# Mathematical Analysis and Clinical Implications of an HIV Model with Adaptive Immunity

Authors: Jaouad Danane; Karam Allali
Journal: Computational and Mathematical Methods in Medicine
 (2019)

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## Abstract

In this paper, a mathematical model describing the human immunodeficiency virus (HIV) pathogenesis with adaptive immune response is presented and studied. The mathematical model includes six nonlinear differential equations describing the interaction between the uninfected cells, the exposed cells, the actively infected cells, the free viruses, and the adaptive immune response. The considered adaptive immunity will be represented by cytotoxic T-lymphocytes cells (CTLs) and antibodies. First, the global stability of the disease-free steady state and the endemic steady states is established depending on the basic reproduction numberR0, the CTL immune response reproduction number R1z, the antibody immune response reproduction number R1w, the antibody immune competition reproduction number R2w, and the CTL immune response competition reproduction number R3z. On the other hand, different numerical simulations are performed in order to confirm numerically the stability for each steady state. Moreover, a comparison with some clinical data is conducted and analyzed. Finally, a sensitivity analysis for R0 is performed in order to check the impact of different input parameters.


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## Body

## 1. Introduction

The human immunodeficiency virus (HIV) is a virus that gradually weakens the immune system since it targets the principal vital immune cells. It is considered as the main cause for several deadly diseases after the resulting acquired immunodeficiency syndrome (AIDS) is reached. With 36.7 million people living with HIV, 1.8 million people becoming newly infected with HIV, and more than 1 million deaths annually, HIV becomes a major global public health issue [1].In the last decades, many mathematical models describing HIV dynamics were developed [2–11]. With the three main dynamics compartments that are free viruses, healthy CD4+ T cells, and infected CD4+ T cells, the first viral dynamics was presented and studied in [2]. Including the exposed cells as a new fourth compartment, a modified HIV viral model was tackled in [8]. More recently, the model describing HIV viral dynamics with another fifth compartment representing the cytotoxic T-lymphocytes (CTL) cells is formulated and studied in [9]. The authors study the global stability of the endemic states and illustrate the numerical simulations in order to show the numerical stability for each problem steady state. Notice that the adaptive immunity has two main arms that are cellular and humoral responses. The first one is mediated by CTL cells that play a crucial role in the infection by killing infected cells, while the second arm is mediated by the antibodies which are proteins that are produced by B cells and are specifically programmed to neutralize the viruses [12]. In this paper, we extend the recent work [9] by incorporating to the model this other main component of the adaptive immune response. The dynamics of the HIV infection model including these two arms of the adaptive immune response will be governed by the following nonlinear system of differential equations:(1)x˙=λ−d1x−k1xvx+v,s˙=k1xvx+v−d2s−k2s,y˙=k2s−d3y−pyz,v˙=ay−d4v−qvw,w˙=gvw−hw,z˙=cyz−bz.With the initial conditions,x0=x0, s0=s0, y0=y0, v0=v0, w0=w0, and z0=z0. In this model, x, s, y, v, w, and z denote the concentration of uninfected cells, exposed cells, infected cells, free viruses, antibodies, and CTL cells, respectively. Susceptible host cells, CD4+ T cells, are produced at a rate λ and die at a rate d1x and become infected by virus at a rate k1xv/x+v. The exposed cells die at a rate d2+k2s. The infected cells increase at rate k2s and decay at rate d3y and are killed by the CTL response at a rate pyz. Free viruses are produced by infected cells at a rate ay and decay at a rate d4v and are killed by the antibodies at a rate qvw. Antibodies develop in response to free viruses at a rate gvw and decay at a rate hw. Finally, CTLs expand in response to viral antigen derived from infected cells at a rate cyz and decay in the absence of antigenic stimulation at a rate bz. Note that this model (1), includes the saturated rate, called the saturated mass action [11], which describes better the rate of viral infection. Such HIV viral dynamics is illustrated in Figure 1. The model (1) extends the recent work [9] by adding a new compartment which is the adaptive immune response. In addition to the mathematical analysis of this new model, we will compare our simulations with some clinical data and we will perform a sensitivity analysis of our parameters.Figure 1
Schematic of the model under consideration.The rest of the paper is organized as follows. The analysis of the model is described in Section2. In Section 3, we illustrate numerical simulations and compare the model solution to some clinical data. We conclude in the last section.


## 2. Analysis of the Model

### 2.1. Positivity and Boundedness

For the problems dealing with cell population evolution, the cell densities should remain nonnegative and bounded. In this section, we will establish the positivity and boundedness of solutions of the model (1). First of all, for biological reasons, the parameters x0, s0, y0, v0, w0, and z0 must be larger than or equal to 0. Hence, we have the following result:Proposition 1.
For any initial conditionsx0,s0,y0,v0,w0,z0, system (1) has a unique solution. Moreover, this solution is nonnegative and bounded for all t≥0.Proof.
By the classical functional differential equations theory (see for instance [13], and the references therein), we can confirm that there is a unique local solution xt,st,yt,vt,wt,zt to system (1) in 0,tm.
We have the following:(2)x˙x=0=λ≥0,s˙y=0=k1xvx+v≥0,y˙s=0=k2s≥0,v˙v=0=ay≥0,w˙w=0=0≥0,z˙z=0=0≥0,this shows the positivity of solutions for t∈0,tm. For the boundedness of the solutions,(3)X=x+s+y+pcz,then, we have(4)X˙=λ−d1x−d2s−d3y−bpcz≤λ−δX,where δ=mind1,d2,d3,b. So,(5)Xt≤X0e−δt+λδ1−e−δt≤X0e−δt+λδ.
Similarly, let us consider(6)V=v+qgw;therefore,(7)V˙=ay−d4v−hqgw≤ay−δV,where α=mind4,h, then,(8)Vt≤V0+aαy∞,this proves that the solutions xt, st, yt, vt, wt, and zt are bounded. Hence, every local solution can be prolonged up to any time tm>0, which means that the solution exists globally.


### 2.2. Steady States

System (1) has an infection-free equilibrium Ef=λ/d1,0,0,0,0,0, corresponding to the maximal level of healthy CD4+ T-cells. By simple calculation, the basic reproduction number of (1) is given by(9)R0=ak1k2d3d4d2+k2=k1k2d2+k2×ad3×1d4,where k2/d2+k2 is the proportion of the exposed cells to become productively infected cells, a/d3 is the number of free virus production by an infected cell, and 1/d4 is the average life of virus. From a biological point of view, R0 stands for the average number of secondary infections generated by one infected cell when all cells are susceptibles. Depending on the value of this basic reproduction number R0; in other words, depending on these three biological proportions, we will study the stability of the free-disease and the endemic equilibria. Indeed, it is easy to see that when R0>1, system (1) has four of them. The first endemic equilibrium is E1=x1,s1,y1,v1,w1,z1, where(10)x1=λd1+k11−1/R0,s1=λk11−1/R0d2+k2d1+k11−1/R0,y1=λd4R0−1ad1+ak11−1/R0,v1=λR0−1d1+k11−1/R0,w1=0,z1=0.We define the antibody immune response reproduction number by(11)R1w=gv1h=gλR0−1d1h+hk11−1/R0,where 1/h is the average life of antibodies cells and v1 is the number of free viruses at E1. For the biological significance, R1w represents the average number of the antibodies activated by virus when the viral infection is successful in the absence of CTL immune response.Furthermore, we introduce the CTL immune response reproduction number given by(12)R1z=cy1b=cλd4R0−1abd1+abk11−1/R0,where 1/b represents the average life of CTL cells and y1 is the number of infected cells at E1. Hence, R1z represents the mean of CTL immune cells activated by an infected cell when the viral infection is successful in the absence of the antibody immune response. The second endemic equilibrium is(13)E2=x2,s2,y2,v2,w2,z2,where(14)x2=−abd1−abk1+λcd4+A2cd1d4,s2=ak1b−abd1−abk1+λcd4+Acd4d2+k2abd1−abk1+λcd4+A,y2=bc,v2=abcd4,w2=0,z2=d2R0−1−ak1b+λcd4+A−abd1R0+1pabd1−ak1b+λcd4+A,with A=abk1−λcd42+a2b2d12+2a2b2d1k1+2λabcd1d4.We introduce the antibody immune competition reproduction number given by(15)R2w=gv2h=abgcd4h,with 1/h represents the average life of antibodies and v2 is the number of free viruses at E2. For biological point of view, R2w represents the average number of the antibodies activated by virus when the viral infection is successful in the absence of CTL response. The third endemic equilibrium is(16)E3=x3,s3,y3,v3,w3,z3,where(17)x3=−d1h−k1h+λg+B2d1g,s3=k1h−d1h−k1h+λg+Bgd2+k2d1h−k1h+λg+B,y3=k1k2h−d1h−k1h+λg+Bd3gd2+k2d1h−k1h+λg+B,v3=hg,w3=d4R0−1−k1h+λg+B−d1hR0+1qd1h−k1h+λg+B,z3=0,with B=hk1−λg2+d12h2+2k1d1h2k1+2λgd1h.We define the CTL immune competition reproduction numberR3z of our model by(18)R3z=cy3b=ck1k2h−d1h−k1h+λg+Bbd3gd2+k2d1h−k1h+λg+B,with 1/b represents the average life of CTL cells and y3 is the number of infected cells at E3. Hence, R3z represents the average number of CTL immune cells activated by an infected cell when the viral infection is successful in the absence of the antibody immune response. The last endemic equilibrium is(19)E4=x4,y4,v4,w4,z4,where(20)x4=−d1h−k1h+λg+B2d1g,s4=k1h−d1h−k1h+λg+Bgd2+k2d1h−k1h+λg+B,y4=bc,v4=hg,w4=abg−cd4hchq=d4qR2w−1,z4=ck1k2h−bd3gd2+k2−k1h+λg+B−d1hck1k2h+bd3gd2+k2pbgd2+k2d1h−k1h+λg+B,=d3pR3z−1.We observe that the second endemic stateE2=x2,y2,v2,w2,z2 exists when R1z>1. We explain the existence of this endemic equilibrium E2 as follows. We recall first that, in this state, both the free viruses and CTL cells are present. Assume that R0>1, in the total absence of CTL immune response, the infected cell load per unit time is λd4R0−1/ad1+ak11−1/R0. Via the six equations of model 1, CTL cells are reproduced due to infected cells stimulated per unit time being cλd4R0−1/ad1+ak11−1/R0=cy1. The CTL load during the lifespan of a CTL cell is cλd4R0−1/abd1+abk11−1/R0=R1z. If cλd4R0−1/abd1+abk11−1/R0>1, we will have the existence of the endemic equilibrium E2. We observe also that the third endemic state E3=x3,y3,v3,w3,z3 exists when R1w>1. We explain the existence of this endemic equilibrium E3 as follows. We recall first that, in this state, both of the free viruses and antibodies are present. Assume that R0>1, in the total absence of the antibody immune response, the viral load per unit time is λR0−1/d1+k11−1/R0. Via the six equations of model 1, antibodies are reproduced due to free viruses stimulation per unit time is gλR0−1/d1+k11−1/R0=gv1. The viral load during the lifespan of virion is gλR0−1/d1h+hk11−1/R0=R1w. If gλR0−1/d1h+hk11−1/R0>1, we will have the existence of the endemic equilibrium E3. Similarly, one can see that E4=x4,y4,v4,w4,z4 exists when R3z>1 and R2w>1.


### 2.3. Global Stability of the Disease-Free Equilibrium

For the global stability of the disease-free equilibrium, we have the following result.Proposition 2.
IfR0≤1, then the endemic point Ef is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(21)ℒx,y,v,w,z=s+d2+k2k2y+d3d2+k2ak2v+qgd3d2+k2ak2w+pcd2+k2k2z.
The time derivative is given by(22)ℒ˙x,s,y,v,w,z=s˙+d2+k2k2y˙+d3d2+k2ak2v˙+qgd3d2+k2ak2w˙+pcd2+k2k2z˙,ℒ˙x,s,y,v,w,z=k1xvx+v−d2+k2s+d2+k2k2k2s−d3y−d2+k2k2pyz+d3d2+k2ak2ay−d4v−qvw+qgd3d2+k2ak2gvw−hw+pcd2+k2k2cyz−bz,ℒ˙x,s,y,v,w,z=k1xvx+v−d3d2av−qhgd3d2+k2ak2w−bpcd2+k2k2z,ℒ˙x,y,s,v,w,z≤k1v−d3d4d2+k2ak2v≤d3d4d2+k2ak2R0−1v.
IfR0<1, then ℒ˙≤0. Moreover, ℒ˙≤0 when v=0. The largest compact invariant is(23)E=x,s,y,v,w,z∣v=0.
According to LaSalle’s invariance principle [14], we have lim+∞vt=0. The limit system of equations is(24)x˙=λ−d1x,s˙=−d2s−k2s,y˙=k2s−d3y−pyz,w˙=−hw,z˙=cyz−bz.
We define(25)ℒx,s,y,w,z=1x0x−x0−x0lnxx0+s+d2+k2k2y+qgd3d2+k2ak2w+d2+k2k2pcz.
Sincex0=λ/d1, then(26)ℒ˙x,s,y,w,z=d12−xx0−x0x−d3d2+k2k2y−qghd3d2+k2ak2w−d2+k2k2pbcz.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(27)2−xx0−x0x≤0.
Therefore,ℒ˙≤0, and the equality holds if x=x0 and s=y=w=z=0, which complete the proof.


### 2.4. Global Stability of Infection Steady States

In this subsection, attention is focused on the stability of the infection steady states.For the first endemic equilibriumE1, we have the following result.Proposition 3.
IfR0>1, R1z≤1, and R1w≤1, then the endemic point E1 is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(28)ℒx,s,y,v,w,z=x−x1+∫x1xd2+k2s1k1uv1/u+v1du+s−s1−s1lnss1+d2+k2k2y−y1−y1lnyy1+d3d2+k2ak2v−v1−v1lnvv1+qgd3d2+k2ak2w+pcd2+k2k2z.we have then(29)ℒ˙x,s,y,v,w,z=x˙−d2+k2s1x+v1k1xv1x˙+s˙−s1ss˙+d2+k2k2y˙−y1yy˙+d3d2+k2ak2v˙−v1vv˙+qgd3d2+k2ak2w˙+pcd2+k2k2z˙.
On the other hand, we have(30)λ=d1x1+d2+k2s1,k1x1v1x1+v1=d2+k2s1,s1v1=d3d4ak2,y1v1=d4a,s1y1=d3k2.
Hence,(31)ℒ˙x,s,y,v,w,z=λ−d1x−k1xvx+v1−d2+k2s1x+v1k1xv1+k1xvx+v−d2+k2s−s1sk1xvx+v−d2+k2s+d2+k2k2k2s−d3y−pyz−d2+k2k2y1yk2s−d3y−pyz+d3d2+k2ak2ay−d4v−qvw−d3d2+k2ak2v1vay−d4v−qvw+qgd3d2+k2ak2gvw−hw+pcd2+k2k2cyz−bz,ℒ˙x,s,y,v,w,z=λ−d1x−d2+k2s1x+v1k1xv1λ−d1x−k1xvx+v−s1sk1xvx+v+d2+k2s1+d2+k2k2py1z+d2+k2d3k2y1−d2+k2sy1y−d3d4d2+k2ak2v+d2+k2d3k2y1−d2+k2sy1y−d3d4d2+k2ak2v−qhgd3d2+k2ak2w+d3d2+k2ak2v1qw−d2+k2k2bpcz.
Thus, this fact implies that(32)ℒ˙x,s,y,v,w,z=λ−d1x−d2+k2s1x+v1k1xv1λ−d1x−k1xvx+v−s1sk1xvx+v+d2s1−d3d4d2+k2ak2v−d3d2+k2k2v1vy+d3d2+k2ak2v1−qhgd3d2+k2ak2w+d3d2+k2ak2qv1w+pzy1−bc.
Since(33)λ−d1x=d1x1+d2+k2s1−d1x,λ−d1x−d2+k2s1x+v1k1xv1λ−d1x−k1xvx+v=d1x11−xx1−x1xx+v1x1+v1+x+v1x1+v1+d2+k2s11−x1xx+v1x1+v1+vv1x+v1x+v−s1sk1xvx+v+d2+k2s1=−s1sxvx1v1x1+v1x+vd2+k2s1+d2+k2s1,
We have(34)ℒ˙x,s,y,v,w,z=d1x11−xx1−x1xx+v1x1+v1+x+v1x1+v1+d2+k2s11−x1xx+v1x1+v1+vv1x+v1x+v+d2+k2s11−s1sxvx1v1x1+v1x+v+d2+k2s11−sy1s1y−vv1+d2+k2s11−v1yy1v+d3d2+k2ak2qwv1−hg+pzd2+k2k2y1−bc.
Therefore,(35)ℒ˙=−d1v1xx1+v1x−x12+d2+k2s1−1−vv1+vv1x+v1x+v+x+vx+v1+d2+k2s15−x1xx+v1x1+v1−s1sxvx1v1x1+v1x+v−sy1s1y−yv1y1v−x+vx+v1+d3d2+k2ak2hgqwR1w−1+d2+k2k2bcpzR1z−1,which implies that(36)ℒ˙=−d1v1xx1+v1x−x12−d2+k2s1xv−v12v1x+v1x+v+d2+k2s15−x1xx+v1x1+v1−s1sxvx1v1x1+v1x+v−sy1s1y−yv1y1v−x+vx+v1+d3d2+k2ak2hgqwR1w−1+d2+k2k2bcpzR1z−1.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(37)5−x1xx+v1x1+v1−s1sxvx1v1x1+v1x+v−sy1s1y−yv1y1v−x+vx+v1≤0,and we know that R1z<1 and R1w<1, then L˙≤0, and the equality holds when x=x1, y=y1, v=v1, w=w1, and z=z1. By the LaSalle invariance principle [14], the endemic point E1 is asymptotically stable when R0>1.
For the second endemic equilibriumE2, we have the following result.Proposition 4.
IfR0>1, R1z>1, and R2w≤1, then the endemic point E2 is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(38)ℒx,s,y,v,w,z=x−x2−∫x2xd2+k2s2k1uv2/u+v2du+s−s2−s2lnss2+d2+k2k2y−y2−y2lnyy2+d3d2+k2+d2+k2pz2ak2×v−v2−v2lnvv2+qgd3d2+k2+d2+k2pz2ak2w+pcd2+k2k2z−z2−z2lnzz2,then, we have(39)ℒ˙x,y,s,v,w,z=x˙−d2+k2s2x+v2k1xv2x˙+s˙−s2ss˙+d2+k2k2y˙−y2yy˙+d3d2+k2+d2+k2pz2ak2v˙−v2vv˙+qgd3d2+k2+d2+k2pz2ak2w˙+pcd2+k2k2z˙−z2zz˙.
We know that(40)λ=d1x2+d2+k2s2,k1x2v2=x2+v2d2+k2s2,s2y2=d3k2+pz2k2,y2v2=d4a,s2v2=d3d4ak2+d4pz2ak2,so, we have(41)ℒ˙x,s,y,v,z=λ−d1x−x2xx+v2x2+v2λ−d1x+d2+k2s2vv2x2+v2x+v−d2+k2s−s2sk1xvx+v−d2+k2s+d2+k2k2k2s−d3y−pyz−d2+k2k2y2yk2s−d3y−pyz+d3d2+k2+d2+k2pz2ak2×ay−d4v−qvw−d3d2+k2+d2+k2pz2ak2v2vay−d4v−qvw+qgd3d2+k2+d2+k2pz2ak2gvw−hw+pcd2+k2k2cyz−bz−pz2czd2+k2k2cyz−bz.
On the other hand, we have(42)λ−d1x=d1x2+d2+k2s2−d1x,λ−d1x−d2+k2s2x+v2k1xv2λ−d1x−k1xvx+v=d1x21−xx2−x2xx+v2x2+v2+x+v2x2+v2+d2+k2s21−x2xx+v2x2+v2+vv2x+v2x+v−s2sk1xvx+v+d2+k2s2=d2+k2s21−s2sxvx2v2x2+v2x+v,d2+k2d3k2y2−d2+k2sy2y−d2+k2d3d4ak2v=d2+k2s21−ss2y2y−vv2+d2+k2k2pz2y2vv2−d2+k2k2pz2y2,d4d3d2+k2ak2v2−d3d2+k2k2v2vy=d2+k2s21−yy2v2v+d2+k2k2pz2yv2v−d2+k2k2pz2y2,d2+k2pz2ak2ay−d4v−d2+k2pz2ak2v2vay−d4v=d2+k2pz2yk2−d2+k2pz2y2k2vv2−d2+k2pz2yk2v2v+d2+k2pz2y2k2,d2+k2pz2ak2ay−d4v−d2+k2pz2ak2v2vay−d4v−z2zpcd2+k2k2cyz−bz+d2+k2k2pz2y2vv2−d2+k2k2pz2y2+d2+k2k2pz2yv2v−d2+k2k2pz2y2=0.
This fact implies that(43)L˙=−d1v2xx2+v2x−x22−d2+k2s2xv−v22v2x+v2x+v+d2+k2s25−x2xx+v2x2+v2−s2sxvx2v2x2+v2x+v−sy2s2y−yv2y2v−x+vx+v2+hgd3d2+k2+d2+k2pz2ak2qwR2w−1.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(44)5−x2xx+v2x2+v2−s2sxvx2v2x2+v2x+v−sy2s2y−yv2y2v−x+vx+v2≤0,and we know that R2w<1 which means that L˙≤0, and the equality holds when x=x2, s=s2, y=y2, v=v2, w=w2, and z=z2. By the LaSalle invariance principle [14], the endemic point E2 is asymptotically stable.
For the third endemic equilibriumE3, we have the following result.Proposition 5.
IfR0>1, R3z≤1, and R1w>1, then the endemic point E3 is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(45)ℒx,s,y,v,w,z=x−x3−∫x3xd2+k2s3k1uv3/u+v3du+s−s3−s3lnss3+d2+k2k2y−y3−y3lnyy3+d3d2+k2ak2×v−v3−v3lnvv3+d3d2+k2ak2qgw−w3−w3lnww3+pcd2+k2k2z.
Then, we have(46)ℒ˙x,s,y,v,w,z=x˙−d2+k2s3x+v3k1xv3x˙+s˙−s3ss˙+d2+k2k2y˙−y3yy˙+d3d2+k2ak2v˙−v3vv˙+d3d2+k2ak2qgw˙−w3ww˙+pcz˙;this fact implies that(47)ℒ˙x,s,y,v,w,z=λ−d1x−d2+k2s3x+v3k1xv3λ−d1x−k1xvx+v−s3sk1xvx+v+d2+k2s3+d3d2+k2k2y3−d2+k2sy3y−d4d3d2+k2ak2v−d3d2+k2k2v3vy+d3d4d2+k2ak2v3−d3d2+k2ak2qw3v+d3d2+k2ak2qv3w3+d2+k2k2pzy3−bc.
We know that(48)λ=d1x3+d2+k2s3,k1x3v3=x3+v3d2+k2s3,s3y3=d3k2,y3v3=d4a+qw3a,s2v2=d3d4ak2+d3qw3ak2.
So, we have(49)λ−d1x=d1x3+d2+k2s3−d1x,λ−d1x−d2+k2s3x+v3k1xv3λ−d1x−k1xvx+v=d1x31−xx3−x3xx+v3x3+v3+x+v3x3+v3+d2+k2s31−x3xx+v3x3+v3+vv3x+v3x+v−s3sk1xvx+v+d2+k2s3=d2+k2s31−s3sxvx3v3x3+v3x+v,d2+k2d3k2y3−d2+k2sy3y=d2+k2s31−ss3y3y,−d3d4d2+k2ak2v−d3d2+k2k2v3vy+d3d4d2+k2ak2v3−d3d2+k2ak2qw3v+d3d2+k2ak2qv3w3=d2+k2s31−vv3−v3vyy3.
Then, we have(50)L˙=−d1v3xx3+v3x−x32−d2+k2s3xv−v32v3x+v3x+v+d2+k2s35−x3xx+v3x3+v3−s3sxvx3v3x3+v3x+v−sy3s3y−yv3y3v−x+vx+v3,+bcd2+k2k2pzR3z−1.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(51)5−x3xx+v3x3+v3−s3sxvx3v3x3+v3x+v−sy3s3y−yv3y3v−x+vx+v3≤0,and we know that R3z<1, then L˙≤0, and the equality holds when x=x3, s=s3, y=y3, v=v3, w=w3, and z=z3. By the LaSalle invariance principle [14], the endemic point E3 is asymptotically stable when R0>1.
Finally, for the last endemic equilibriumE4, we have the following result.Proposition 6.
IfR0>1, R3z>1, and R2w>1, then the endemic point E4 is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(52)ℒx,s,y,v,w,z=x−x4−∫x4xd2+k2s4k1uv4/u+v4du+s−s4−s4lnss4+d2+k2k2y−y4−y4lnyy4+d3d2+k2+d2+k2pz4ak2×v−v4−v4lnvv4+d3d2+k2+d2+k2pz4ak2qgw−w4−w4lnww4+pcd2+k2k2z−z4−z4lnzz4.
Then, we have(53)ℒ˙x,s,y,v,w,z=λ−d1x−d2+k2s4x+v4k1xv4λ−d1x−k1xvx+v−s4sk1xvx+v+d2+k2s4+d3d2+k2k2y4−d2+k2y4ys−d3d4d2+k2+d4d2+k2pz4ak2v−d3d2+k2+d2+k2pz4k2v4vy+d3d4d2+k2+d4d2+k2pz4ak2v4−d3d2+k2+d2+k2pz4ak2qw4v+d3d2+k2+d2+k2pz4ak2qv4w4+d2+k2k2pz4y4.
We know that(54)λ=d1x4+d2+k2s4,k1x4v4=x4+v4d2+k2s4,s4y4=d3k2+pz4k2,y4v4=d4a+qw4a,s4v4=d3d4ak2+d4pz2ak2+d3qw4ak2+pqz4w4ak2,then,(55)λ−d1x=d1x4+d2+k2s4−d1x,λ−d1x−d2+k2s4x+v4k1xv4λ−d1x−k1xvx+v=d1x41−xx4−x4xx+v4x4+v4+x+v4x4+v4+d2+k2s41−x4xx+v4x4+v4+vv4x+v4x+v−s4sk1xvx+v+d2+k2s4=d2+k2s41−s4sxvx4v4x4+v4x+v,d3d2+k2k2y4+d2+k2k2py4z4−d2+k2y4ys=d2+k2s41−y4yss4,−d3d4d2+k2+d4d2+k2pz4ak2v−d3d2+k2+d2+k2pz4k2v4vy+d3d4d2+k2+d4d2+k2pz4ak2v4−d3d2+k2+d2+k2pz4ak2qw4v+d3d2+k2+d2+k2pz4ak2qv4w4=d2+k2s41−v4vyy4−vv4.
This fact implies that(56)ℒ˙x,s,y,v,w,z=−d1v4xx4+v4x−x42−d2+k2s4xv−v42v4x+v4x+v+d2+k2s45−x4xx+v4x4+v4−s4sxvx4v4x4+v4x+v−sy4s4y−yv4y4v−x+vx+v4.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(57)5−x4xx+v4x4+v4−s4sxvx4v4x4+v4x+v−sy4s4y−yv4y4v−x+vx+v4≤0,which means that ℒ˙≤0, and the equality holds when x=x4, s=s4, y=y4, v=v4, w=w4, and z=z4. By the LaSalle invariance principle [14], the endemic point E4 is globally asymptotically stable when R0>1.


## 2.1. Positivity and Boundedness

For the problems dealing with cell population evolution, the cell densities should remain nonnegative and bounded. In this section, we will establish the positivity and boundedness of solutions of the model (1). First of all, for biological reasons, the parameters x0, s0, y0, v0, w0, and z0 must be larger than or equal to 0. Hence, we have the following result:Proposition 1.
For any initial conditionsx0,s0,y0,v0,w0,z0, system (1) has a unique solution. Moreover, this solution is nonnegative and bounded for all t≥0.Proof.
By the classical functional differential equations theory (see for instance [13], and the references therein), we can confirm that there is a unique local solution xt,st,yt,vt,wt,zt to system (1) in 0,tm.
We have the following:(2)x˙x=0=λ≥0,s˙y=0=k1xvx+v≥0,y˙s=0=k2s≥0,v˙v=0=ay≥0,w˙w=0=0≥0,z˙z=0=0≥0,this shows the positivity of solutions for t∈0,tm. For the boundedness of the solutions,(3)X=x+s+y+pcz,then, we have(4)X˙=λ−d1x−d2s−d3y−bpcz≤λ−δX,where δ=mind1,d2,d3,b. So,(5)Xt≤X0e−δt+λδ1−e−δt≤X0e−δt+λδ.
Similarly, let us consider(6)V=v+qgw;therefore,(7)V˙=ay−d4v−hqgw≤ay−δV,where α=mind4,h, then,(8)Vt≤V0+aαy∞,this proves that the solutions xt, st, yt, vt, wt, and zt are bounded. Hence, every local solution can be prolonged up to any time tm>0, which means that the solution exists globally.


## 2.2. Steady States

System (1) has an infection-free equilibrium Ef=λ/d1,0,0,0,0,0, corresponding to the maximal level of healthy CD4+ T-cells. By simple calculation, the basic reproduction number of (1) is given by(9)R0=ak1k2d3d4d2+k2=k1k2d2+k2×ad3×1d4,where k2/d2+k2 is the proportion of the exposed cells to become productively infected cells, a/d3 is the number of free virus production by an infected cell, and 1/d4 is the average life of virus. From a biological point of view, R0 stands for the average number of secondary infections generated by one infected cell when all cells are susceptibles. Depending on the value of this basic reproduction number R0; in other words, depending on these three biological proportions, we will study the stability of the free-disease and the endemic equilibria. Indeed, it is easy to see that when R0>1, system (1) has four of them. The first endemic equilibrium is E1=x1,s1,y1,v1,w1,z1, where(10)x1=λd1+k11−1/R0,s1=λk11−1/R0d2+k2d1+k11−1/R0,y1=λd4R0−1ad1+ak11−1/R0,v1=λR0−1d1+k11−1/R0,w1=0,z1=0.We define the antibody immune response reproduction number by(11)R1w=gv1h=gλR0−1d1h+hk11−1/R0,where 1/h is the average life of antibodies cells and v1 is the number of free viruses at E1. For the biological significance, R1w represents the average number of the antibodies activated by virus when the viral infection is successful in the absence of CTL immune response.Furthermore, we introduce the CTL immune response reproduction number given by(12)R1z=cy1b=cλd4R0−1abd1+abk11−1/R0,where 1/b represents the average life of CTL cells and y1 is the number of infected cells at E1. Hence, R1z represents the mean of CTL immune cells activated by an infected cell when the viral infection is successful in the absence of the antibody immune response. The second endemic equilibrium is(13)E2=x2,s2,y2,v2,w2,z2,where(14)x2=−abd1−abk1+λcd4+A2cd1d4,s2=ak1b−abd1−abk1+λcd4+Acd4d2+k2abd1−abk1+λcd4+A,y2=bc,v2=abcd4,w2=0,z2=d2R0−1−ak1b+λcd4+A−abd1R0+1pabd1−ak1b+λcd4+A,with A=abk1−λcd42+a2b2d12+2a2b2d1k1+2λabcd1d4.We introduce the antibody immune competition reproduction number given by(15)R2w=gv2h=abgcd4h,with 1/h represents the average life of antibodies and v2 is the number of free viruses at E2. For biological point of view, R2w represents the average number of the antibodies activated by virus when the viral infection is successful in the absence of CTL response. The third endemic equilibrium is(16)E3=x3,s3,y3,v3,w3,z3,where(17)x3=−d1h−k1h+λg+B2d1g,s3=k1h−d1h−k1h+λg+Bgd2+k2d1h−k1h+λg+B,y3=k1k2h−d1h−k1h+λg+Bd3gd2+k2d1h−k1h+λg+B,v3=hg,w3=d4R0−1−k1h+λg+B−d1hR0+1qd1h−k1h+λg+B,z3=0,with B=hk1−λg2+d12h2+2k1d1h2k1+2λgd1h.We define the CTL immune competition reproduction numberR3z of our model by(18)R3z=cy3b=ck1k2h−d1h−k1h+λg+Bbd3gd2+k2d1h−k1h+λg+B,with 1/b represents the average life of CTL cells and y3 is the number of infected cells at E3. Hence, R3z represents the average number of CTL immune cells activated by an infected cell when the viral infection is successful in the absence of the antibody immune response. The last endemic equilibrium is(19)E4=x4,y4,v4,w4,z4,where(20)x4=−d1h−k1h+λg+B2d1g,s4=k1h−d1h−k1h+λg+Bgd2+k2d1h−k1h+λg+B,y4=bc,v4=hg,w4=abg−cd4hchq=d4qR2w−1,z4=ck1k2h−bd3gd2+k2−k1h+λg+B−d1hck1k2h+bd3gd2+k2pbgd2+k2d1h−k1h+λg+B,=d3pR3z−1.We observe that the second endemic stateE2=x2,y2,v2,w2,z2 exists when R1z>1. We explain the existence of this endemic equilibrium E2 as follows. We recall first that, in this state, both the free viruses and CTL cells are present. Assume that R0>1, in the total absence of CTL immune response, the infected cell load per unit time is λd4R0−1/ad1+ak11−1/R0. Via the six equations of model 1, CTL cells are reproduced due to infected cells stimulated per unit time being cλd4R0−1/ad1+ak11−1/R0=cy1. The CTL load during the lifespan of a CTL cell is cλd4R0−1/abd1+abk11−1/R0=R1z. If cλd4R0−1/abd1+abk11−1/R0>1, we will have the existence of the endemic equilibrium E2. We observe also that the third endemic state E3=x3,y3,v3,w3,z3 exists when R1w>1. We explain the existence of this endemic equilibrium E3 as follows. We recall first that, in this state, both of the free viruses and antibodies are present. Assume that R0>1, in the total absence of the antibody immune response, the viral load per unit time is λR0−1/d1+k11−1/R0. Via the six equations of model 1, antibodies are reproduced due to free viruses stimulation per unit time is gλR0−1/d1+k11−1/R0=gv1. The viral load during the lifespan of virion is gλR0−1/d1h+hk11−1/R0=R1w. If gλR0−1/d1h+hk11−1/R0>1, we will have the existence of the endemic equilibrium E3. Similarly, one can see that E4=x4,y4,v4,w4,z4 exists when R3z>1 and R2w>1.


## 2.3. Global Stability of the Disease-Free Equilibrium

For the global stability of the disease-free equilibrium, we have the following result.Proposition 2.
IfR0≤1, then the endemic point Ef is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(21)ℒx,y,v,w,z=s+d2+k2k2y+d3d2+k2ak2v+qgd3d2+k2ak2w+pcd2+k2k2z.
The time derivative is given by(22)ℒ˙x,s,y,v,w,z=s˙+d2+k2k2y˙+d3d2+k2ak2v˙+qgd3d2+k2ak2w˙+pcd2+k2k2z˙,ℒ˙x,s,y,v,w,z=k1xvx+v−d2+k2s+d2+k2k2k2s−d3y−d2+k2k2pyz+d3d2+k2ak2ay−d4v−qvw+qgd3d2+k2ak2gvw−hw+pcd2+k2k2cyz−bz,ℒ˙x,s,y,v,w,z=k1xvx+v−d3d2av−qhgd3d2+k2ak2w−bpcd2+k2k2z,ℒ˙x,y,s,v,w,z≤k1v−d3d4d2+k2ak2v≤d3d4d2+k2ak2R0−1v.
IfR0<1, then ℒ˙≤0. Moreover, ℒ˙≤0 when v=0. The largest compact invariant is(23)E=x,s,y,v,w,z∣v=0.
According to LaSalle’s invariance principle [14], we have lim+∞vt=0. The limit system of equations is(24)x˙=λ−d1x,s˙=−d2s−k2s,y˙=k2s−d3y−pyz,w˙=−hw,z˙=cyz−bz.
We define(25)ℒx,s,y,w,z=1x0x−x0−x0lnxx0+s+d2+k2k2y+qgd3d2+k2ak2w+d2+k2k2pcz.
Sincex0=λ/d1, then(26)ℒ˙x,s,y,w,z=d12−xx0−x0x−d3d2+k2k2y−qghd3d2+k2ak2w−d2+k2k2pbcz.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(27)2−xx0−x0x≤0.
Therefore,ℒ˙≤0, and the equality holds if x=x0 and s=y=w=z=0, which complete the proof.


## 2.4. Global Stability of Infection Steady States

In this subsection, attention is focused on the stability of the infection steady states.For the first endemic equilibriumE1, we have the following result.Proposition 3.
IfR0>1, R1z≤1, and R1w≤1, then the endemic point E1 is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(28)ℒx,s,y,v,w,z=x−x1+∫x1xd2+k2s1k1uv1/u+v1du+s−s1−s1lnss1+d2+k2k2y−y1−y1lnyy1+d3d2+k2ak2v−v1−v1lnvv1+qgd3d2+k2ak2w+pcd2+k2k2z.we have then(29)ℒ˙x,s,y,v,w,z=x˙−d2+k2s1x+v1k1xv1x˙+s˙−s1ss˙+d2+k2k2y˙−y1yy˙+d3d2+k2ak2v˙−v1vv˙+qgd3d2+k2ak2w˙+pcd2+k2k2z˙.
On the other hand, we have(30)λ=d1x1+d2+k2s1,k1x1v1x1+v1=d2+k2s1,s1v1=d3d4ak2,y1v1=d4a,s1y1=d3k2.
Hence,(31)ℒ˙x,s,y,v,w,z=λ−d1x−k1xvx+v1−d2+k2s1x+v1k1xv1+k1xvx+v−d2+k2s−s1sk1xvx+v−d2+k2s+d2+k2k2k2s−d3y−pyz−d2+k2k2y1yk2s−d3y−pyz+d3d2+k2ak2ay−d4v−qvw−d3d2+k2ak2v1vay−d4v−qvw+qgd3d2+k2ak2gvw−hw+pcd2+k2k2cyz−bz,ℒ˙x,s,y,v,w,z=λ−d1x−d2+k2s1x+v1k1xv1λ−d1x−k1xvx+v−s1sk1xvx+v+d2+k2s1+d2+k2k2py1z+d2+k2d3k2y1−d2+k2sy1y−d3d4d2+k2ak2v+d2+k2d3k2y1−d2+k2sy1y−d3d4d2+k2ak2v−qhgd3d2+k2ak2w+d3d2+k2ak2v1qw−d2+k2k2bpcz.
Thus, this fact implies that(32)ℒ˙x,s,y,v,w,z=λ−d1x−d2+k2s1x+v1k1xv1λ−d1x−k1xvx+v−s1sk1xvx+v+d2s1−d3d4d2+k2ak2v−d3d2+k2k2v1vy+d3d2+k2ak2v1−qhgd3d2+k2ak2w+d3d2+k2ak2qv1w+pzy1−bc.
Since(33)λ−d1x=d1x1+d2+k2s1−d1x,λ−d1x−d2+k2s1x+v1k1xv1λ−d1x−k1xvx+v=d1x11−xx1−x1xx+v1x1+v1+x+v1x1+v1+d2+k2s11−x1xx+v1x1+v1+vv1x+v1x+v−s1sk1xvx+v+d2+k2s1=−s1sxvx1v1x1+v1x+vd2+k2s1+d2+k2s1,
We have(34)ℒ˙x,s,y,v,w,z=d1x11−xx1−x1xx+v1x1+v1+x+v1x1+v1+d2+k2s11−x1xx+v1x1+v1+vv1x+v1x+v+d2+k2s11−s1sxvx1v1x1+v1x+v+d2+k2s11−sy1s1y−vv1+d2+k2s11−v1yy1v+d3d2+k2ak2qwv1−hg+pzd2+k2k2y1−bc.
Therefore,(35)ℒ˙=−d1v1xx1+v1x−x12+d2+k2s1−1−vv1+vv1x+v1x+v+x+vx+v1+d2+k2s15−x1xx+v1x1+v1−s1sxvx1v1x1+v1x+v−sy1s1y−yv1y1v−x+vx+v1+d3d2+k2ak2hgqwR1w−1+d2+k2k2bcpzR1z−1,which implies that(36)ℒ˙=−d1v1xx1+v1x−x12−d2+k2s1xv−v12v1x+v1x+v+d2+k2s15−x1xx+v1x1+v1−s1sxvx1v1x1+v1x+v−sy1s1y−yv1y1v−x+vx+v1+d3d2+k2ak2hgqwR1w−1+d2+k2k2bcpzR1z−1.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(37)5−x1xx+v1x1+v1−s1sxvx1v1x1+v1x+v−sy1s1y−yv1y1v−x+vx+v1≤0,and we know that R1z<1 and R1w<1, then L˙≤0, and the equality holds when x=x1, y=y1, v=v1, w=w1, and z=z1. By the LaSalle invariance principle [14], the endemic point E1 is asymptotically stable when R0>1.
For the second endemic equilibriumE2, we have the following result.Proposition 4.
IfR0>1, R1z>1, and R2w≤1, then the endemic point E2 is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(38)ℒx,s,y,v,w,z=x−x2−∫x2xd2+k2s2k1uv2/u+v2du+s−s2−s2lnss2+d2+k2k2y−y2−y2lnyy2+d3d2+k2+d2+k2pz2ak2×v−v2−v2lnvv2+qgd3d2+k2+d2+k2pz2ak2w+pcd2+k2k2z−z2−z2lnzz2,then, we have(39)ℒ˙x,y,s,v,w,z=x˙−d2+k2s2x+v2k1xv2x˙+s˙−s2ss˙+d2+k2k2y˙−y2yy˙+d3d2+k2+d2+k2pz2ak2v˙−v2vv˙+qgd3d2+k2+d2+k2pz2ak2w˙+pcd2+k2k2z˙−z2zz˙.
We know that(40)λ=d1x2+d2+k2s2,k1x2v2=x2+v2d2+k2s2,s2y2=d3k2+pz2k2,y2v2=d4a,s2v2=d3d4ak2+d4pz2ak2,so, we have(41)ℒ˙x,s,y,v,z=λ−d1x−x2xx+v2x2+v2λ−d1x+d2+k2s2vv2x2+v2x+v−d2+k2s−s2sk1xvx+v−d2+k2s+d2+k2k2k2s−d3y−pyz−d2+k2k2y2yk2s−d3y−pyz+d3d2+k2+d2+k2pz2ak2×ay−d4v−qvw−d3d2+k2+d2+k2pz2ak2v2vay−d4v−qvw+qgd3d2+k2+d2+k2pz2ak2gvw−hw+pcd2+k2k2cyz−bz−pz2czd2+k2k2cyz−bz.
On the other hand, we have(42)λ−d1x=d1x2+d2+k2s2−d1x,λ−d1x−d2+k2s2x+v2k1xv2λ−d1x−k1xvx+v=d1x21−xx2−x2xx+v2x2+v2+x+v2x2+v2+d2+k2s21−x2xx+v2x2+v2+vv2x+v2x+v−s2sk1xvx+v+d2+k2s2=d2+k2s21−s2sxvx2v2x2+v2x+v,d2+k2d3k2y2−d2+k2sy2y−d2+k2d3d4ak2v=d2+k2s21−ss2y2y−vv2+d2+k2k2pz2y2vv2−d2+k2k2pz2y2,d4d3d2+k2ak2v2−d3d2+k2k2v2vy=d2+k2s21−yy2v2v+d2+k2k2pz2yv2v−d2+k2k2pz2y2,d2+k2pz2ak2ay−d4v−d2+k2pz2ak2v2vay−d4v=d2+k2pz2yk2−d2+k2pz2y2k2vv2−d2+k2pz2yk2v2v+d2+k2pz2y2k2,d2+k2pz2ak2ay−d4v−d2+k2pz2ak2v2vay−d4v−z2zpcd2+k2k2cyz−bz+d2+k2k2pz2y2vv2−d2+k2k2pz2y2+d2+k2k2pz2yv2v−d2+k2k2pz2y2=0.
This fact implies that(43)L˙=−d1v2xx2+v2x−x22−d2+k2s2xv−v22v2x+v2x+v+d2+k2s25−x2xx+v2x2+v2−s2sxvx2v2x2+v2x+v−sy2s2y−yv2y2v−x+vx+v2+hgd3d2+k2+d2+k2pz2ak2qwR2w−1.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(44)5−x2xx+v2x2+v2−s2sxvx2v2x2+v2x+v−sy2s2y−yv2y2v−x+vx+v2≤0,and we know that R2w<1 which means that L˙≤0, and the equality holds when x=x2, s=s2, y=y2, v=v2, w=w2, and z=z2. By the LaSalle invariance principle [14], the endemic point E2 is asymptotically stable.
For the third endemic equilibriumE3, we have the following result.Proposition 5.
IfR0>1, R3z≤1, and R1w>1, then the endemic point E3 is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(45)ℒx,s,y,v,w,z=x−x3−∫x3xd2+k2s3k1uv3/u+v3du+s−s3−s3lnss3+d2+k2k2y−y3−y3lnyy3+d3d2+k2ak2×v−v3−v3lnvv3+d3d2+k2ak2qgw−w3−w3lnww3+pcd2+k2k2z.
Then, we have(46)ℒ˙x,s,y,v,w,z=x˙−d2+k2s3x+v3k1xv3x˙+s˙−s3ss˙+d2+k2k2y˙−y3yy˙+d3d2+k2ak2v˙−v3vv˙+d3d2+k2ak2qgw˙−w3ww˙+pcz˙;this fact implies that(47)ℒ˙x,s,y,v,w,z=λ−d1x−d2+k2s3x+v3k1xv3λ−d1x−k1xvx+v−s3sk1xvx+v+d2+k2s3+d3d2+k2k2y3−d2+k2sy3y−d4d3d2+k2ak2v−d3d2+k2k2v3vy+d3d4d2+k2ak2v3−d3d2+k2ak2qw3v+d3d2+k2ak2qv3w3+d2+k2k2pzy3−bc.
We know that(48)λ=d1x3+d2+k2s3,k1x3v3=x3+v3d2+k2s3,s3y3=d3k2,y3v3=d4a+qw3a,s2v2=d3d4ak2+d3qw3ak2.
So, we have(49)λ−d1x=d1x3+d2+k2s3−d1x,λ−d1x−d2+k2s3x+v3k1xv3λ−d1x−k1xvx+v=d1x31−xx3−x3xx+v3x3+v3+x+v3x3+v3+d2+k2s31−x3xx+v3x3+v3+vv3x+v3x+v−s3sk1xvx+v+d2+k2s3=d2+k2s31−s3sxvx3v3x3+v3x+v,d2+k2d3k2y3−d2+k2sy3y=d2+k2s31−ss3y3y,−d3d4d2+k2ak2v−d3d2+k2k2v3vy+d3d4d2+k2ak2v3−d3d2+k2ak2qw3v+d3d2+k2ak2qv3w3=d2+k2s31−vv3−v3vyy3.
Then, we have(50)L˙=−d1v3xx3+v3x−x32−d2+k2s3xv−v32v3x+v3x+v+d2+k2s35−x3xx+v3x3+v3−s3sxvx3v3x3+v3x+v−sy3s3y−yv3y3v−x+vx+v3,+bcd2+k2k2pzR3z−1.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(51)5−x3xx+v3x3+v3−s3sxvx3v3x3+v3x+v−sy3s3y−yv3y3v−x+vx+v3≤0,and we know that R3z<1, then L˙≤0, and the equality holds when x=x3, s=s3, y=y3, v=v3, w=w3, and z=z3. By the LaSalle invariance principle [14], the endemic point E3 is asymptotically stable when R0>1.
Finally, for the last endemic equilibriumE4, we have the following result.Proposition 6.
IfR0>1, R3z>1, and R2w>1, then the endemic point E4 is globally asymptotically stable.Proof.
Let the following Lyapunov functional be(52)ℒx,s,y,v,w,z=x−x4−∫x4xd2+k2s4k1uv4/u+v4du+s−s4−s4lnss4+d2+k2k2y−y4−y4lnyy4+d3d2+k2+d2+k2pz4ak2×v−v4−v4lnvv4+d3d2+k2+d2+k2pz4ak2qgw−w4−w4lnww4+pcd2+k2k2z−z4−z4lnzz4.
Then, we have(53)ℒ˙x,s,y,v,w,z=λ−d1x−d2+k2s4x+v4k1xv4λ−d1x−k1xvx+v−s4sk1xvx+v+d2+k2s4+d3d2+k2k2y4−d2+k2y4ys−d3d4d2+k2+d4d2+k2pz4ak2v−d3d2+k2+d2+k2pz4k2v4vy+d3d4d2+k2+d4d2+k2pz4ak2v4−d3d2+k2+d2+k2pz4ak2qw4v+d3d2+k2+d2+k2pz4ak2qv4w4+d2+k2k2pz4y4.
We know that(54)λ=d1x4+d2+k2s4,k1x4v4=x4+v4d2+k2s4,s4y4=d3k2+pz4k2,y4v4=d4a+qw4a,s4v4=d3d4ak2+d4pz2ak2+d3qw4ak2+pqz4w4ak2,then,(55)λ−d1x=d1x4+d2+k2s4−d1x,λ−d1x−d2+k2s4x+v4k1xv4λ−d1x−k1xvx+v=d1x41−xx4−x4xx+v4x4+v4+x+v4x4+v4+d2+k2s41−x4xx+v4x4+v4+vv4x+v4x+v−s4sk1xvx+v+d2+k2s4=d2+k2s41−s4sxvx4v4x4+v4x+v,d3d2+k2k2y4+d2+k2k2py4z4−d2+k2y4ys=d2+k2s41−y4yss4,−d3d4d2+k2+d4d2+k2pz4ak2v−d3d2+k2+d2+k2pz4k2v4vy+d3d4d2+k2+d4d2+k2pz4ak2v4−d3d2+k2+d2+k2pz4ak2qw4v+d3d2+k2+d2+k2pz4ak2qv4w4=d2+k2s41−v4vyy4−vv4.
This fact implies that(56)ℒ˙x,s,y,v,w,z=−d1v4xx4+v4x−x42−d2+k2s4xv−v42v4x+v4x+v+d2+k2s45−x4xx+v4x4+v4−s4sxvx4v4x4+v4x+v−sy4s4y−yv4y4v−x+vx+v4.
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that(57)5−x4xx+v4x4+v4−s4sxvx4v4x4+v4x+v−sy4s4y−yv4y4v−x+vx+v4≤0,which means that ℒ˙≤0, and the equality holds when x=x4, s=s4, y=y4, v=v4, w=w4, and z=z4. By the LaSalle invariance principle [14], the endemic point E4 is globally asymptotically stable when R0>1.


## 3. Numerical Results

For our numerical simulations, system (1) is solved using the Runge–Kutta method iterative scheme. The numerical ranges of our parameters are given in Table 1. Figure 2 shows the behavior of disease during the first 60 days of observation. From this figure, we observe that the solution converges to the point Ef=827.22,0,0,0,0,0. With these chosen parameters, we have R0=0.22<1, which proves that Ef is stable. This supports our theoretical findings. Figure 3 shows the behavior of the disease during 60 first days. From this figure, we observe that the solution of (1) converges towards the point E1=33.17,1.33,2.54,4.24×102,0,0. With these chosen parameters, we have R0=13.81>1, R1z=7.64×10−1<1, and R1w=4.24×10−8<1. This fact supports that E1 is stable. Figure 4 shows the behavior of disease during 60 days. We observe that the solution of (1) converges towards the endemic point E2=1.96×102,6.32,3.33,5.55×102,0,6.28×102. In this figure, we have R0=13.81>1, R1z=3.81>1, and R2w=5.55×10−8<1, which supports the fact that E2 is stable. Figure 5 shows the behavior of disease during the first 60 days of observation. We clearly see that the solution of (1) converges towards the endemic point E3=32.07,1.35,2.57,1000,4.45,0. With the chosen parameters, we have R0=2.39×102>1, R3z=0.77<1, and R1w=9.61>1; this supports the stability of E3. In addition, Figure 6 shows the behavior of disease for the first 60 days. We remark that the solution converges towards the last endemic point E4=1.77×102,6.55,3.33,1000,4.46,6.61×102. With the used parameters, we have R0=75.31>1, R3z=3.74>1, and R2w=3.03>1; this confirms the theoretical result concerning the stability of E4.Table 1
Parameters and their symbols and default values used in the suggested HIV model.
Parameters
Units
Meaning
Value
References
Λ
cells
μ
l
−
1
day
−
1
Source rate of CD4+ T cells
0,10
[15]
k
1
μ
l
virion
−
1
day
−
1
Average of infection
2.5
×
10
−
4
,
0.5
[9]
d
1
day
−
1
Decay rate of healthy cells
0.0139
[9]
d
2
day
−
1
Death rate of exposed CD4+ T cells
0.0495
[9]
k
2
day
−
1
The rate that exposed cells become infected CD4+ T cells
1.1
[9]
d
3
day
−
1
Death rate of infected CD4+ T cells, not by CTL killing
0.5776
[9]
a
day
−
1
The rate of production the virus by infected CD4+ T cells
2,1250
[9]
d
4
day
−
1
Clearance rate of virus
0.3466
,
2.4
[9]
Q
μ
l virion
days
−
1
Killing rate of antibody
0.5
[16]
G
μ
l virion
days
−
1
Activation rate CTL cells
10
−
11
,
10
−
4
[16]
H
day
−
1
Death rate of antibody
0.1
[16]
P
μ
l
cell
−
1
day
−
1
Clearance rate of infection
0.0024
[17]
C
cells
cell
−
1
day
−
1
Activation rate CTL cells
0.15
[17]
b
day
−
1
Death rate of CTL cells
0.5
[17]Figure 2
The behavior of the disease forλ=10, d1=0.0139, k1=0.04, d2=0.0495, k2=1.1, d3=0.5776, a=2, d4=0.6, q=0.05, g=10−11, h=0.1, p=0.0024, c=0.15, and b=0.5.
(a)
(b)
(c)
(d)
(e)
(f)Figure 3
The behavior of the disease forλ=2, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=100, d4=0.6, q=0.05, g=10−11, h=0.1, p=0.0024, c=0.15, and b=0.5.
(a)
(b)
(c)
(d)
(e)
(f)Figure 4
The behavior of the disease forλ=10, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=100, d4=0.6, q=0.05, g=10−11, h=0.1, p=0.0024, c=0.15, and b=0.5.
(a)
(b)
(c)
(d)
(e)
(f)Figure 5
The behavior of the disease forλ=2, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=500, d4=0.6, q=0.05, g=10−4, h=0.1, p=0.0024, c=0.15, and b=0.5.
(a)
(b)
(c)
(d)
(e)
(f)Figure 6
The behavior of the disease forλ=10, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=800, d4=0.6, q=0.5, g=10−4, h=0.1, p=0.0024, c=0.15, and b=0.5.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.1. Comparison with the Clinical Data

First, define the following objective function:(58)J=1n∑i=1nlogvti−logv^ti2,where vti represents the virus concentration at time ti using the mathematical model (1) and v^ti represents the virus concentration clinical data at time ti [18].The numerical simulations are performed and compared to three patients’ data picked from [18]. The data were from the University of Washington study [7] and from the Aaron Diamond AIDS Research Center (see Table 2).Table 2
The used clinical data [18] for Figure 7 (A), for Figure 8 (B), and for Figure 9 (C).
Clinical day test
Viral load (virions perμl)
A
22
27.7
43
210
78
85.9
106
81.1
B
0
228.8
2
599.2
14
169.6
21
93.7
42
165.6
98
127
C
0
1350.6
9
337.2
12
340.6
16
202.3
19
169.7
23
141.4
26
56.48
30
182.75
50
267
60
182.7In Figure7, the dots show the evolution of the infection during the first 120 days for the first patient [18], while the solid curve represents the numerical simulation of our suggested model. The error between the numerical simulation and the clinical data is approximately J≈2.378×10−1 which indicates that the numerical simulation is a good approximation of the clinical data. Figures 8 and 9 show a comparison between the clinical data (dots) and the mathematical model (solid line), and the error is approximately J≈8.43×10−2 and J≈1.64×10−1, respectively. These three results indicate that our mathematical model can fit the clinical data of different patients for the first days of observations. However, the limit of our model is to predict a long time behavior of the infection disease.Figure 7
Comparison between the mathematical model (solid line) and the clinical data of the first patient [18] (dots). The used parameters for the model are λ=10, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=850, d4=0.6, q=0.5, g=1.2×10−3, h=0.1, p=0.0024, c=0.15, and b=0.5.Figure 8
Comparison between the mathematical model (solid line) and the clinical data of the fifth patient [18] (dots). The used parameters for the model are λ=10, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=650, d4=0.6, q=0.5, g=10−3, h=0.12, p=0.0024, c=0.15, and b=0.5.Figure 9
Comparison between the mathematical model (solid line) and the clinical data of the seventh patient [18] (dots). The used parameters for the model are λ=10, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=600, d4=0.6, q=0.5, g=10−3, h=0.182, p=0.0024, c=0.15, and b=0.5.


### 3.2. Sensitivity Analysis

Using the method outlined in [19], we perform a sensitivity analysis using partial rank correlation coefficients (PRCC) to identify the main drivers of the basic reproduction number R0. Parameters were tested within the ranges given in Table 1.In Figure10, we observe that a and k1 is highly positively correlated with R0. However, d4 has a strong negative correlation with R0. The other parameters k2, d2, and d3 present a weak correlation with R0. From the biological point of view, the sensitivity analysis shows that an increase of production rate of the virus by infected cells a or an increase of the infection rate k1 leads to an increase of the basic reproduction number R0. However, an increase in the clearance rate of virus d4 leads to a significant decease of the basic reproduction number R0.Figure 10
Sensitivity analysis ofR0 to different input parameters of the model.


## 3.1. Comparison with the Clinical Data

First, define the following objective function:(58)J=1n∑i=1nlogvti−logv^ti2,where vti represents the virus concentration at time ti using the mathematical model (1) and v^ti represents the virus concentration clinical data at time ti [18].The numerical simulations are performed and compared to three patients’ data picked from [18]. The data were from the University of Washington study [7] and from the Aaron Diamond AIDS Research Center (see Table 2).Table 2
The used clinical data [18] for Figure 7 (A), for Figure 8 (B), and for Figure 9 (C).
Clinical day test
Viral load (virions perμl)
A
22
27.7
43
210
78
85.9
106
81.1
B
0
228.8
2
599.2
14
169.6
21
93.7
42
165.6
98
127
C
0
1350.6
9
337.2
12
340.6
16
202.3
19
169.7
23
141.4
26
56.48
30
182.75
50
267
60
182.7In Figure7, the dots show the evolution of the infection during the first 120 days for the first patient [18], while the solid curve represents the numerical simulation of our suggested model. The error between the numerical simulation and the clinical data is approximately J≈2.378×10−1 which indicates that the numerical simulation is a good approximation of the clinical data. Figures 8 and 9 show a comparison between the clinical data (dots) and the mathematical model (solid line), and the error is approximately J≈8.43×10−2 and J≈1.64×10−1, respectively. These three results indicate that our mathematical model can fit the clinical data of different patients for the first days of observations. However, the limit of our model is to predict a long time behavior of the infection disease.Figure 7
Comparison between the mathematical model (solid line) and the clinical data of the first patient [18] (dots). The used parameters for the model are λ=10, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=850, d4=0.6, q=0.5, g=1.2×10−3, h=0.1, p=0.0024, c=0.15, and b=0.5.Figure 8
Comparison between the mathematical model (solid line) and the clinical data of the fifth patient [18] (dots). The used parameters for the model are λ=10, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=650, d4=0.6, q=0.5, g=10−3, h=0.12, p=0.0024, c=0.15, and b=0.5.Figure 9
Comparison between the mathematical model (solid line) and the clinical data of the seventh patient [18] (dots). The used parameters for the model are λ=10, d1=0.0139, k1=0.05, d2=0.0495, k2=1.1, d3=0.5776, a=600, d4=0.6, q=0.5, g=10−3, h=0.182, p=0.0024, c=0.15, and b=0.5.


## 3.2. Sensitivity Analysis

Using the method outlined in [19], we perform a sensitivity analysis using partial rank correlation coefficients (PRCC) to identify the main drivers of the basic reproduction number R0. Parameters were tested within the ranges given in Table 1.In Figure10, we observe that a and k1 is highly positively correlated with R0. However, d4 has a strong negative correlation with R0. The other parameters k2, d2, and d3 present a weak correlation with R0. From the biological point of view, the sensitivity analysis shows that an increase of production rate of the virus by infected cells a or an increase of the infection rate k1 leads to an increase of the basic reproduction number R0. However, an increase in the clearance rate of virus d4 leads to a significant decease of the basic reproduction number R0.Figure 10
Sensitivity analysis ofR0 to different input parameters of the model.


## 4. Conclusion

In this paper, we have presented and studied a mathematical model describing HIV viral infection with saturated rate in the presence of the adaptive immune response. This adaptive immunity is represented by CTL immune response and antibodies. By using suitable Lyapunov functionals, the global stability of each equilibrium has been established. More precisely, the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is below unity (R0≤1). Also, the endemic steady state E1 is globally asymptotically stable when R0≥1, R1z≤1, and R1w≤1. In presence of the adaptive immune response governed by competition between CTL and antibody responses, system (1) admits three infection steady states. The first infection steady state E2 is with only the presence of CTL response which is globally asymptotically stable if R1z≥1 and R2w≤1. The second infection steady state E3 is with only the presence of the antibody response which is globally asymptotically stable if R1w≥1 and R3z≤1. The third infection steady state is E4 with the activation of the antibodies and the CTL response at the same time. In this case, this equilibrium E4 is globally asymptotically stable when R2w≥1 and R3z≥1. In addition, different numerical simulations are performed in order to confirm the theoretical findings and to show that the adaptive immune response is responsible to reduce the viral load, increase the uninfected cells, and decrease the infected cells. Moreover, a comparison with some clinical data shows that our suggested model can be considered as a good approximation of the clinical tests especially for the first days of observation.



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*Source: 7673212-2019-11-16.xml*