# A Fractional Order Model for Viral Infection with Cure of Infected Cells and Humoral Immunity

Authors: Adnane Boukhouima; Khalid Hattaf; Noura Yousfi
Journal: International Journal of Differential Equations
 (2018)

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## Abstract

In this paper, we study the dynamics of a viral infection model formulated by five fractional differential equations (FDEs) to describe the interactions between host cells, virus, and humoral immunity presented by antibodies. The infection transmission process is modeled by Hattaf-Yousfi functional response which covers several forms of incidence rate existing in the literature. We first show that the model is mathematically and biologically well-posed. By constructing suitable Lyapunov functionals, the global stability of equilibria is established and characterized by two threshold parameters. Finally, some numerical simulations are presented to illustrate our theoretical analysis.


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## Body

## 1. Introduction

The immune response plays an important role to control the dynamics of viral infections such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), and human T-cell leukemia virus (HTLV). Therefore, many mathematical models have been developed to incorporate the role of immune response in viral infections. Some of these models considered the cellular immune response mediated by cytotoxic T lymphocytes (CTL) cells that attack and kill the infected cells [1–5] and the others considered the humoral immune response based on the antibodies which are produced by the B-cells and are programmed to neutralize the viruses [6–11]. However, all these models have been formulated by using ordinary differential equations (ODEs) in which the memory effect is neglected while the immune response involves memory [12, 13].Fractional derivative is a generalization of integer derivative and it is a suitable tool to model real phenomena with memory which exists in most biological systems [14–16]. The fractional derivative is a nonlocal operator in contrast to integer derivative. This means that if we want to compute the fractional derivative at some point t=t1, it is necessary to take into account the entire history from the starting point t=t0 up to the point t=t1. For these reasons, modeling some real process by using fractional derivative has drawn attention of several authors in various fields [17–22]. In biology, it has been shown that the fractional derivative is useful to analyse the rheological proprieties of cells [23]. Furthermore, it has been deduced that the membranes of cells of biological organism have fractional order electrical conductance [24]. Recently, much works have been done on modeling the dynamics of viral infections with FDEs [25–31]. These works ignored the impact of the immune response and the majority of them deal only with the local stability.In some viral infections, the humoral immune response is more effective than cellular immune response [32]. For this reason, we improve the above ODE and FDE models by proposing a new fractional order model that describes the interactions between susceptible host cells, viral particles, and the humoral immune response mediated by the antibodies; that is,(1)Dαxt=λ-dx-fx,vv+ρl,Dαlt=fx,vv-m+ρ+γl,Dαyt=γl-ay,Dαvt=ky-μv-qvw,Dαwt=gvw-hw,where x(t), l(t), y(t), v(t), and w(t) are the concentrations of susceptible host cells, latently infected cells (infected cells which are not yet able to produce virions), productive infected cells, free virus particles, and antibodies at time t, respectively. Susceptible host cells are assumed to be produced at a constant rate λ, die at the rate dx, and become infected by virus at the rate f(x,v)v. Latently infected cells die at the rate ml and return to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus at the rate ρl. Productive infected cells are produced from latently infected cells at the rate γl and die at the rate ay. Free virus particles are produced from productive infected cells at the rate ky, cleared at the rate μv, and are neutralized by antibodies at the rate qvw. Antibodies are activated against virus at the rate gvw and die at the rate hw.In system (1), Dα represents the Caputo fractional derivative of order α defined for an arbitrary function φ by (2)Dαφt=1Γ1-α∫0tφ′ut-uαdu,with 0<α≤1 [33]. Further, the infection transmission process in (1) is modeled by Hattaf-Yousfi functional response [34] which was recently used in [35, 36] and has the form f(x,v)=βx/(α0+α1x+α2v+α3xv), where α0,α1,α2,α3≥0 are the saturation factors measuring the psychological or inhibitory effect and β>0 is the infection rate. In addition, this functional response generalizes many common types existing in the literature such as the specific functional response proposed by Hattaf et al. in [37] and used in [2, 31] when α0=1; the Crowley-Martin functional response introduced in [38] and used in [39] when α0=1 and α3=α1α2; and the Beddington-DeAngelis functional response proposed in [40, 41] and used in [3, 4, 10] when α0=1 and α3=0. Also, the Hattaf-Yousfi functional response is reduced to the saturated incidence rate used in [9] when α0=1 and α1=α3=0 and the standard incidence function used in [27] when α0=α3=0 and α1=α2=1, and it was simplified to the bilinear incidence rate used in [5, 6] when α0=1 and α1=α2=α3=0.On the other hand, system (1) becomes a model with ODEs when α=1, which improves and generalizes the ODE model with bilinear incidence rate [42], the ODE model with saturated incidence rate [43], and the ODE model with specific functional response [44].The rest of the paper is organized as follows. The next section deals with some basic proprieties of the solutions and the existence of equilibria. The global stability of equilibria is established in Section3. To verify our theoretical results, we provide some numerical simulations in Section 4, and we conclude in Section 5.


## 2. Basic Properties and Equilibria

In this section, we will show that our model is well-posed and we discuss the existence of equilibria.Since system (1) describes the evolution of cells, then we need to prove that the cell numbers should remain nonnegative and bounded. For biological considerations, we assume that the initial conditions of (1) satisfy(3)x0≥0,l0≥0,y0≥0,v0≥0,w0≥0.Then we have the following result.Theorem 1.
Assume that the initial conditions satisfy (3). Then there exists a unique solution of system (1) defined on 0,+∞. Moreover, this solution remains nonnegative and bounded for all t≥0.Proof.
First, system (1) can be written as follows:(4)DαXt=FX,where(5)Xt=xtltytvtwtandFX=λ-dx-fx,vv+ρlfx,vv-m+ρ+γlγl-ayky-μv-qvwgvw-hw.
It is important to note that whenα=1, (4) becomes a system with ODEs. In this case, we refer the reader to [45] for the existence of solutions and to the works [46–50] for the stability of equilibria. In the case of FDEs, we will use Lemma 2.4 in [31] to prove the existence and uniqueness of solutions. Hence, we put (6)ζ=λ0000,A=-dρ0000-m+ρ+γ0000γ-a0000k-μ00000-handC=0000000000000000000-q0000g.We discuss four cases:(i)
Ifα0≠0, F(X) can be formulated as follows: (7)FX=ζ+AX+α0α0+α1x+α2v+α3xvvB0X+vCX,
where(8)B0=-βα00000βα00000000000000000000.Hence,(9)FX≤ζ+A+vB0+CX.(ii)
Ifα1≠0, we can write F(X) in the form (10)FX=ζ+AX+α1xα0+α1x+α2v+α3xvB1X+vCX,
where(11)B1=000-βα10000βα10000000000000000.Moreover, we get(12)FX≤ζ+A+B1+vCX.(iii)
Ifα2≠0, we have (13)FX=ζ+AX+α2vα0+α1x+α2v+α3xvB2X+vCX,
where(14)B2=-βα20000βα20000000000000000000.Further, we obtain(15)FX≤ζ+A+B2+vCX.(iv)
Ifα3≠0, we have (16)FX=ζ+AX+α3xvα0+α1x+α2v+α3xvB3+vCX,
where(17)B3=-βα3βα3000.Then(18)FX≤ζ+B3+A+vCX.Hence, the conditions of Lemma 2.4 in [31] are verified. Then system (1) has a unique solution on 0,+∞. Now, we show the nonnegativity of solutions. By (1), we have (19)Dαxtx=0=λ+ρl≥0,Dαltl=0=fx,vv≥0,Dαyty=0=γl≥0,Dαvtv=0=ky≥0,Dαwtw=0=0≥0.As in [31, Theorem 2.7], we deduce that the solution of (1) is nonnegative.
Finally, we prove the boundedness of solutions. We define the function(20)Tt=xt+lt+yt+a2kvt+aq2kgwt.Then, we have (21)DαTt=Dαxt+Dαlt+Dαyt+a2kDαvt+aq2kgDαwt=λ-dxt-mlt-a2yt-aμ2kvt-aqh2kgwt≤λ-δTt,where δ=min⁡{d,m,a/2,μ,h}. Thus, we obtain (22)Tt≤T0Eα-δtα+λδ1-Eα-δtα.Since 0≤Eα(-δtα)≤1, we get (23)Tt≤T0+λδ.This completes the proof.Now, we discuss the existence of equilibria. It is clear that system (1) has always an infection-free equilibrium E0λ/d,0,0,0,0. Then the basic reproduction number of (1) is as follows: (24)R0=kβλγaμm+ρ+γdα0+λα1.To find the other equilibria, we solve the following system:(25)λ-dx-fx,vv+ρl=0,(26)fx,vv-m+ρ+γl=0,(27)γl-ay=0,(28)ky-μv-qvw=0,(29)gvw-hw=0.From (29), we get w=0 or v=h/g. Then we discuss two cases.Ifw=0, by (25)-(28), we have l=(λ-dx)/(m+γ), y=γ(λ-dx)/a(m+γ), v=kγ(λ-dx)/aμ(m+γ), and(30)fx,kγλ-dxaμm+γ=aμm+ρ+γkγ.Since l≥0, y≥0, and v≥0, then x≤λ/d. Consequently, there is no equilibrium when x>λ/d.We define the functionh1 on 0,λ/d by (31)h1x=fx,kγλ-dxaμm+γ-aμm+ρ+γkγ.We have h1(0)=-aμ(m+ρ+γ)/kγ<0, h1′(x)=∂f/∂x-(kγd/aμm+γ)(∂f/∂v)>0, and h1λ/d=(aμm+ρ+γ/kγ)(R0-1).Hence ifR0>1, (30) has a unique root x1∈0,λ/d. As a result, when R0>1 there exists an equilibrium E1(x1,l1,y1,v1,0) satisfying x1∈0,λ/d, l1=(λ-dx1)/(m+γ), y1=γ(λ-dx1)/a(m+γ), and v1=kγ(λ-dx1)/aμ(m+γ).Ifw≠0, then v=h/g. By (25)-(27), we obtain l=(λ-dx)/(m+γ), y=γ(λ-dx)/a(m+γ), w=kγg(λ-dx)/aqh(m+γ)-μ/q, and(32)fx,hg=gm+ρ+γhm+γλ-dx.Since l≥0, y≥0, and w≥0, we have x≤λ/d-ahμ(m+γ)/dkgγ. Hence, there is no equilibrium if x>λ/d-ahμ(m+γ)/dkgγ.We define the functionh2 on 0,λ/d-ahμ(m+γ)/dkgγ by (33)h2x=fx,hg-gm+ρ+γhm+γλ-dx.We have h2(0)=-gλ(m+ρ+γ)/h(m+γ)<0, h2′(x)=∂f/∂x+gd(m+ρ+γ)/h(m+γ)>0, and h2λ/d-ahμ(m+γ)/dkgγ=h1λ/d-ahμ(m+γ)/dkgγ.Let us introduce the reproduction number for humoral immunity as follows:(34)R1=gv1h,which 1/h denotes the average life expectancy of antibodies and v1 is the number of free viruses at E1. For the biological significance, R1 represents the average number of the antibodies activated by virus.IfR1<1, we have x1>λ/d-ahμ(m+γ)/dkgγ and (35)h2λd-ahμm+γdkgγ<h1x1=0.Therefore, there is no equilibrium when R1<1.IfR1>1, then x1<λ/d-ahμ(m+γ)/dkgγ and (36)h2λd-ahμm+γdkgγ>h1x1=0.In this case, (32) has one root x2∈0,λ/d-ahμ(m+γ)/dkgγ. Consequently, when R1>1, there exists an equilibrium E2(x2,l2,y2,v2,w2) satisfying x2∈0,λ/d-ahμ(m+γ)/dkgγ, l2=(λ-dx2)/(m+γ), y2=γ(λ-dx2)/a(m+γ), v2=h/g, and w2=kγg(λ-dx2)/aqh(m+γ)-μ/q. When R1=1, E1=E2.We summarize the above discussions in the following theorem.Theorem 2.
(i)
IfR0≤1, then system (1) has one infection-free equilibrium of the form E0x0,0,0,0,0, where x0=λ/d.(ii)
IfR0>1, then system (1) has an infection equilibrium without humoral immunity of the form E1(x1,l1,y1,v1,0), where x1∈0,λ/d, l1=(λ-dx1)/(m+γ), y1=γ(λ-dx1)/a(m+γ), and v1=kγ(λ-dx1)/aμ(m+γ).(iii)
IfR1>1, then system (1) has an infection equilibrium with humoral immunity of the form E2(x2,l2,y2,v2,w2), where x2∈0,λ/d-ahμ(m+γ)/dkgγ, l2=(λ-dx2)/(m+γ), y2=γ(λ-dx2)/a(m+γ), v2=h/g, and w2=kγg(λ-dx1)/aqh(m+γ)-μ/q.


## 3. Global Stability of Equilibria

In this section, we focus on the global stability of equilibria.Theorem 3.
IfR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable and it becomes unstable if R0>1.Proof.
The proof of the first part of this theorem is based on the construction of a suitable Lyapunov functional that satisfies the conditions given in [51, Lemma 4.6]. Hence, we define a Lyapunov functional as follows: (37)L0t=α0α0+α1x0x0Φxx0+ρα02d+m+γα0+α1x0x0x-x0+l2+l+m+ρ+γγy+am+ρ+γkγv+aqm+ρ+γkgγw,where Φ(x)=x-1-ln(x) for x>0. It is not hard to show that the functional L0 is nonnegative. In fact, the function Φ has a global minimum at x=1. Consequently, Φ(x)≥0 for all x>0.
Calculating the fractional derivative ofL0(t) along solutions of system (1) and using the results in [52], we get (38)DαL0t≤α0α0+α1x01-x0xDαx+ρα0d+m+γα0+α1x0x0x-x0+lDαx+Dαl+Dαl+m+ρ+γγDαy+am+ρ+γkγDαv+aqm+ρ+γkgγDαw.Using λ=dx0, we obtain (39)DαL0t≤-dα0x-x02α0+α1x0x-α0α0+α1x01-x0xfx,vv+ρα0α0+α1x01-x0xlρα0x-x0+ld+m+γα0+α1x0x0dx0-x-m+γl+fx,vv-aμm+ρ+γkγv-aqhm+ρ+γkgγw≤-1x+ρd+m+γx0dα0x-x02α0+α1x0-ρα0m+γl2d+m+γα0+α1x0x0-ρα0x-x02lα0+α1x0xx0+aμm+ρ+γkγR0-1v-aqhm+ρ+γkgγw.Hence if R0≤1, then DαL0(t)≤0. In addition, the equality holds if and only if x=x0,l=0,y=0,w=0, and (R0-1)v=0. If R0<1, then v=0. If R0=1, from (1), we get f(x0,v)v=0 which implies that v=0. Consequently, the largest invariant set of {(x,l,y,v,w)∈R+5:DαL0(t)=0} is the singleton {E0}. Therefore, by the LaSalle’s invariance principle [51], E0 is globally asymptotically stable.
The proof of the instability ofE0 is based on the computation of the Jacobean matrix of system (1) and the results presented in [53–55]. The Jacobean matrix of (1) at any equilibrium E(x,l,y,v,w) is given by(40)-d-∂f∂xvρ0-∂f∂vv-fx,v0∂f∂xv-m+ρ+γ0∂f∂vv+fx,v00γ-a0000k-μ-qw-qv000gwgv-h.We recall that E is locally asymptotically stable if the all eigenvalues ξi of (40) satisfy the following condition [53–55]: (41)arg⁡ξi>απ2.From (40), the characteristic equation at E0 is given as follows:(42)d+ξh+ξg0ξ=0,where (43)g0ξ=m+ρ+γ+ξa+ξμ+ξ-kγβλdα0+α1λ.Obviously, (42) has the roots ξ1=-d and ξ2=-h. If R0>1, we have g0(0)=aμ(m+ρ+γ)(1-R0)<0 and limξ→+∞⁡g0(ξ)=+∞. Then, there exists ξ∗>0 satisfying g0(ξ∗)=0. In addition, we have argξ∗=0<απ/2. Consequently, when R0>1, E0 is unstable.Theorem 4.
(i)
The infection equilibrium without humoral immunityE1 is globally asymptotically stable if R0>1, R1≤1, and(44)R0≤1+m+ρ+γα0adμm+ρ+dkλγα2+kργα3λ2aρμm+ρ+γα0d+λα1.(ii)
WhenR1>1, E1 is unstable.Proof.
Define a Lyapunov functional as follows:(45)L1t=α0+α2v1α0+α1x1+α2v1+α3x1v1x1Φxx1+l1Φll1+ρα0+α2v12d+m+γα0+α1x1+α2v1+α3x1v1x1x-x1+l-l12+m+ρ+γγy1Φyy1+am+ρ+γkγv1Φvv1+aqm+ρ+γkgγw.Calculating the fractional derivative of L1(t), we get (46)DαL1t=α0+α2v1α0+α1x1+α2v1+α3x1v11-x1xDαx+1-l1lDαl+ρα0+α2v1x-x1+l-l1d+m+γα0+α1x1+α2v1+α3x1v1x1Dαx+Dαl+m+ρ+γγ1-y1yDαy+am+ρ+γkγ1-v1vDαv+aqm+ρ+γkgγw.Using λ=dx1+(m+γ)l1, f(x1,v1)v1=(m+ρ+γ)l1, γl1=ay1, ky1=μv1, and 1-fxi,vi/fx,vi=(α0+α2vi/α0+α1xi+α2vi+α3xivi)1-xi/x∀i∈{1,2}, we obtain (47)DαL1t≤d1-fx1,v1fx,v1x1-x+m+ρ+γl11-fx1,v1fx,v1+vv1fx,vfx,v1+m+ρ+γl11-l1fx,vvlfx1,v1v1+m+ρ+γl11-ly1l1y+m+ρ+γl11-vv1-yv1y1v+ρl-l11-fx1,v1fx,v1-ρα0+α2v1dx-x12+m+γl-l12+d+m+γx-x1l-l1d+m+γα0+α1x1+α2v1+α3x1v1x1+aqhm+ρ+γkgγgv1h-1w.Hence,(48)DαL1t≤-α0+α2v1x-x12xx1α0+α1x1+α2v1+α3x1v1dx1-ρl1+ρl+dρxd+m+γ-ρα0+α2v1m+γl-l12m+ρ+γα0+α1x1+α2v1+α3x1v1x1+m+ρ+γl15-fx1,v1fx,v1-l1fx,vvlfx1,v1v1-ly1l1y-yv1y1v-fx,v1fx,v-m+ρ+γl1α0+α1xα2+α3xv-v12v1α0+α1x+α2v+α3xvα0+α1x+α2v1+α3xv1+aqhm+ρ+γkgγR1-1w.Using the arithmetic-geometric inequality, we have(49)5-fxi,vifx,vi-lifx,vvlfxi,vivi-lyiliy-yviyiv-fx,vifx,v≤0.Since R1≤1, we have DαL1(t)≤0 if dx1≥ρl1. It is easy to see that this condition is equivalent to (44). Furthermore, DαL1(t)=0 if and only if x=x1,l=l1,y=y1,v=v1, and R1-1w=0. We discuss two cases: If R1<1, then w=0. If R1=1, from (1), we get Dαv1=0=ky1-μv1-qv1w, and then w=0. Hence, the largest invariant set of {(x,l,y,v,w)∈R+5:DαL1(t)=0} is the singleton {E1}. By the LaSalle’s invariance principle, E1 is globally asymptotically stable.
AtE1, the characteristic equation of (40) is given as follows:(50)gv1-h-ξg1ξ=0,where(51)g1ξ=-d-∂f∂xx1,v1v1-ξρ0-∂f∂vx1,v1v1-fx1,v1∂f∂xx1,v1v1-m+ρ+γ-ξ0∂f∂vx1,v1v1+fx1,v10γ-a-ξ000k-μ-ξ.We can easily see that (50) has the root ξ1=gv1-h. Then, when R1>1, we have ξ1>0. In this case, E1 is unstable.Theorem 5.
The infection equilibrium with humoral immunityE2 is globally asymptotically stable if R1>1 and(52)ρβh≤dm+ρ+γα0g+α2h+ρλα1g+α3h.Proof.
Consider the following Lyapunov functional:(53)
L
2
t
=
α
0
+
α
2
v
2
α
0
+
α
1
x
2
+
α
2
v
2
+
α
3
x
2
v
2
x
2
Φ
x
x
2
+
l
2
Φ
l
l
2
+
ρ
α
0
+
α
2
v
2
2
d
+
m
+
γ
α
0
+
α
1
x
2
+
α
2
v
2
+
α
3
x
2
v
2
x
2
x
-
x
2
+
l
-
l
2
2
+
m
+
ρ
+
γ
γ
y
2
Φ
y
y
2
+
a
m
+
ρ
+
γ
k
γ
v
2
Φ
v
v
2
+
a
q
m
+
ρ
+
γ
k
g
γ
w
2
Φ
w
w
2
.Computing the fractional derivative of L2(t) and using λ=dx2+(m+γ)l2, f(x2,v2)v2=(m+ρ+γ)l2, γl2=ay2, ky2=(μ+qw2)v2, and v2=h/g, we get (54)DαL2t≤d1-fx2,v2fx,v2x2-x+m+ρ+γl21-fx2,v2fx,v2+fx,vvfx,v2v2+m+ρ+γl21-l2fx,vvlfx2,v2v2+m+ρ+γl21-ly2l2y+m+ρ+γl21-vv2-yv2y2v+ρl-l21-fx2,v2fx,v2-ρα0+α2v2dx-x22+m+γl-l22+d+m+γx-x2l-l2d+m+γα0+α1x2+α2v2+α3x2v2x2≤-α0+α2v2x-x22xx2α0+α1x2+α2v2+α3x2v2dx2-ρl2+ρl+dρxd+m+γ-ρα0+α2v2m+γl-l22m+ρ+γα0+α1x2+α2v2+α3x2v2x2+m+ρ+γl25-fx2,v2fx,v2-l2fx,vvlfx2,v2v2-ly2l2y-yv2y2v-fx,v2fx,v-m+ρ+γl2α0+α1xα2+α3xv-v22v2α0+α1x+α2v+α3xvα0+α1x+α2v2+α3xv2.From (49), we have DαL2(t)≤0 when dx2≥ρl2. This condition is equivalent to (52). In addition, DαL2(t)=0 if x=x2,l=l2,y=y2, and v=v2. Further, Dαv2=0=ky2-μv2-qv2w; then w=w2. Consequently, the largest invariant set of {(x,l,y,v,w)∈R+5:DαL2(t)=0} is the singleton {E2}. By the LaSalle’s invariance principle, E2 is globally asymptotically stable.It is important to note that whenρ is sufficiently small or γ is sufficiently large, the two conditions (44) and (52) are satisfied. Then, we have the following corollary.Corollary 6.
Assume thatR0>1. When ρ is sufficiently small or γ is sufficiently large, then we have the following:(i)
The infection equilibrium without humoral immunityE1 is globally asymptotically stable if R1≤1.(ii)
The infection equilibrium with humoral immunityE2 is globally asymptotically stable if R1>1.


## 4. Numerical Simulations

In this section, we validate our theoretical results to HIV infection. Firstly, we take the parameter values as shown in Table1.Table 1
Parameter values of system (1).
parameters values parameters values parameters values λ 10 a 0.27 h 0.2 d 0.0139 γ 0.01 g 0.0001 β 0.00024 k 800 α 0 1 ρ 0.01 μ 3 α 1 0.1 m 0.0347 q 0.01 α 2 0.01 α 3 0.00001By calculation, we haveR0=0.4274≤1. Then system (1) has an infection-free equilibrium E0(719.4245,0,0,0,0). By Theorem 3, the solution of (1) converges to E0 (see Figure 1). Consequently, the virus is cleared and the infection dies out.Figure 1
Stability of the infection-free equilibriumE0.Now, we chooseβ=0.0012 and we keep the other parameter values. Hence, we obtain R0=2.137, R1=0.8334, and (55)1+m+ρ+γα0adμm+ρ+dkλγα2+kργα3λ2aρμm+ρ+γα0d+λα1=2.5934.Consequently, condition (44) is satisfied. Therefore, the infection equilibrium without humoral immunity E1(176.6853,168.7712,6.2508,1666.9,0) is globally asymptotically stable. Figure 2 demonstrates this result. In this case, the infection becomes chronic.Figure 2
Stability of the infection equilibrium without humoral immunityE1.Next, we takeg=0.0004 and do not change the other parameter values. In this case, we have R1=3.3338, ρβh=0.0000024, and d(m+ρ+γ)(α0g+α2h)+ρλ(α1g+α3h)=0.000006. Hence, condition (52) is satisfied. Consequently, system (1) has an infection equilibrium with humoral immunity E2(423.4261,92.0442,3.4090,500,245.4473) which is globally asymptotically stable. Figure 3 illustrates this result. We can observe that the activation of the humoral immune response increases the healthy cells and decreases the productive infected cells and viral load to a lower levels but it is not able to eradicate the infection.Figure 3
Stability of the infection equilibrium with humoral immunityE2.


## 5. Conclusion

In the present paper, we have studied the dynamics of a viral infection model by taking into account the memory effect represented by the Caputo fractional derivative and the humoral immunity. We have proved that the solutions of the model are nonnegative and bounded which assure the well-posedness. We have shown that the proposed model has three infection equilibriums, namely, the infection-free equilibriumE0, the infection equilibrium without humoral immunity E1, and the infection equilibrium with humoral immunity E2. By constructing suitable Lyapunov functionals, the global stability of these equilibria is fully determined by two threshold parameters R0 and R1. More precisely, when R0≤1, E0 is globally asymptotically stable, whereas if R0>1, it becomes unstable and another equilibrium point appears, that is, E1, which is globally asymptotically stable whenever R1≤1 and condition (44) is satisfied. In the case that R1>1, E1 becomes unstable and there exists another equilibrium point E2 which is globally asymptotically stable when condition (52) is satisfied. In addition, we remarked that when ρ is sufficiently small or γ is sufficiently large, conditions (44) and (52) are verified, and then the global stability of E1 and E2 is characterized only by R0 and R1.From our theoretical and numerical results, we deduce that the order of the fractional derivativeα has no effect on the dynamics of the model. However, when the value of α decreases (long memory), the solutions of our model converge rapidly to the steady states (see Figures 1–3). This behavior can be explained by the memory term 1/Γ(1-α)(t-u)α included in the fractional derivative which represents the time needed for the interaction between cells and viral particles and the time needed for the activation of humoral immune response. In fact, the knowledge about the infection and the activation of the humoral immune response in an early stage can help us to control the infection.



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*Source: 1019242-2018-12-02.xml*