We study the global dynamics of an HIV infection model describing the interaction of the HIV with CD4+ T cells and macrophages. The incidence rate of virus infection and the growth rate of the uninfected CD4+ T cells and macrophages are given by general functions. We have incorporated two types of distributed delays into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious (matures) virus particles. We have established a set of conditions which are sufficient for the global stability of the steady states of the model. Using Lyapunov functionals and LaSalle's invariant principle, we have proven that if the basic reproduction number R0 is less than or equal to unity, then the uninfected steady state is globally asymptotically stable (GAS), and if the infected steady state exists, then it is GAS.
1. Introduction
Recently, many mathematical models have been developed to describe the interaction of human immunodeficiency virus (HIV) with the immune system [1]. Those mathematical models can provide some insights into the dynamics of HIV viral load in vivo and may play a significant role in the development of a better understanding of HIV/AIDS and drug therapies. For example, they provided a quantitative understanding of the level of virus production during the long asymptomatic stage of HIV infection [2].
The basic mathematical model describing the dynamics of HIV infection of CD4+ T cells is given by [3]
(1)x˙=λ-dx-βxv,(2)y˙=βxv-ay,(3)v˙=ky-cv,
where x, y, and v represent the populations of the uninfected CD4+ T cells, infected cells, and free virus particles, respectively. The uninfected cells are generated from sources within the body at a rate λ. The parameter d is the death rate constant of the uninfected cells. The incidence rate is given by the bilinear form βxv, where β is the rate constant characterizing infections of the cells. Equation (2) describes the population dynamics of the infected cells and shows that they die with rate constant a. The virus particles are produced by the infected cells with a rate constant k and are cleared from plasma with a rate constant c.
Many researchers suggested that the bilinear incidence rate is insufficient to describe the infection process in detail [4–8]. Therefore, different forms of the incidence rate of infection have been proposed such as saturated incidence rate βxv/(1+γv) [4], Holling type II functional response βxv/(1+αx) [5], Beddington-DeAngelis infection rate βxv/(1+αx+γv) [6], Crowley-Martin functional response βxv/((1+αx)(1+γv)) [7, 8], where α, γ≥0, and a general function ψ(x,v) [9].
In model (1)–(3), it is assumed that the infection could occur and the viruses are produced from infected cells instantaneously, once the uninfected cells are contacted by the virus particles. However, this assumption is unrealistic. Therefore, more realistic HIV dynamics models incorporate the delay between the time of viral entry into the uninfected cell and the time for the production of new virus particles, modeled with discrete time delay or distributed time delay using functional differential equations (see, e.g., [2, 5, 10–18]). As pointed by Li and Shu [18], the period between the time for HIV to enter the target cell and the time for new virions to be produced from the infected cell needs the following stages: (i) the period between the viral entry of a target cell and integration of viral DNA into the host genome, (ii) the period from the integration of viral DNA to the transcription of viral RNA and translation of viral proteins such as reverse transcriptase, integrase, and protease, and (iii) the period between the transcription of viral RNA and the release and maturation of virus [18].
All of the afore mentioned delayed HIV infection models are mainly modeled with the interaction of the HIV with one class of target cells, CD4+ T cells. More accurate modeling was developed in 1997 when Perelson et al. [19] observed that, after the rapid first phase of decay during the initial 1-2 weeks of antiretroviral treatment, plasma virus levels declined at a considerably slower rate. This second phase of viral decay was attributed to the turnover of a longer-lived virus reservoir of infected cells. These cells are called macrophages and considered as the second target cell for the HIV. To model the second class of target cells, two additional equations describing the population dynamics of the uninfected and infected macrophages have to be added to the basic model (1)–(3) (see [20, 21]). In [20–27], the HIV models have been proposed to describe the HIV dynamics with the CD4+ T cells and macrophages. The global stability analysis of these models has been investigated in [24–27]. Elaiw [24] studied the global properties of HIV infection model with bilinear and nonlinear incidence rates. In [25], Beddington-DeAngelis functional response has been considered. In [24, 25], the effect of time delay is neglected. Elaiw et al. [26] studied the global stability of HIV model with Beddington-DeAngelis functional response and one kind of discrete time delay. Elaiw [27] studied the global dynamics of a delay HIV model with saturated functional response.
In this paper, we study the global dynamics of HIV infection of CD4+ T cells and macrophages. We assume that the incidence rate and the growth rate of the uninfected cells are given by general functions. We incorporate two types of distributed delays into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious (mature) virus particles. We established a set of conditions which are sufficient for the global stability of the steady states of the model. Using Lyapunov functionals and LaSalle’s invariant principle, we prove that if the basic reproduction number R0 is less than or equal to unity, then the uninfected steady state is globally asymptotically stable (GAS), and if the infected steady state exists, then it is GAS.
2. The Model
In this sections we propose an HIV dynamics model which describes the interaction of the HIV with the CD4+ T cells and macrophages. Two types of distributed time delays are incorporated into the model
(4)x˙i=ri(xi)-ψi(xi,v),i=1,2,(5)y˙i=βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi,i=1,2,(6)v˙=∑i=12pi∫0ligi(ω)yi(t-ω)dω-cv,
where xi and yi represent the populations of the uninfected and infected cells, respectively, where i=1 and 2 correspond to CD4+ T cells and macrophages. Parameters βi′ and pi satisfy βi′<1 and pi<ki, i=1,2. All the variables and other parameters of the model have the same meanings as given in model (1)–(3). To take into account the delay between viral infection of an uninfected target cell and the production of an actively infected target cell, we let τ be the random variable that describes the time between viral entry and the transcription of viral RNA (stages (i) and (ii)) with a probability distribution fi(τ) over the interval [0,hi] and hi is limit superior of this delay. On the other hand, to consider the delay between viral RNA transcription and viral release and maturation, we let ω be the random variable that is the time between these two events with a probability distribution gi(ω) over the interval [0,li] and li is limit superior of this delay [18]. The probability distribution functions fi(τ):[0,hi]→ℝ+ and gi(ω):[0,li]→ℝ+ are integral functions with
(7)∫0hifi(τ)dτ=∫0ligi(ω)dω=1,i=1,2.
Function ψi(xi,v) represents the rate for the uninfected target cell of class i to be infected by the mature viruses. Special forms of function ψi(xi,v) have been presented in the literature as follows:
The growth rate of the uninfected cells is given by general function ri(xi). The following particular forms of function ri(xi) have widely been used in the literature of HIV dynamics:
(8)r¯i(xi)=λi-dxi,r~i(xi)=λi-dxi+bixi(1-xixi,max),
where bi is the maximum proliferation rate of the target cells of class i and xi,max is the maximum level of uninfected cells population in the body [13, 16, 31].
Initial Conditions. The initial conditions for system (4)–(6) take the form
(9)x1(θ)=φ1(θ),x2(θ)=φ2(θ),y1(θ)=φ3(θ),y2(θ)=φ4(θ),v(θ)=φ5(θ),φj(θ)≥0,θ∈-[ℓ,0),φj(0)>0,j=1,…,5,ℓ=max{h1,h2,l1,l2},
where (φ1(θ),φ2(θ),…,φ5(θ))∈C and C=C([-ℓ,0],ℝ+5) is the Banach space of continuous functions mapping the interval [-ℓ,0] into ℝ+5. By the fundamental theory of functional differential equations [32], system (4)–(6) has a unique solution (x1(t),x2(t),y1(t),y2(t),v(t)) satisfying initial conditions (9).
We assume that functions ri and ψi satisfy the following assumptions.
Assumption 1.
Function ri:[0,∞)→ℝ satisfies the following:
ri(xi) is continuous and differentiable and ri(0)>0,
there exits an xi0>0 such that
(10)ri(xi0)=0,ri′(xi0)<0,(xi-xi0)ri(xi)≤0,xi≠xi0.
Assumption 2.
(i) ψi(xi,v) is positive, continuous, and differentiable for all xi>0, and v>0,
(ii) ψi(0,v)=ψi(xi,0)=0, ∂ψi(xi,0)/∂v>0 for any x>0, v>0, i=1,2,
(iii) ∂ψi(xi,v)/∂xi>0, ∂ψi(xi,v)/∂v>0 for all xi>0, v>0, i=1,2.
2.1. Nonnegativity and Boundedness of Solutions
In the following, we establish the nonnegativity and boundedness of solutions of (4)–(6) with initial conditions (9).
Proposition 3.
Assume that Assumptions 1 and 2 are satisfied. Let X(t)=(x1(t),x2(t),y1(t),y2(t),v(t))T be any solution of (4)–(6) satisfying the initial conditions (9); then X(t) is nonnegative for t≥0 and ultimately bounded.
Proof.
First, we prove that xi(t)>0, i=1,2, for all t≥0. Assume that xi(t) loses its nonnegativity on some local existence interval [0,ρ] for some constant ρ and let ti*∈[0,ρ] be such that xi(ti*)=0, i=1,2. From (4), we have
(11)x˙i(ti*)=ri(xi(ti*))-ψi(xi(ti*),v(ti*))=ri(0)-ψi(0,v(ti*))=ri(0)>0,i=1,2.
Hence, xi(t)>0 for some t∈(ti*,ti*+ε), where ε>0 is sufficiently small. This leads to a contradiction and hence xi(t)>0 for all t≥0. Further, from (5) and (6), we have
(12)yi(t)=yi(0)e-ait+βi′∫0te-ai(t-η)×∫0hifi(τ)ψi(xi(η-τ),v(η-τ))dτdη,i=1,2,v(t)=v(0)e-ct+∑i=12pi∫0te-c(t-η)∫0ligi(ω)yi(η-ω)dωdη,
confirming that yi(t)≥0, i=1,2, and v(t)≥0 for all t∈[0,ℓ]. By a recursive argument, we obtain yi(t)≥0, i=1,2, and v(t)≥0 for all t≥0.
Now, we show the boundedness of the solutions of (4)–(6). Assumption 1 and (4) imply that limsupt→∞xi(t)≤xi0. It follows that
(13)∫0hifi(τ)xi(t-τ)dτ≤xi0.
Let Ti(t)=βi′∫0hifi(τ)xi(t-τ)dτ+yi(t), Si=supxi∈[0,xi0]ri(xi) and a¯i≤min{ai,Si/xi0}, i=1,2; then
(14)T˙i(t)=∫0hiβi′fi(τ)(ri(xi(t-τ))-ψi(xi(t-τ),v(t-τ)))dτ+β′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi(t)≤βi′Si-aiyi(t)≤βi′Si-aiyi(t)+βi′Si-a¯iβi′∫0hifi(τ)xi(t-τ)dτ≤2βi′Si-a¯iTi(t).
Hence, limsupt→∞Ti(t)≤Li, where Li=2βi′Si/a¯i. Since ∫0hifi(τ)xi(t-τ)dτ>0, we get limsupt→∞yi(t)≤Li. On the other hand,
(15)v˙(t)≤∑i=12piLi∫0ligi(τ)dτ-cv=∑i=12piLi-cv.
Then limsupt→∞v(t)≤L*, where L*=∑i=12(piLi/c). Therefore, X(t) is ultimately bounded.
2.2. Steady States
Let Assumptions 1 and 2 be satisfied; then system (4)–(6) has an uninfected steady state E0(x10,x20,y10,y20,v0), where xi0 is defined in Assumption 1, yi0=0, i=1,2, and v0=0. The system can also have another steady state E*(x1*,x2*,y1*,y2*,v*) which is the infected steady state; the coordinates of the infected steady state, if they exist, satisfy the equalities
(16)ri(xi*)=ψi(xi*,v*),i=1,2,aiyi*=βi′ψi(xi*,v*),i=1,2,cv*=∑i=12piyi*.
We define the basic reproduction number for system (4)–(6) as
(17)R0=∑i=12Ri=∑i=12piβi′aic∂ψi(xi0,0)∂v.
The term ∂ψi(xi0,0)/∂v denotes the maximal average number of target cells i that each virus infects, and Ri is the basic reproduction number for the dynamics of the virus and the uninfected cell of class i.
2.3. Global Stability Analysis
In this section, we establish a set of conditions which are sufficient for the global stability of the uninfected and infected steady states of system (4)–(6). The strategy of the proofs is to use suitable Lyapunov functionals which are similar in nature to those used in [33, 34]. We will use the following notation: z=z(t) for any z∈{xi,yi,v,i=1,2}. We also define a function H:(0,∞)→[0,∞) as
(18)H(z)=z-1-lnz.
It is clear that H(z)≥0 for any z>0 and H has the global minimum H(1)=0.
Assumption 4.
ψi(xi,v)≤v(∂ψi(xi,0)/∂v), for all v>0.
Assumption 5.
(1-(∂ψi(xi0,0)/∂v)/(∂ψi(xi,0)/∂v))ri(xi)≤0, xi>0.
The global stability of the uninfected steady state will be established in the next theorem.
Theorem 6.
If Assumptions 1–5 hold true and R0≤1, then E0 is GAS.
Proof.
Define a Lyapunov functional W0 as follows:
(19)W0=∑i=12piai[βi′(xi-xi0-∫xi0xilimv→0+ψi(xi0,v)ψi(θ,v)dθ)+yi+βi′∫0hiδfi,hi(τ)ψi(xi(t-τ),v(t-τ))dτ+ai∫0liδgi,li(ω)yi(t-ω)dωψi(xi0,v)ψi(θ,v)]+v,
where
(20)δfi,hi(τ)=∫τhifi(σ)dσ,δgi,li(ω)=∫ωligi(σ)dσ,i=1,2.
Functions δfi,hi and δgi,li satisfy
(21)δfi,hi(0)=1,δfi,hi(hi)=0,dδfi,hi(τ)dτ=-fi(τ),δgi,li(0)=1,δgi,li(li)=0,dδgi,li(ω)dω=-gi(ω).
We note that W0 is defined and continuous for all (x1,x2,y1,y2,v)>0. Also, the global minimum W0=0 occurs at the uninfected steady state E0. The time derivative of W0 along the solution of (4)–(6) is given by
(22)dW0dt=∑i=12piai[βi′(1-limv→0+ψi(xi0,v)ψi(xi,v))(ri(xi)-ψi(xi,v))+βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi+βi′∫0hifi(τ)ψi(xi,v)-ψi(xi(t-τ),v(t-τ))dτ+ai∫0ligi(ω)(yi-yi(t-ω))dωβi′(1-limv→0+ψi(xi0,v)ψi(xi,v))]+∑i=12pi∫0ligi(ω)yi(t-ω)dω-cv=∑i=12piai[βi′(1-limv→0+ψi(xi0,v)ψi(xi,v))ri(xi)+βi′ψi(xi,v)limv→0+ψi(xi0,v)ψi(xi,v)]-cv=∑i=12piai[βi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+βi′ψi(xi,v)∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v]-cv≤∑i=12piai[βi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+βi′v∂ψi(xi,0)∂v(∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)]-cv=∑i=12piaiβi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+(∑i=12piβi′aic∂ψi(xi0,0)∂v-1)cv=∑i=12piaiβi′(1-∂ψi(xi0,0)/∂v∂ψi(xi,0)/∂v)ri(xi)+(R0-1)cv.
The first term of (22) is less than or equal to zero according to Assumption 5, so it can be seen that if R0≤1, then dW0/dt≤0 for all xi,v>0, i=1,2. By Theorem 5.3.1 in [32], the solutions of system (4)–(6) limit to M, the largest invariant subset of {dW0/dt=0}. Clearly, it follows from (22) that dW0/dt=0 if and only if xi=xi0, i=1,2, and v=0. Noting that M is invariant, for each element of M, we have v=0, and then v˙=0. From (6), we derive that
(23)0=v˙=∑i=12pi∫0lig(ω)yi(t-ω)dω.
Since yi(t-θ)≥0 for all θ∈[0,ℓ], then ∑i=12pi∫0lig(ω)yi(t-ω)dω=0 if and only if yi(t-ω)=0, i=1,2. Hence, dW0/dt=0 if and only if xi=xi0, yi=0, i=1,2, and v=0. From LaSalle’s invariance principle, E0 is GAS.
To establish the global stability of the infected steady state, we need the following conditions.
Assume that Assumptions 1, 2, 7, and 8 hold true and E* exists; then E* is GAS.
Proof.
Define Lyapunov functional W1 as follows:
(24)W1=∑i=12piai[βi′(xi-xi*-∫xi*xiψi(xi*,v*)ψi(θ,v*)dθ)+yi*(yiyi*-1-lnyiyi*)+βi′ψi(xi*,v*)×∫0hiδfi,hi(τ)H(ψi(xi(t-τ),v(t-τ))ψi(xi*,v*))dτ+aiyi*∫0liδgi,li(ω)H(yi(t-ω)yi*)dω]+v*(vv*-1-lnvv*).
By calculating the time derivative along (4)–(6), we get
(25)dW1dt=∑i=12piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ψi(xi,v))+(1-yi*yi)×(βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi)+βi′ψi(xi*,v*)×∫0hifi(τ)(ψi(xi,v)ψi(xi*,v*)-ψi(xi(t-τ),v(t-τ))ψi(xi*,v*)+lnψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ+aiyi*∫0ligi(ω)(yiyi*-yi(t-ω)yi*+lnyi(t-ω)yi)dω]+(1-v*v)(∑i=12pi∫0ligi(ω)yi(t-ω)dω-cv).
Collecting terms of (25), we obtain
(26)dW1dt=∑i=12piai×[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+βi′ri(xi*)(1-ψi(xi*,v*)ψi(xi,v*))-βi′ψi(xi,v)+βi′ψi(xi*,v*)ψi(xi,v)ψi(xi,v*)+βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ-aiyi-yi*yiβi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ+aiyi*+βi′ψi(xi,v)-βi′∫0hifi(τ)ψi(xi(t-τ),v(t-τ))dτ+βi′ψi(xi*,v*)×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ+aiyi-ai∫0ligi(ω)(yi(t-ω))dω+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω]+∑i=12pi∫0ligi(ω)yi(t-ω)dω-cv-∑i=12pi∫0ligi(ω)v*yi(t-ω)vdω+cv*.
Using the infected steady state conditions (16), we get
(27)dW1dt=∑i=12piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+aiyi*(1-ψi(xi*,v*)ψi(xi,v*))+aiyi*ψi(xi,v)ψi(xi,v*)-aiyi*∫0hifi(τ)yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*)dτ+2aiyi*+aiyi*×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ-aiyi*∫0ligi(ω)v*yi(t-ω)vyi*dτ+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω-aiyi*vv*]=∑i=12piai[βi′(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)-ri(xi*))+aiyi*×(-1+ψi(xi,v)ψi(xi,v*)-vv*+vv*ψi(xi,v*)ψi(xi,v))-aiyi*ψi(xi*,v*)ψi(xi,v*)-aiyi*vv*ψi(xi,v*)ψi(xi,v)-aiyi*∫0hifi(τ)yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*)dτ+4aiyi*+aiyi*×∫0hifi(τ)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))dτ-aiyi*∫0ligi(ω)v*yi(t-ω)vyi*dτ+aiyi*∫0ligi(ω)ln(yi(t-ω)yi)dω].
Using the following equalities:
(28)ln(ψi(xi(t-τ),v(t-τ))ψi(xi,v))=ln(yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*))+ln(ψi(xi*,v*)ψi(xi,v*))+ln(vv*ψi(xi,v*)ψi(xi,v))+ln(v*yivyi*),ln(yi(t-ω)yi)=ln(vyi*v*yi)+ln(v*yi(t-ω)vyi*),
we obtain
(29)dW1dt=∑i=12piai[aiyi*(1-ψi(xi*,v*)ψi(xi,v*))(ri(xi)ri(xi*)-1)+aiyi*(ψi(xi,v)ψi(xi,v*)-vv*)(1-ψi(xi,v*)ψi(xi,v))-aiyi*H(ψi(xi*,v*)ψi(xi,v*))-aiyi*H(vv*ψi(xi,v*)ψi(xi,v))-aiyi*∫0hifi(τ)×H(yi*ψi(xi(t-τ),v(t-τ))yiψi(xi*,v*))dτ-aiyi*∫0ligi(ω)H(v*yi(t-ω)vyi*)dω].
Clearly, if E* exists, then dW1/dt≤0 for all xi,yi,v>0, i=1,2, and dW1/dt=0 if and only if xi=xi*, yi=yi*, and v=v*, which is the infected steady state E*. It follows that E* is GAS.
3. Conclusion
In this paper, we have proposed an HIV dynamics model describing the interaction of the HIV with CD4+ T cells and macrophages. The incidence rate of virus infection and the growth rate of the uninfected cells are given by general functions. Two types of distributed delays have been incorporated into the model to account for the time delay between the time the uninfected cells are contacted by the virus particle and the time for the emission of infectious (mature) virus particles. We have established a set of conditions which are sufficient for the global stability of the steady states of the model. Using Lyapunov functionals and LaSalle's invariant principle, we have proven that if R0≤1, then the uninfected steady state is GAS, and if the infected steady state exists, then it is GAS.
Acknowledgments
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks the DSR technical and financial support. The authors are grateful to the anonymous reviewers for constructive suggestions and valuable comments, which improve the quality of the paper.
NowakM. A.MayR. M.2000Oxford, UKOxford University PressMR2009143ZBL1206.83102YuanZ.ZouX.Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays2013102483498MR301944310.3934/mbe.2013.10.483ZBL1260.92088NowakM. A.BanghamC. R. M.Population dynamics of immune responses to persistent viruses1996272525874792-s2.0-0029985351SongX.NeumannA. U.Global stability and periodic solution of the viral dynamics2007329128129710.1016/j.jmaa.2006.06.064MR2306802ZBL1105.92011HuangG.YokoiH.TakeuchiY.KajiwaraT.SasakiT.Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics201128338341110.1007/s13160-011-0045-xMR2846182ZBL1226.92049HuangG.MaW.TakeuchiY.Global properties for virus dynamics model with Beddington-DeAngelis functional response200922111690169310.1016/j.aml.2009.06.004MR2569065ZBL1178.37125XuS.Global stability of the virus dynamics model with Crowley-Martin functional response201310.1002/mma.2895MR2878794ZhouX.CuiJ.Global stability of the viral dynamics with Crowley-Martin functional response201148355557410.4134/BKMS.2011.48.3.555MR2827765KorobeinikovA.Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence, and non-linear incidence rate200926225239CulshawR. V.RuanS.A delay-differential equation model of HIV infection of CD4+ T-cells2000165127392-s2.0-003410866210.1016/S0025-5564(00)00006-7NelsonP. W.MurrayJ. D.PerelsonA. S.A model of HIV-1 pathogenesis that includes an intracellular delay2000163220121510.1016/S0025-5564(99)00055-3MR1740580ZBL0942.92017DixitN. M.PerelsonA. S.Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay200422619510910.1016/j.jtbi.2003.09.002MR2068328LiuS.WangL.Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy20107367568510.3934/mbe.2010.7.675MR2740568ZBL1260.92065HuangG.MaW.TakeuchiY.Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response20112471199120310.1016/j.aml.2011.02.007MR2784182ZBL1217.34128NelsonP. W.PerelsonA. S.Mathematical analysis of delay differential equation models of HIV-1 infection20021791739410.1016/S0025-5564(02)00099-8MR1908737ZBL0992.92035XuR.Global dynamics of an HIV-1 infection model with distributed intracellular delays20116192799280510.1016/j.camwa.2011.03.050MR2795403ZBL1221.37204HuangG.TakeuchiY.MaW.Lyapunov functionals for delay differential equations model of viral infections20107072693270810.1137/090780821MR2678058ZBL1209.92035LiM. Y.ShuH.Impact of intracellular delays and target-cell dynamics on in vivo viral infections20107072434244810.1137/090779322MR2678046ZBL1209.92037PerelsonA. S.EssungerP.CaoY.VesanenM.HurleyA.SakselaK.MarkowitzM.HoD. D.Decay characteristics of HIV-1-infected compartments during combination therapy199738766291881912-s2.0-003095247910.1038/387188a0CallawayD. S.PerelsonA. S.HIV-1 infection and low steady state viral loads200264129642-s2.0-003638111010.1006/bulm.2001.0266PerelsonA. S.NelsonP. W.Mathematical analysis of HIV-1 dynamics in vivo199941134410.1137/S0036144598335107MR1669741ZBL1078.92502ElaiwA. M.XiaX.HIV dynamics: analysis and robust multirate MPC-based treatment schedules2009359128530110.1016/j.jmaa.2009.05.038MR2542175ElaiwA. M.ShehataA. M.Stability and feedback stabilization of HIV infection model with two classes of target cells201220122010.1155/2012/963864963864MR2965724ZBL1248.93134ElaiwA. M.Global properties of a class of HIV models20101142253226310.1016/j.nonrwa.2009.07.001MR2661895ZBL1197.34073ElaiwA. M.AzozS. A.Global properties of a class of HIV infection models with Beddington-DeAngelis functional response201336438339410.1002/mma.2596MR3032352ElaiwA.HassanienI.AzozS.Global stability of HIV infection models with intracellular delays201249477979410.4134/JKMS.2012.49.4.779MR2976099ZBL1256.34068ElaiwA. M.Global dynamics of an HIV infection model with two classes of target cells and distributed delays201220121310.1155/2012/253703253703MR2965727ZBL1253.37082ElaiwA. M.Global properties of a class of virus infection models with multitarget cells2012691-242343510.1007/s11071-011-0275-0MR2929883ZBL1254.92064ElaiwA. M.AlghamdiM. A.Global properties of virus dynamics models with multitarget cells and discrete-time delays201120111920127410.1155/2011/201274MR2861950ZBL1233.92059BakrA. A.ElaiwA. M.RaizahZ. A. S.Mathematical analysis of virus dynamics model with multitarget cells in vivo201323ObaidM. A.Global analysis of a virus infection model with multitarget cells and distrib-
uted intracellular delays20129415001508HaleJ. K.Verduyn LunelS. M.199399New York, NY, USASpringerApplied Mathematical SciencesMR1243878McCluskeyC. C.Complete global stability for an SIR epidemic model with delay—distributed or discrete2010111555910.1016/j.nonrwa.2008.10.014MR2570523ZBL1185.37209KajiwaraT.SasakiT.TakeuchiY.Construction of Lyapunov functionals for delay differential equations in virology and epidemiology20121341802182610.1016/j.nonrwa.2011.12.011MR2891011ZBL1257.34053