopf bifurcation at P1 when τ2=τn2 for nN.

    Now, assume (H2) holds. Let ˜z2<˜z1 be the only positive real zeros of h(z), which are also simple. Similarly as for the case of (H1), we can get two increasing positive sequences {τn21} and {τn22}, associated with ˜z1 and ˜z2, respectively. Since h(˜z1)>0>h(˜z2), we easily see that τ021<τ022. Since ˜ω1=˜z1>˜z2=˜ω2, we have 2π˜ω1<2π˜ω2. Thus we define

    k=min{lN:τl+121=τ021+2π(l+1)˜ω1τ022+2πl˜ω2=τl22}.

    Such k exists due to τl+121τl22 as l. Then the first few Hopf bifurcation values can be ordered as

    τ021<τ022<τ121<τ122<<τk121<τk122<τk21<τk+121τk22<.

    Theorem 3.8. Assume τ1=0 and (H2) holds. Given nN and j{1,2}, system (1.1) undergoes Hopf bifurcation at τ2=τn2j if τn2jτl2(3j) for all lN. Furthermore, the stability of P1 switches off (namely, it becomes unstable) when τ2 crosses τ021, , τk21 and switches on (namely, it becomes stable) when τ2 crosses τ022, , τk122. In other words, P1 is stable when τ2[0,τ021)(τ022,τ121)(τk122,τk21) and unstable when τ2(τ021,τ022)(τk121,τk122)(τk21,).

    We mention that we can study the global continuation of Hopf bifurcation in Theorem 3.8 as in Li and Shu [34]. We believe that the Hopf bifurcation branches are bounded and each joins a pair of τn21 and τn22 for nN. As a result, for τ2(τk+121,τk22), there will be two stable periodic orbits.

    Also in Theorem 3.8, we exclude the situation where τn21=τl22 for some n, lN. If this happens, then n>l since τ021<τ022 and 2π˜ω1<2π˜ω2. In this critical situation, as τ2 crosses this common critical value, two pairs of purely imaginary eigenvalues ±i˜ω1 and ±i˜ω2 appear and all other eigenvalues have nonzero real parts. Therefore, a double Hopf bifurcation occurs.

    Theorem 3.9. Assume τ1=0 and (H2) holds. If there exist integers n>l0 such that τn21=τl22=τ20, then (1.1) undergoes a double Hopf bifurcation at P1 when τ2=τ20.

    When (H3) holds, we can similarly get three sequences of critical values for τ2. Similar results as those in Theorem 3.8 and Theorem 3.9 can be obtained. Moreover, the global Hopf bifurcation associated with the third sequence is unbounded (one can refer to Li et al. [35] for a similar discussion).

    In this paper, we rigorously analyzed an HIV-infection model with CTL-immune response and two time delays. The model incorporates a logistic growth term for the target cell growth and a natural resource for the immune effectors. The basic reproduction number R0 played an important in the infection dynamics. If R0<1 then the infection-free steady state is globally asymptotically stable. Note that R0 explicitly depends on τ1. It follows that if τ1 is large enough then the virus will be cleared. We emphasize that this has not been noted in most existing study. Of course, in the real situation, this delay between viral entry and subsequent viral production usually is not very big. This leads to the complicated dynamics when R0>1. In this case, the unique infected steady state could be stable or unstable, depending on the parameter values. In particular, we focused on the effects of time delays. Theoretical results indicate that there can be Hopf bifurcation, double Hopf bifurcation and stability switches.

    We conclude this paper with some numerical simulations to illustrate the above mentioned main results. The ranges of the parameters except the delays are summarized in Table 2.

    Table 2.  Parameter values for simulation.
    ParameterUnitRangeReferences
    scells ml1day1010[28]
    dday10.00010.2[23]
    rday10.033[36]
    Tmaxcells ml16001600[36]
    kml1day14.6×1080.5[7]
    αday1α[d,δ][7]
    δday10.000191.4[23]
    dxml1day10.00014.048[7,23]
    Nviron cells16.2523599.9[23]
    cday1236[23,26]
    λEcells ml1day10[37]
    pday10.00513.912[23]
    dEday10.0048.087[23,37]
    τ1days01.5[7]
    τ2days035[7]

     | Show Table
    DownLoad: CSV

    For simplicity, we use the same initial condition (T0,T0,V0,E0)=(100,0,102,0.6) in all simulations.

    First, we take s=10, d=0.01, r=0.03, Tmax=1500, α=0.02, δ=0.3, dx=0.01, N=21, c=3, λE=1, p=0.3, dE=0.1, τ1=1.2 and k=2.4×105. Then R0=0.1680<1. By Theorem 3.2, the infection-free steady state P0=(1366,0,0,10) is globally asymptotically stable (see Figure 1 with τ2=4).

    Figure 1.  When R0<1 the infection-free steady state P0 is globally asymptotically stable. See the text for the parameter values.

    Next, we only change k to k=2.4×103 and keep the others as above. In this case, we have ˆτ1=142.2801. Take τ1=1.2[0,ˆτ1). Then R0=16.8038>1. Through numerical calculations, we get the first few critical values τ021=2.9940 and τ121=32.9360 associated with ω1=0.2098 and τ022=28.8271 associated with ω2=0.1066. By Theorem 3.5, the infected steady state P1=(171.7615,14.8383,31.1604,54.5149) is locally asymptotically stable for τ2<τ021 (see Figure 2 with τ2=2).

    Figure 2.  The infected steady state P1 is locally stable. We refer to the text for the parameter values.

    In fact, numerical simulations indicates that there is Hopf bifurcation for τ2(τ021,τ022) (see Figure 3 with τ2=5) and there is a stability switch at τ2=τ022, that is, P1 is locally asymptotically stable for τ2(τ022,τ121) (see Figure 4 with τ2=32).

    Figure 3.  There is a periodic solution bifurcated from the infected steady state P1 through Hopf bifurcation. Parameter values are given in the text.
    Figure 4.  The infected steady state P1 gains stability and this indicates a stability switch. See the text for the parameter values.

    In the following we illustrate this more clearly with the special case where τ1=0. We distinguish three cases.

    Case 1: (ⅱ) of Proposition 3.1 holds. We take s=5, d=0.2, r=0.03, Tmax=1500, α=0.2, δ=0.3, dx=0.01, N=2800, c=15, λE=1, p=0.3, dE=0.1, τ1=0 and k=2.4×103. Then R0=9.8484>1 and system (1.1) has the unique infected steady state P1=(4.5277,6.9510,389.2547,30.8529). In this case, Δ=4.1507×106>0, b0=0.6078>0, and z1=1.9553×102<0. This means that (ⅱ-1) of Proposition 3.1 holds. It follows from Theorem 3.6 that the infected steady state P1 is locally asymptotically stable for all τ20 (see Figure 5 with τ2=1).

    Figure 5.  The infected steady state P1 is locally asymptotically stable. For parameter values, see the text.

    Case 2: Assumption (H1) holds. We choose the parameter values s=10, d=0.01, r=0.25, Tmax=1500, α=0.02, δ=0.3, dx=0.01, N=21, c=3, λE=1, p=0.3, dE=0.1 and k=2.4×104. Then R0=1.8655>1 and system (1.1) has the unique infected steady state P1=(1448.2000,10.9961,23.0918,42.9882). Note that Δ=0.2247<0, b0=6.0220×104<0, and ˆz2=0.0442<0, namely, assumption (H1) (ⅰ-4) holds. We can get ω=0.1519 and τj2=4.3175+2jπω for jN. It follows from Theorem 3.7 that the infected steady state P1 is locally asymptotically stable for τ2[0,τ02) (see Figure 6 with τ2=4<τ02) and unstable for τ>τ02. Moreover, system (1.1) undergoes Hopf bifurcation at τ2=τj2 for jN. Figure 7 supports this with τ2=5>τ02. Figure 8 provides the bifurcation diagram.

    Figure 6.  The infected steady state P1 is locally asymptotically stable. For parameter values, we refer to the text.
    Figure 7.  There is a periodic orbit bifurcated through Hopf bifurcation at the infected steady state P1 when τ2=5>τ02. See the text for parameter values.
    Figure 8.  The bifurcation diagram at P1 with τ2 as the bifurcation parameter. See the text for the other parameter values.

    Case 3: Assumption (H2) holds. This time we replace k with k=2.4×103, and r with r=0.03, and keep the other parameter values as in Case 2. It follows that R0=17.2119>1 and system (1.1) has the unique infected steady state P1=(168.9145,15.0443,31.5930,55.1329). Moreover, Δ=0.4441<0, b0=3.0321×104>0, ˆz1=11.1936<0<ˆz2=0.0018>0, and h(z3)=9.3650×104<0. Therefore, assumption (H2) (ⅱ-6) holds. In this case, we have ˜ω1=0.2845, ˜ω2=0.1401, τj21=2.0033+2jπ˜ω1, and τj22=22.3237+2jπ˜ω2 for jN. Then the first few Hopf bifurcation values are ordered as τ021<τ022<τ121<τ221<τ122<. By Theorem 3.8, the infected steady state P1 is locally asymptotically stable for τ2[0,τ021)(τ022,τ121) (see Figure 9 and Figure 10 with τ2=1.5<τ022 and τ2=24(τ022,τ121), respectively) and is unstable for τ2(τ021,τ022)(τ121,). Thus there are stability switches for P1. Moreover, there are supercritical Hopf bifurcation at τ2=τj21 and subcritical Hopf bifurcation at τ2=τj22 (see Figures 11 and 12). The numerical simulations also strongly indicate that the global Hopf bifurcation branches are bounded and each branch connects a pair τj21 and τj22, which we will confirm in a future work.

    Figure 9.  The infected steady state P1 is locally asymptotically stable for τ2=1.5<τ021=2.0033. See the text for the values of the other parameters.
    Figure 10.  The infected steady state P1 is asymptotically stable for τ022<τ2=24<τ121. We refer to the text for values of the other parameters.
    Figure 11.  There is a periodic orbit bifurcated from the infected steady state P1 through supercritical Hopf bifurcation when τ021<τ2=2.5<τ022. See the text for the values of the other parameters.
    Figure 12.  There is a periodic orbit bifurcated from the infected steady state P1 through subcritical Hopf bifurcation when τ021<τ2=19<τ022. See the text for the values of the other parameters.

    The authors are grateful to the anonymous reviewers for their constructive comments. JW is supported by the Nanhu Scholar Program for Young Scholars of Xinyang Normal University. CQ is supported by Scientific Research Foundation of Graduate School of Xinyang Normal University (No. 2018KYJJ12). XW is supported by the NSFC (No. 11771374), the Nanhu Scholar Program for Young Scholars of Xinyang Normal University, the Program for Science and Technology Innovation Talents in Universities of Henan Province (17HASTIT011). YC is supported by NSERC of Canada.

    All authors declare no conflicts of interest in this paper.



    [1] A. L. Cunningham, H. Donaghy, A. N. Harman, et al., Manipulation of dendritic cell function by viruses, Curr. Opin. Microbiol., 13 (2010), 524–529.
    [2] A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), 96.
    [3] R. M. Riberio and A. S. Perelson, The analysis of HIV dynamics using mathematical modeling, in AIDS and other manisfestation of HIV infection, (Edited by G.P. Wormser), San Diego, Elsevier, (2004), 905–912.
    [4] P. De Leenheer and H. L. Smith, Virus dynamics: a global analysis, SIAM J. Appl. Math., 63 (2003), 1313–1327.
    [5] A. M. Elaiw and S. A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci., 36 (2013), 383–394.
    [6] X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563–584.
    [7] K. A. Pawelek, S. Liu, F. Pahlevani, et al., A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98–109.
    [8] A. S. Perelson, D. E. Kirschner and R. de Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81–125.
    [9] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3–44.
    [10] P. K. Srivastava, M. Banerjee and P. Chandra, A primary infection model for HIV and immune response with two discrete time delays, Differ. Equ. Dyn. Syst., 18 (2010), 385–399.
    [11] P. K. Srivastava and P. Chandra, Hopf bifurcation and periodic solutions in a dynamical model for HIV and immune response, Differ. Equ. Dyn. Syst., 16 (2008), 77–100.
    [12] H. Wang, R. Xu, Z. Wang, et al., Global dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Anal. Model. Control, 20 (2015), 21–37.
    [13] X. Wang, Y. Lou and X. Song, Age-structured within-host HIV dynamics with multiple target cells, Stud. Appl. Math., 138 (2017), 43–76.
    [14] X. Wang, G. Mink, D. Lin, et al., Influence of raltegravir intensification on viral load and 2-LTR dynamics in HIV patients on suppressive antiretroviral therapy, J. Theor. Biol., 416 (2017), 16–27.
    [15] X. Wang, X. Song, S. Tang, et al., Dynamics of an HIV model with multiple infection stages and treatment with different drug classes, Bull. Math. Biol., 78 (2016), 322–349.
    [16] Y. Wang, Y. Zhou, F. Brauer, et al., Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901–934.
    [17] N. M. Dixit, M. Markowitz, D. D. Ho, et al., Estimates of intracellular delay and average drug efficacy from viral load data of HIV-infected individuals under antiretroviral therapy, Antivir. Ther., 9 (2004), 237–246.
    [18] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, et al., Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. Nat. Acad. Sci. USA, 93 (1996), 7247– 7251.
    [19] K. Allali, S. Harroudi and D. F. M. Torres, Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, Math. Comput. Sci., 12 (2018), 111–127.
    [20] M. S. Ciupe, B. L. Bivort, D. M. Bortz, et al., Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1–27.
    [21] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T cells, ath. Biosci., 165 (2000), 27–39.
    [22] R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell HIV-1 that include a time delay, J. Math. Biol., 46 (2003), 425–444.
    [23] B. Li, Y. Chen, X. Lu, et al., A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135–157.
    [24] Y. Liu and C. Wu, Global dynamics for an HIV infection model with Crowley-Martin functional response and two distributed delays, J. Syst. Sci. Complex., 31 (2018), 385–395.
    [25] Y. Wang, Y. Zhou, J. Wu, et al., Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci., 219 (2009), 104–112.
    [26] H. Zhu and X. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511–524.
    [27] N. Tarfulea, A. Blink, E. Nelson, et al., A CTL-inclusive mathematical model for antiretroviral treatment of HIV infection, Int. J. Biomath., 4 (2011), 1–22.
    [28] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that Includes an intracellular delay, Math. Biosci., 163 (2000), 201–215.
    [29] J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer- Verlag, New York, 1993.
    [30] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188–211.
    [31] J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, with Applications, Academic Press, New York, 1961.
    [32] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144–1165.
    [33] X. Yan andW. Li, Stability and bifurcation in a simplified four-neural BAM network with multiple delays, Discrete Dyn. Nat. Soc., 2006 (2006), 1–29.
    [34] M. Y. Li and H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774–1793.
    [35] M. Y. Li, X. Lin and H. Wang, Global Hopf branches in a delayed model for immune response to HTLV-1 infections: coexistence of multiple limit cycles, Can. Appl. Math. Q., 20 (2012), 39–50.
    [36] A. Debadatta and B. Nandadulal, Analysis and computation of multi-pathways and multi-delays HIV-1 infection model, Appl. Math. Model., 54 (2018), 517–536.
    [37] B. M. Adams, H. T. Banks, M. Davidian, et al., HIV dynamics: modeling, data analysis, and optimal treatment protocols, J. Comput. Appl. Math., 184 (2005), 10–49.
  • This article has been cited by:

    1. A. M. Elaiw, S. F. ALSHEHAIWEEN, A. D. HOBINY, GLOBAL PROPERTIES OF HIV DYNAMICS MODELS INCLUDING IMPAIRMENT OF B-CELL FUNCTIONS, 2020, 28, 0218-3390, 1, 10.1142/S0218339020500011
    2. A. M. Elaiw, S. F. Alshehaiween, Global stability of delay‐distributed viral infection model with two modes of viral transmission and B‐cell impairment, 2020, 43, 0170-4214, 6677, 10.1002/mma.6408
    3. A. M. Elaiw, N. H. AlShamrani, A. D. Hobiny, I. A. Abbas, Global stability of an adaptive immunity HIV dynamics model with silent and active cell-to-cell transmissions, 2020, 10, 2158-3226, 085216, 10.1063/5.0017214
    4. N. H. AlShamrani, A. M. Elaiw, H. Dutta, Stability of a delay-distributed HIV infection model with silent infected cell-to-cell spread and CTL-mediated immunity, 2020, 135, 2190-5444, 10.1140/epjp/s13360-020-00594-3
    5. Ahmed M. Elaiw, Safiya F. Alshehaiween, Aatef D. Hobiny, Global Properties of a Delay-Distributed HIV Dynamics Model Including Impairment of B-Cell Functions, 2019, 7, 2227-7390, 837, 10.3390/math7090837
    6. A. M. Elaiw, N. H. AlShamrani, Impact of adaptive immune response and cellular infection on delayed virus dynamics with multi-stages of infected cells, 2020, 13, 1793-5245, 2050003, 10.1142/S1793524520500035
    7. Ahmed Elaiw, Noura AlShamrani, Stability of a general adaptive immunity virus dynamics model with multistages of infected cells and two routes of infection, 2020, 43, 0170-4214, 1145, 10.1002/mma.5923
    8. A.M. Elaiw, N.H. AlShamrani, Global stability of a delayed adaptive immunity viral infection with two routes of infection and multi-stages of infected cells, 2020, 86, 10075704, 105259, 10.1016/j.cnsns.2020.105259
    9. A. M. Elaiw, M. A. Alshaikh, Global stability of discrete virus dynamics models with humoural immunity and latency, 2019, 13, 1751-3758, 639, 10.1080/17513758.2019.1683630
    10. N.H. AlShamrani, Stability of a general adaptive immunity HIV infection model with silent infected cell-to-cell spread, 2021, 09600779, 110422, 10.1016/j.chaos.2020.110422
    11. A. M. Elaiw, N. H. AlShamrani, STABILITY OF A DELAYED ADAPTIVE IMMUNITY HIV INFECTION MODEL WITH SILENT INFECTED CELLS AND CELLULAR INFECTION, 2021, 11, 2156-907X, 964, 10.11948/20200124
    12. A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta, Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection, 2020, 0, 1937-1179, 0, 10.3934/dcdss.2020441
    13. A. M. Elaiw, N. H. AlShamrani, Modeling and stability analysis of HIV/HTLV-I co-infection, 2021, 14, 1793-5245, 2150030, 10.1142/S1793524521500303
    14. N. H. AlShamrani, Stability of an HTLV-HIV coinfection model with multiple delays and CTL-mediated immunity, 2021, 2021, 1687-1847, 10.1186/s13662-021-03416-7
    15. Yu Yang, Gang Huang, Yueping Dong, Stability and Hopf bifurcation of an HIV infection model with two time delays, 2022, 20, 1551-0018, 1938, 10.3934/mbe.2023089
    16. Dianhong Wang, Yading Liu, Xiaojie Gao, Chuncheng Wang, Dejun Fan, Dynamics of an HIV infection model with two time delays, 2023, 0, 1531-3492, 0, 10.3934/dcdsb.2023069
    17. Liang Hong, Jie Li, Libin Rong, Xia Wang, Global dynamics of a delayed model with cytokine-enhanced viral infection and cell-to-cell transmission, 2024, 9, 2473-6988, 16280, 10.3934/math.2024788
    18. B. S. Alofi, S. A. Azoz, Stability of general pathogen dynamic models with two types of infectious transmission with immune impairment, 2021, 6, 2473-6988, 114, 10.3934/math.2021009
    19. Zitong Li, Zhe Zhang, Stabilization Control for a Class of Fractional-Order HIV-1 Infection Model with Time Delays, 2023, 12, 2075-1680, 695, 10.3390/axioms12070695
    20. Hanane Hmarrass, Redouane Qesmi, Global stability and Hopf bifurcation of a delayed HIV model with macrophages, CD4+T cells with latent reservoirs and immune response, 2025, 140, 2190-5444, 10.1140/epjp/s13360-025-06001-z
    21. Lili Lv, Junxian Yang, Zihao Hu, Dongmei Fan, Dynamics Analysis of a Delayed HIV Model With Latent Reservoir and Both Viral and Cellular Infections, 2025, 48, 0170-4214, 6063, 10.1002/mma.10655
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6348) PDF downloads(850) Cited by(21)

Figures and Tables

Figures(12)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog