# Global Dynamics of Virus Infection Model with Antibody Immune Response and Distributed Delays

Authors: A. M. Elaiw; A. Alhejelan; M. A. Alghamdi
Journal: Discrete Dynamics in Nature and Society
 (2013)

---

## Abstract

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 numberR0 and antibody immune response reproduction number R1. We have proven that if R0≤1, then the uninfected steady state is globally asymptotically stable (GAS), if  R1≤1<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.


---

## Body

## 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) [1–14], 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 [19].Mathematical models for virus dynamics with the antibody immune response have been developed in [20–26]. The basic virus dynamics model with antibody immune response was introduced by Murase et al. [21] 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., [7–14, 27]). In [7–14, 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 [31]. In [26], 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 [32]. 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 R0≤1, then the uninfected steady state is globally asymptotically stable (GAS), if R1≤1<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)=β∫0hf(τ)e-mτx(t-τ)v(t-τ)1+αv(t-τ)dτ-δy(t),(4)v˙(t)=Nδ∫0ωg(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 functionsf(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-mτdτ≤1,form≥0,0<∫0ωg(τ)e-nτdτ≤1,forn≥0.
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],ℝ+4), where C([-ρ,0],ℝ+4) is the Banach space of continuous functions mapping the interval [-ρ,0]into ℝ+4. By the fundamental theory of functional differential equations [33], 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 t≥0 and ultimately bounded.Proof.
First, we prove thatx(t)>0 for all t≥0. Assume that x(t) loses its non-negativity on some local existence interval [0,ℓ] for some constant ℓ, and let t1∈[0,ℓ] 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 t≥0. Now from (3), (4), and (5) we have
(9)y(t)=y(0)e-δt+β∫0te-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+Nδ∫0te-∫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 t≥0.
Next we show the boundedness of the solutions. From (2) we have x˙(t)≤λ-dx(t). This implies limsupt→∞x(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)=λ∫0hf(τ)e-mτdτ-d×∫0hf(τ)e-mτx(t-τ)dτ-δy(t)≤λ∫0hf(τ)e-mτdτ-σ1×[∫0hf(τ)e-mτx(t-τ)dτ+y(t)]≤λ-σ1X(t),
where σ1=min{d,δ}. Hence, limsupt→∞X(t)≤L1, where L1=λ/σ1. Since ∫0hf(τ)e-mτx(t-τ)dτ>0, then limsupt→∞y(t)≤L1. On the other hand, let Z(t)=v(t)+(q/r)z(t); then
(11)Z˙(t)=Nδ∫0ωg(τ)e-nτy(t-τ)dτ-cv(t)-qμrz(t)≤NδL1∫0ωg(τ)e-nτdτ-σ2(v(t)+qrz(t))=NδL1∫0ωg(τ)e-nτdτ-σ2Z(t)≤NδL1-σ2Z(t),
where σ2=min{c,μ}. Hence, limsupt→∞Z(t)≤L2, where L2=NδL1/σ2. Since v(t)≥0, and v(t)≥0, then limsupt→∞v(t)≤L2 and limsupt→∞z(t)≤L2.
Therefore,x(t),y(t),v(t), and z(t) are ultimately bounded.


### 2.2. Steady States

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:(i)
ifR0>1, then there exists a positive steady state E1(x1,y1,v1,0);(ii)
ifR1>1, then there exists a positive steady state E2(x2,y2,v2,z2).


## 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 t≥0 and ultimately bounded.Proof.
First, we prove thatx(t)>0 for all t≥0. Assume that x(t) loses its non-negativity on some local existence interval [0,ℓ] for some constant ℓ, and let t1∈[0,ℓ] 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 t≥0. Now from (3), (4), and (5) we have
(9)y(t)=y(0)e-δt+β∫0te-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+Nδ∫0te-∫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 t≥0.
Next we show the boundedness of the solutions. From (2) we have x˙(t)≤λ-dx(t). This implies limsupt→∞x(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)=λ∫0hf(τ)e-mτdτ-d×∫0hf(τ)e-mτx(t-τ)dτ-δy(t)≤λ∫0hf(τ)e-mτdτ-σ1×[∫0hf(τ)e-mτx(t-τ)dτ+y(t)]≤λ-σ1X(t),
where σ1=min{d,δ}. Hence, limsupt→∞X(t)≤L1, where L1=λ/σ1. Since ∫0hf(τ)e-mτx(t-τ)dτ>0, then limsupt→∞y(t)≤L1. On the other hand, let Z(t)=v(t)+(q/r)z(t); then
(11)Z˙(t)=Nδ∫0ωg(τ)e-nτy(t-τ)dτ-cv(t)-qμrz(t)≤NδL1∫0ωg(τ)e-nτdτ-σ2(v(t)+qrz(t))=NδL1∫0ωg(τ)e-nτdτ-σ2Z(t)≤NδL1-σ2Z(t),
where σ2=min{c,μ}. Hence, limsupt→∞Z(t)≤L2, where L2=NδL1/σ2. Since v(t)≥0, and v(t)≥0, then limsupt→∞v(t)≤L2 and limsupt→∞z(t)≤L2.
Therefore,x(t),y(t),v(t), and z(t) are ultimately bounded.


## 2.2. Steady States

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:(i)
ifR0>1, then there exists a positive steady state E1(x1,y1,v1,0);(ii)
ifR1>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 [32] 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.
IfR0≤1, then E0 is GAS.Proof.
Define a Lyapunov functionalW0 as follows:
(16)W0=NFG[x0H(xx0)+1Fy+βF×∫0h‍f(τ)e-mτ×∫0τ‍x(t-θ)v(t-θ)1+αv(t-θ)dθdτ+δFG∫0ω‍g(τ)e-nτ∫0τ‍y(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)+βF∫0hf(τ)e-mτx(t-τ)v(t-τ)1+αv(t-τ)dτ-δFy+βF∫0hf(τ)e-mτ×(xv1+αv-x(t-τ)v(t-τ)1+αv(t-τ))dτ+δFG∫0ωg(τ)e-nτ(y-y(t-τ))dτ]+Nδ∫0ωg(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 R0≤1, then dW0/dt≤0 for all x,v,z>0. By Theorem 5.3.1 in [33], 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˙=Nδ∫0ω‍g(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.
IfR1≤1<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+δy1FG∫0ω‍g(τ)e-nτ∫0τ‍H(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)×(β∫0h‍f(τ)e-mτx(t-τ)v(t-τ)1+αv(t-τ)dτ-δy)+βF∫0h‍f(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+δFG∫0ω‍g(τ)e-nτ×(y-y(t-τ)+y1ln(y(t-τ)y))dτ]+(1-v1v)(Nδ∫0ω‍g(τ)e-nτy(t-τ)dτ-cv-qzvNδ∫0ω‍)+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+δy1FG∫0ω‍g(τ)e-nτln(y(t-τ)y)dτ(x-x1)2x]+Nδy1∫0ω‍g(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-δy1FG∫0w‍g(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-δFGy1∫0ω‍g(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-δFGy1∫0ω‍g(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 R1≤1, then dW1/dt≤0 for all x,y,v>0. By Theorem 5.3.1 in [33], 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=1foralmostallτ∈[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.
IfR1>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+δy2FG∫0ω‍g(τ)e-nτ∫0τ‍H(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)×(β∫0h‍f(τ)e-mτx(t-τ)v(t-τ)1+αv(t-τ)dτ-δy)+βF∫0h‍f(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+δFG∫0ω‍g(τ)e-nτ×(y2ln(y(t-τ)y)y-y(t-τ)+y2ln(y(t-τ)y))dτ]+(1-v2v)×(Nδ∫0ω‍g(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+δy2FG∫0ω‍g(τ)e-nτln(y(t-τ)y)dτ-d(x-x2)2x]-Nδy2∫0ω‍g(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-δy2FG∫0ω‍g(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-δFGy2∫0ω‍g(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/dt≤0. By Theorem 5.3.1 in [33], 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=1foralmostallτ∈[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 parametersR0 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 R0≤1, then the uninfected steady state is GAS, and the viruses are cleared. If R1≤1<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<F≤1 and 0<G≤1, then the intracellular delay can reduce the parameters R0 and R1. As a consequence, ignoring the delay will produce overestimation of R0.



---

*Source: 781407-2013-12-03.xml*