DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 781407 10.1155/2013/781407 781407 Research Article Global Dynamics of Virus Infection Model with Antibody Immune Response and Distributed Delays Elaiw A. M. 1, 2 Alhejelan A. 1, 3 Alghamdi M. A. 1 Kozyakin Victor S. 1 Department of Mathematics Faculty of Science King Abdulaziz University P.O. Box 80203, Jeddah 21589 Saudi Arabia kau.edu.sa 2 Department of Mathematics Faculty of Science Al-Azhar University, Assiut Egypt 3 Department of Mathematics Faculty of Arts and Science Qassim University, Buraidah 71511 Saudi Arabia qu.edu.sa 2013 3 12 2013 2013 08 08 2013 07 10 2013 2013 Copyright © 2013 A. M. Elaiw et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present qualitative behavior of virus infection model with antibody immune response. The incidence rate of infection is given by saturation functional response. Two types of distributed delays are incorporated into the model to account for the time delay between the time when uninfected cells are contacted by the virus particle and the time when emission of infectious (matures) virus particles. Using the method of Lyapunov functional, we have established that the global stability of the steady states of the model is determined by two threshold numbers, the basic reproduction number R0 and antibody immune response reproduction number R1. We have proven that if R01, then the uninfected steady state is globally asymptotically stable (GAS), if R11<R0, then the infected steady state without antibody immune response is GAS, and if R1>1, then the infected steady state with antibody immune response is GAS.

1. Introduction

In the past ten years there has been a growing interest in modeling viral infections for the study and characterization of host infection dynamics. The mathematical models, based on biological interactions, present a framework which can be used to obtain new insights into the viral dynamics and to interpret experimental data. Many authors have formulated mathematical models to describe the population dynamics of several viruses such as, human immunodeficiency virus (HIV) , hepatitis B virus (HBV) [15, 16], and hepatitis C virus (HCV) [17, 18]. During viral infections, the host immune system reacts with antigen-specific immune response. In particular, B cells play a critical role in antiviral defense by attacking free virus particles by making antibodies to clear antigens circulating in blood and lymph. The antibody immune response is more effective than the cell-mediated immune in some diseases like in malaria infection .

Mathematical models for virus dynamics with the antibody immune response have been developed in . The basic virus dynamics model with antibody immune response was introduced by Murase et al.  as (1)x˙(t)=λ-dx(t)-βx(t)v(t),y˙(t)=βx(t)v(t)-δy(t),v˙(t)=Nδy(t)-cv(t)-qv(t)z(t),z˙(t)=rv(t)z(t)-μz(t), where x(t), y(t), v(t), and z(t) represent the populations of uninfected cells, infected cells, virus, and B cells at time t, respectively; λ and d are the recruited rate and death rate constants of uninfected cells, respectively; β is the infection rate constant; N is the number of free virus produced during the average infected cell life span; δ is the death rate constant of infected cells; c is the clearance rate constant of the virus; r and μ are the recruited rate and death rate constants of B cells; and q is the B cells neutralizion rate. Model (1) is based on the assumption that the infection could occur and that the viruses are produced from infected cells instantaneously, once the uninfected cells are contacted by the virus particles. Other accurate models incorporate the delay between the time viral entry into the uninfected cell and the time the production of new virus particles, modeled with discrete time delay or distributed time delay using functional differential equations (see e.g., [714, 27]). In [714, 27], the viral infection models are presented without taking into consideration the antibody immune response. In [25, 28], global stability of viral infection models with antibody immune response and discrete delays has been studied.

In model (1) the infection rate is assumed to be bilinear in x and v however, this bilinear incidence rate associated with the mass action principle is insufficient to describe the infection process in detail [29, 30]. For example, a less than linear response in v could occur due to saturation at high virus concentration, where the infectious fraction is high so that exposure is very likely. Thus, it is reasonable to assume that the infection rate is given by saturation functional response . In , a virus infection model with antibody immune response and with saturation incidence rate has been considered. However, the time delay was not considered.

In this paper, we assume that the infection rate is given by saturation functional response. We incorporate two types of distributed delays into the model to account for the time delay between the time when uninfected cells are contacted by the virus particle and the time of emission of infectious (matures) virus particles. The global stability of this model is established using Lyapunov functionals, which are similar in nature to those used in . We prove that the global dynamics of this model is determined by the basic reproduction number R0 and antibody immune response reproduction number R1. If R01, then the uninfected steady state is globally asymptotically stable (GAS), if R11<R0, then the infected steady state without antibody immune response is GAS, and if R1>1, then the infected steady state with antibody immune response is GAS.

2. The Model

In this section we propose a delay mathematical model of viral infection with saturation functional response which describes the interaction of the virus with uninfected and infected cells, taking into account the effect of antibody immune response. Consider (2)x˙(t)=λ-dx(t)-βx(t)v(t)1+αv(t),(3)y˙(t)=β0hf(τ)e-mτx(t-τ)v(t-τ)1+αv(t-τ)dτ-δy(t),(4)v˙(t)=Nδ0ωg(τ)e-nτy(t-τ)dτ-cv(t)-qv(t)z(t),(5)z˙(t)=rv(t)z(t)-μz(t), where α is a positive constant, and all the variables and parameters of the model have the same meanings as given in model (1). To account for the time lag between viral contacting a target cell and the production of new virus particles, two distributed intracellular delays are introduced. It is assumed that the uninfected cells that are contacted by the virus particles at time t-τ become infected cells at time t, where τ is a random variable with a probability distribution f(τ) over the interval [0,h] and h is limit superior of this delay. The factor e-mτ accounts for the probability of surviving the time period of delay, where m is the death rate of infected cells but not yet virus producer cells. On the other hand, it is assumed that a cell infected at time t-τ starts to yield new infectious virus at time t where τ is distributed according to a probability distribution g(τ) over the interval [0,ω] and ω is limit superior of this delay. The factor e-nτ accounts for the probability of surviving during the time period of delay, where n is constant. All the parameters are supposed to be positive.

The probability distribution functions f(τ) and g(τ) are assumed to satisfy f(τ)>0 and g(τ)>0, and (6)0hf(τ)dτ=0ωg(τ)dτ=1,0hf(u)esudu<,0ωg(u)esudu<, where s is a positive number. Then (7)0<0hf(τ)e-mτdτ1,for  m0,0<0ωg(τ)e-nτdτ1,for  n0. Let F=0hf(τ)e-mτdτ and G=0ωg(τ)e-nτdτ. The initial conditions for system (2)–(5) take the form (8)x(θ)=φ1(θ),y(θ)=φ2(θ),v(θ)=φ3(θ),z(θ)=φ4(θ),φj(θ)0,θ[-ρ,0),  j=1,,4,φj(0)>0,j=1,,4, where ρ=max{h,ω},(φ1(θ),φ2(θ),,φ4(θ))C([-ρ,0],+4), where C([-ρ,0],+4) is the Banach space of continuous functions mapping the interval [-ρ,0]    into +4. By the fundamental theory of functional differential equations , system (2)–(5) has a unique solution satisfying the initial conditions (8).

2.1. Nonnegativity and Boundedness of Solutions

In the following, we establish the nonnegativity and boundedness of solutions of (2)–(5) with initial conditions (8).

Proposition 1.

Let (x(t),y(t),v(t),z(t)) be any solution of (2)–(5) satisfying the initial conditions (8); then x(t),y(t),v(t), and z(t) are all nonnegative for t0 and ultimately bounded.

Proof.

First, we prove that x(t)>0 for all t0. Assume that x(t) loses its non-negativity on some local existence interval [0,] for some constant , and let t1[0,] be such that x(t1)=0. From (2) we have x˙(t1)=λ>0. Hence, x(t)>0 for some t(t1,t1+ɛ), where ɛ>0 is sufficiently small. This leads to contradiction, and hence x(t)>0 for all t0. Now from (3), (4), and (5) we have (9)y(t)=y(0)e-δt+β0te-δ(t-η)×0hf(τ)e-mτx(η-τ)v(η-τ)1+αv(η-τ)dτdη,v(t)=v(0)e-0t(c+qz(ξ))dξ+Nδ0te-ηt(c+qz(ξ))dξ×0ωg(τ)e-nτy(η-τ)dτdη,z(t)=z(0)e-0t(μ-rv(ξ))dξ, confirming that y(t)0,v(t)0, and z(t)0 for all t[0,ρ]. By a recursive argument, we obtain y(t)0,v(t)0, and z(t)0 for all t0.

Next we show the boundedness of the solutions. From (2) we have x˙(t)λ-dx(t). This implies limsuptx(t)λ/d. Let X(t)=0hf(τ)e-mτx(t-τ)dτ+y(t); then (10)X˙(t)=0hf(τ)e-mτ(βx(t-τ)v(t-τ)1+αv(t-τ)λ-dx(t-τ)-βx(t-τ)v(t-τ)1+αv(t-τ))dτ+0hf(τ)e-mτβx(t-τ)v(t-τ)1+αv(t-τ)dτ-δy(t)=λ0hf(τ)e-mτdτ-d×0hf(τ)e-mτx(t-τ)dτ-δy(t)λ0hf(τ)e-mτdτ-σ1×[0hf(τ)e-mτx(t-τ)dτ+y(t)]λ-σ1X(t), where σ1=min{d,δ}. Hence, limsuptX(t)L1, where L1=λ/σ1. Since 0hf(τ)e-mτx(t-τ)dτ>0, then limsupty(t)L1. On the other hand, let Z(t)=v(t)+(q/r)z(t); then (11)Z˙(t)=Nδ0ωg(τ)e-nτy(t-τ)dτ-cv(t)-qμrz(t)NδL10ωg(τ)e-nτdτ-σ2(v(t)+qrz(t))=NδL10ωg(τ)e-nτdτ-σ2Z(t)NδL1-σ2Z(t), where σ2=min{c,μ}. Hence, limsuptZ(t)L2, where L2=NδL1/σ2. Since v(t)0, and v(t)0, then limsuptv(t)L2 and limsuptz(t)L2.

Therefore, x(t),y(t),v(t), and z(t) are ultimately bounded.

We define the basic reproduction number for system (2)–(5) as (12)R0=NFGβλcd, and the antibody immune response reproduction number (13)R1=R01+((μαd+μβ)/rd). Clearly R1<R0. It can be seen that system (2)–(5) has an uninfected steady state E0=(x0,0,0,0), where x0=λ/d. In addition to E0, the system has an infected steady state without immune response E1(x1,y1,v1,0) and infected steady state with immune response E2(x2,y2,v2,z2), where (14)x1=x0(1+αv1)R0,y1=cNδGv1,v1=dαd+β(R0-1),x2=λd+(βv2/(1+αv2)),y2=Fβλv2δ(d(1+αv2)+βv2),v2=μr,z2=cq(R1-1). From the above we have the following:

if R0>1, then there exists a positive steady state E1(x1,y1,v1,0);

if R1>1, then there exists a positive steady state E2(x2,y2,v2,z2).

3. Global Stability

In this section, we prove the global stability of the steady states of system (2)–(5) employing the method of Lyapunov functional which is used in  for SIR epidemic model with distributed delay. Next we will use the following notation: b=b(t) for any b{x,y,v,z}. We also define a function H:(0,)[0,) as (15)H(b)=b-1-ln(b). It is clear that H(b)0 for any b>0 and H has the global minimum H(1)=0.

Theorem 2.

If R01, then E0 is GAS.

Proof.

Define a Lyapunov functional W0 as follows: (16)W0=NFG[x0H(xx0)+1Fy+βF×0hf(τ)e-mτ×0τx(t-θ)v(t-θ)1+αv(t-θ)dθdτ+δFG0ωg(τ)e-nτ0τy(t-θ)dθdτ]+v+qrz. The time derivative of W0 along the trajectories of (2)–(5) satisfies (17)dW0dt=NFG[(1-x0x)(λ-dx-βxv1+αv)+βF0hf(τ)e-mτx(t-τ)v(t-τ)1+αv(t-τ)dτ-δFy+βF0hf(τ)e-mτ×(xv1+αv-x(t-τ)v(t-τ)1+αv(t-τ))dτ+δFG0ωg(τ)e-nτ(y-y(t-τ))dτ]+Nδ0ωg(τ)e-nτy(t-τ)dτ-cv-qvz+qvz-qμrz=NFG[-d(x-x0)2x+βx0v1+αv]-cv-qμrz=-NFGd(x-x0)2x-cv+cvR0(1+αv-αv)1+αv-qμrz=-NFGdx(x-x0)2+cv(R0-1)-R0cαv21+αv-qμrz. If R01, then dW0/dt0 for all x,v,z>0. By Theorem 5.3.1 in , the solutions of system (2)–(5) limit to M, the largest invariant subset of {dW0/dt=0}. Clearly, it follows from (17) that dW0/dt=0 if and only if x=x0, v=0, and z=0. Noting that M is invariant for each element of M, we have v=0, and z=0, and then v˙=0. From (4) we derive that (18)0=v˙=Nδ0ωg(τ)e-nτy(t-τ)dτ. This yields y=0. Hence, dW0/dt=0 if and only if x=x0, y=0, v=0, and z=0. From LaSalle’s Invariance Principle, E0 is GAS.

Theorem 3.

If R11<R0, then E1 is GAS.

Proof.

We construct the following Lyapunov functional: (19)W1=NFG[0τx1H(xx1)+1Fy1H(yy1)+1Fβx1v1(1+αv1)×0hf(τ)e-mτ×0τH(x(t-θ)v(t-θ)(1+αv1)x1v1(1+αv(t-θ)))dθdτ+δy1FG0ωg(τ)e-nτ0τH(y(t-θ)y1)dθdτ]+v1H(vv1)+qrz. The time derivative of W1 along the trajectories of (2)–(5) is given by (20)dW1dt=NFG[(y-y(t-τ)+y1ln(y(t-τ)y))(1-x1x)(λ-dx-βxv1+αv)+1F(1-y1y)×(β0hf(τ)e-mτx(t-τ)v(t-τ)1+αv(t-τ)dτ-δy)+βF0hf(τ)e-mτ×(xv1+αv-x(t-τ)v(t-τ)1+αv(t-τ)+x1v11+αv1×ln(x(t-τ)v(t-τ)(1+αv)xv(1+αv(t-τ))))dτ+δFG0ωg(τ)e-nτ×(y-y(t-τ)+y1ln(y(t-τ)y))dτ]+(1-v1v)(Nδ0ωg(τ)e-nτy(t-τ)dτ-cv-qzvNδ0ω)+qvz-qμrz. Using the steady state conditions for E1: (21)λ=dx1+βx1v11+αv1,δy1=Fβx1v11+αv1,cv1=NδGy1, we have (22)dW1dt=NFG[-d(x-x1)2x+βx1v11+αv1-βx1v11+αv1x1x+βx1v1+αv-1Fβx1v1(1+αv1)×0hf(τ)e-mτ×y1x(t-τ)v(t-τ)(1+αv1)yx1v1(1+αv(t-τ))dτ+δFy1+1Fβx1v11+αv1×0hf(τ)e-mτ×ln(x(t-τ)v(t-τ)(1+αv)xv(1+αv(t-τ)))dτ+δy1FG0ωg(τ)e-nτln(y(t-τ)y)dτ(x-x1)2x]+Nδy10ωg(τ)e-nτv1y(t-τ)vy1dτ-cv+cv1+qv1z-qμrz. Using the following equalities: (23)ln(x(t-τ)v(t-τ)(1+αv)xv(1+αv(t-τ)))=ln(y1x(t-τ)v(t-τ)(1+αv1)yx1v1(1+αv(t-τ)))+ln(x1x)+ln(v1yvy1)+ln(1+αv1+αv1),ln(y(t-τ)y)=ln(vy1v1y)+ln(v1y(t-τ)vy1), then collecting terms of (22), we obtain (24)dW1dt=NFG[-d(x-x1)2x-δy1F(x1x-1-lnx1x)+δy1Fv(1+αv1)v1(1+αv)-δy1F2×0hf(τ)e-mτ×(y1x(t-τ)v(t-τ)(1+αv1)yx1v1(1+αv(t-τ))-1-ln(y1x(t-τ)v(t-τ)(1+αv1)yx1v1(1+αv(t-τ))))dτ-δy1FG0wg(τ)e-nτ×(v1y(t-τ)vy1-1-ln(v1y(t-τ)vy1))dτ+δy1F(ln(1+αv1+αv1)-vv1)(x-x1)2x]+qz(v1-μr)=NFG[-d(x-x1)2x+δy1F×(-1+v(1+αv1)v1(1+αv)-vv1+1+αv1+αv1)-δy1FH(x1x)-δy1F2×0hf(τ)e-mτ×H(y1x(t-τ)v(t-τ)(1+αv1)yx1v1(1+αv(t-τ)))dτ-δFGy10ωg(τ)e-nτH(v1y(t-τ)vy1)dτ-δy1FH(1+αv1+αv1)]+qz(v1-μr)=NFG[-d(x-x1)2x-δy1Fα(v-v1)2v1(1+αv)(1+αv1)-δy1FH(x1x)-δy1F2×0hf(τ)e-mτ×H(y1x(t-τ)v(t-τ)(1+αv1)yx1v1(1+αv(t-τ)))dτ-δFGy10ωg(τ)e-nτH(v1y(t-τ)vy1)dτ-δy1FH(1+αv1+αv1)]+qdαd+β(1+μαd+μβrd)(R1-1)z. Thus, if R0>1, then x1,y1, and v1>0, and hence, if R11, then dW1/dt0 for all x,y,v>0. By Theorem 5.3.1 in , the solutions of system (2)–(5) limit to M, the largest invariant subset of {dW1/dt=0}. It can be seen that dW1/dt=0 if and only if x=x1,v=v1, and H=0; that is, (25)y1x(t-τ)v(t-τ)(1+αv1)yx1v1(1+αv(t-τ))=v1y(t-τ)vy1=1for  almost  all  τ[0,ρ]. From (25), if v=v1, then y=y1, and hence dW1/dt equals zero at E1. LaSalle’s Invariance Principle implies global stability of E1.

Theorem 4.

If R1>1, then E2 is GAS.

Proof.

We construct the following Lyapunov functional: (26)W2=NFG[x2H(xx2)+1Fy2H(yy2)+1Fβx2v2(1+αv2)×0hf(τ)e-mτ×0τH(x(t-θ)v(t-θ)(1+αv2)x2v2(1+αv(t-θ)))dθdτ+δy2FG0ωg(τ)e-nτ0τH(y(t-θ)y2)dθdτ]+v2H(vv2)+qrz2H(zz2). The time derivative of W2 along the trajectories of (2)–(5) is given by (27)dW2dt=NFG[(y-y(t-τ)+y2ln(y(t-τ)y))(1-x2x)(λ-dx-βxv1+αv)+1F(1-y2y)×(β0hf(τ)e-mτx(t-τ)v(t-τ)1+αv(t-τ)dτ-δy)+βF0hf(τ)e-mτ×(xv1+αv-x(t-τ)v(t-τ)1+αv(t-τ)+x2v21+αv2×ln(x(t-τ)v(t-τ)(1+αv)xv(1+αv(t-τ))))dτ+δFG0ωg(τ)e-nτ×(y2ln(y(t-τ)y)y-y(t-τ)+y2ln(y(t-τ)y))dτ]+(1-v2v)×(Nδ0ωg(τ)e-nτy(t-τ)dτ-cv-qzv)+(1-z2z)(qvz-qμrz). Using the steady state conditions for E2: (28)λ=dx2+βx2v21+αv2,cv2=NFG(δFy2)-qv2z2,μ=rv2, we obtain (29)dW2dt=NFG[-d(x-x2)2x+βx2v21+αv2-βx2v21+αv2x2x+βx2v1+αv-1Fβx2v21+αv2×0hf(τ)e-mτ×y2x(t-τ)v(t-τ)(1+αv2)yx2v2(1+αv(t-τ))dτ+δFy2+1Fβx2v21+αv2×0hf(τ)e-mτ×ln(x(t-τ)v(t-τ)(1+αv)xv(1+αv(t-τ)))dτ+δy2FG0ωg(τ)e-nτln(y(t-τ)y)dτ-d(x-x2)2x]-Nδy20ωg(τ)e-nτv2y(t-τ)vy2dτ-cv+cv2+qv2z-qvz2-qμrz+qμrz2. Using the following equalities: (30)cv=cv2vv2=NFG(δFy2vv2)-qvz2,ln(x(t-τ)v(t-τ)(1+αv)xv(1+αv(t-τ)))=ln(y2x(t-τ)v(t-τ)(1+αv2)yx2v2(1+αv(t-τ)))+ln(x2x)+ln(v2yvy2)+ln(1+αv1+αv2),ln(y(t-τ)y)=ln(vy2v2y)+ln(v2y(t-τ)vy2), and collecting terms of (29), we obtain (31)dW2dt=NFG[-d(x-x2)2x-δy1F(x2x-1-lnx2x)+δy2Fv(1+αv2)v2(1+αv)-δy2F2×0hf(τ)e-mτ×((y2x(t-τ)v(t-τ)(1+αv2)yx2v2(1+αv(t-τ)))y2x(t-τ)v(t-τ)(1+αv2)yx2v2(1+αv(t-τ))-1-ln(y2x(t-τ)v(t-τ)(1+αv2)yx2v2(1+αv(t-τ))))dτ-δy2FG0ωg(τ)e-nτ(v2y(t-τ)vy2-1-ln(v2y(t-τ)vy2))dτ+δy2F(ln(1+αv1+αv2)-vv2)-d(x-x2)2x-δy1F(x2x-1-lnx2x)]=NFG[-d(x-x2)2x-δy2Fα(v-v2)2v2(1+αv)(1+αv2)-δy2FH(x2x)-δy2F2×0hf(τ)e-mτ×H(y2x(t-τ)v(t-τ)(1+αv2)yx2v2(1+αv(t-τ)))dτ-δFGy20ωg(τ)e-nτH(v2y(t-τ)vy2)dτ-δy2FH(1+αv1+αv2)]. Thus, if R1>1, then x2,y2,v2, and z2>0 and dW2/dt0. By Theorem 5.3.1 in , the solutions of system (2)–(5) limit to M, the largest invariant subset of {dW2/dt=0}. It can be seen that dW2/dt=0 if and only if x=x2,v=v2, and H=0; that is, (32)y2x(t-τ)v(t-τ)(1+αv2)yx2v2(1+αv(t-τ))=v2y(t-τ)vy2=1+αv1+αv2=1for  almost  all  τ[0,ρ]. Then from (32) y=y2, and hence dW2/dt=0 at the steady state E2. LaSalle’s Invariance Principle implies global stability of E2.

4. Conclusion

In this paper, we have proposed a virus infection model which describes the interaction of the virus with the uninfected and infected cells taking into account the antibody immune response. The infection rate is given by saturation functional response. Two types of distributed time delays have been incorporated into the model to describe the time needed for infection of uninfected cell and virus replication. Using the method of Lyapunov functional, we have established that the global dynamics of the model is determined by two threshold parameters R0 and R1. The basic reproduction number viral infection R0 determines whether a chronic infection can be established, and the antibody immune response reproduction number R1 determines whether a persistent antibody immune response can be established. We have proven that if R01, then the uninfected steady state is GAS, and the viruses are cleared. If R11<R0, then the infected steady state without antibody immune response is GAS, and the infection becomes chronic but without persistent antibody immune response. If R1>1, then the infected steady state with antibody immune response is GAS, and the infection is chronic with persistent antibody immune response. We note that the effect of the time delay appears in the parameters F and G. Since 0<F1 and 0<G1, then the intracellular delay can reduce the parameters R0 and R1. As a consequence, ignoring the delay will produce overestimation of R0.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are also grateful to Professor Victor Kozyakin and to the anonymous reviewers for constructive suggestions and valuable comments, which improved the quality of the article.

Nowak M. A. May R. M. Virus Dynamicss: Mathematical Principles of Immunology and Virology 2000 Oxford, UK Oxford University Press MR2009143 Perelson A. S. Nelson P. W. Mathematical analysis of HIV-1 dynamics in vivo SIAM Review 1999 41 1 3 44 10.1137/S0036144598335107 MR1669741 ZBL1078.92502 Callaway D. S. Perelson A. S. HIV-1 infection and low steady state viral loads Bulletin of Mathematical Biology 2002 64 1 29 64 2-s2.0-0036381110 10.1006/bulm.2001.0266 Roy P. K. Chatterjee A. N. Greenhalgh D. Khan Q. J. A. Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model Nonlinear Analysis: Real World Applications 2013 14 3 1621 1633 10.1016/j.nonrwa.2012.10.021 MR3004525 ZBL06141787 Nelson P. W. Murray J. D. Perelson A. S. A model of HIV-1 pathogenesis that includes an intracellular delay Mathematical Biosciences 2000 163 2 201 215 10.1016/S0025-5564(99)00055-3 MR1740580 ZBL0942.92017 Nelson P. W. Perelson A. S. Mathematical analysis of delay differential equation models of HIV-1 infection Mathematical Biosciences 2002 179 1 73 94 10.1016/S0025-5564(02)00099-8 MR1908737 ZBL0992.92035 Culshaw R. V. Ruan S. A delay-differential equation model of HIV infection of CD4+ T-cells Mathematical Biosciences 2000 165 1 27 39 2-s2.0-0034108662 10.1016/S0025-5564(00)00006-7 Mittler J. E. Sulzer B. Neumann A. U. Perelson A. S. Influence of delayed viral production on viral dynamics in HIV-1 infected patients Mathematical Biosciences 1998 152 2 143 163 2-s2.0-0032168279 10.1016/S0025-5564(98)10027-5 Elaiw A. M. Alsheri A. S. Global Dynamics of HIV Infection of CD4+ T Cells and Macrophages Discrete Dynamics in Nature and Society 2013 2013 8 264759 10.1155/2013/264759 Elaiw A. M. Azoz S. A. Global properties of a class of HIV infection models with Beddington-DeAngelis functional response Mathematical Methods in the Applied Sciences 2013 36 4 383 394 10.1002/mma.2596 MR3032352 ZBL06163079 Elaiw A. Hassanien I. Azoz S. Global stability of HIV infection models with intracellular delays Journal of the Korean Mathematical Society 2012 49 4 779 794 10.4134/JKMS.2012.49.4.779 MR2976099 ZBL1256.34068 Elaiw A. M. Alghamdi M. A. Global properties of virus dynamics models with multitarget cells and discrete-time delays Discrete Dynamics in Nature and Society 2011 2011 19 201274 10.1155/2011/201274 MR2861950 ZBL1233.92059 Elaiw A. M. Global properties of a class of virus infection models with multitarget cells Nonlinear Dynamics 2012 69 1-2 423 435 10.1007/s11071-011-0275-0 MR2929883 ZBL1254.92064 Elaiw A. M. Global dynamics of an HIV infection model with two classes of target cells and distributed delays Discrete Dynamics in Nature and Society 2012 2012 13 253703 MR2965727 ZBL1253.37082 Nowak M. A. Bangham C. R. M. Population dynamics of immune responses to persistent viruses Science 1996 272 5258 74 79 2-s2.0-0029985351 Wang K. Fan A. Torres A. Global properties of an improved hepatitis B virus model Nonlinear Analysis: Real World Applications 2010 11 4 3131 3138 10.1016/j.nonrwa.2009.11.008 MR2661975 ZBL1197.34081 Zeuzem S. Schmidt J. M. Lee J.-H. Rüster B. Roth W. K. Effect of interferon alfa on the dynamics of hepatitis C virus turnover in vivo Hepatology 1996 23 2 366 371 2-s2.0-0030039045 10.1053/jhep.1996.v23.pm0008591865 Neumann A. U. Lam N. P. Dahari H. Gretch D. R. Wiley T. E. Layden T. J. Perelson A. S. Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-α therapy Science 1998 282 5386 103 107 2-s2.0-0032475822 10.1126/science.282.5386.103 Deans J. A. Cohen S. Immunology of malaria Annual Review of Microbiology 1983 37 25 49 2-s2.0-0020653624 Anderson R. M. May R. M. Gupta S. Non-linear phenomena in host-parasite interactions Parasitology 1989 99 S59 S79 2-s2.0-0024349915 Murase A. Sasaki T. Kajiwara T. Stability analysis of pathogen-immune interaction dynamics Journal of Mathematical Biology 2005 51 3 247 267 10.1007/s00285-005-0321-y MR2206233 ZBL1086.92029 Wodarz D. May R. M. Nowak M. A. The role of antigen-independent persistence of memory cytotoxic T lymphocytes International Immunology 2000 12 4 467 477 2-s2.0-0034037955 Chiyaka C. Garira W. Dube S. Modelling immune response and drug therapy in human malaria infection Computational and Mathematical Methods in Medicine 2008 9 2 143 163 10.1080/17486700701865661 MR2412334 ZBL1140.92012 Perelson A. S. Modelling viral and immune system dynamics Nature Reviews Immunology 2002 2 1 28 36 2-s2.0-0036371443 Wang S. Zou D. Global stability of in-host viral models with humoral immunity and intracellular delays Applied Mathematical Modelling 2012 36 3 1313 1322 10.1016/j.apm.2011.07.086 MR2872899 ZBL1243.34106 Huo H. F. Tang Y. L. Feng L. X. A virus dynamics model with saturation infection and humoral immunity International Journal of Mathematical Analysis 2012 6 40 1977 1983 Dixit N. M. Perelson A. S. Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay Journal of Theoretical Biology 2004 226 1 95 109 10.1016/j.jtbi.2003.09.002 MR2068328 Wang X. Liu S. A class of delayed viral models with saturation infection rate and immune response Mathematical Methods in the Applied Sciences 2013 36 2 125 142 10.1002/mma.2576 MR3008329 ZBL06133223 Korobeinikov A. Global properties of infectious disease models with nonlinear incidence Bulletin of Mathematical Biology 2007 69 6 1871 1886 10.1007/s11538-007-9196-y MR2329184 ZBL05265715 Huang G. Takeuchi Y. Ma W. Lyapunov functionals for delay differential equations model of viral infections SIAM Journal on Applied Mathematics 2010 70 7 2693 2708 10.1137/090780821 MR2678058 ZBL1209.92035 Song X. Neumann A. U. Global stability and periodic solution of the viral dynamics Journal of Mathematical Analysis and Applications 2007 329 1 281 297 10.1016/j.jmaa.2006.06.064 MR2306802 ZBL1105.92011 McCluskey C. C. Complete global stability for an SIR epidemic model with delay—distributed or discrete Nonlinear Analysis: Real World Applications 2010 11 1 55 59 10.1016/j.nonrwa.2008.10.014 MR2570523 ZBL1185.37209 Hale J. K. Verduyn Lunel S. M. Introduction to Functional-Differential Equations 1993 99 New York, NY, USA Springer MR1243878