Two virus infection models with antibody immune response and chronically infected cells are proposed and analyzed. Bilinear incidence rate is considered in the first model, while the incidence rate is given by a saturated functional response in the second one. One main feature of these models is that it includes both short-lived infected cells and chronically infected cells. The chronically infected cells produce much smaller amounts of virus than the short-lived infected cells and die at a much slower rate. Our mathematical analysis establishes that the global dynamics of the two models are determined by two threshold parameters R0 and R1. By constructing Lyapunov functions and using LaSalle's invariance principle, we have established the global asymptotic stability of all steady states of the models. We have proven that, the uninfected steady state is globally asymptotically
stable (GAS) if R0<1, the infected steady state without antibody immune response exists and it
is GAS if R1<1<R0, and the infected steady state with antibody immune response exists and it is GAS if R1>1. We check our theorems with numerical simulation in the end.
1. Introduction
In recent years, many mathematical models have been proposed to study the dynamics of viral infections such as the human immunodeficiency virus (HIV), the hepatitis C virus (HCV), and the hepatitis B virus (HBV) (see, e.g., [1–17]). Such virus infection models can be very useful in the control of epidemic diseases and provide insights into the dynamics of viral load in vivo. Therefore, mathematical analysis of the virus infection models can play a significant role in the development of a better understanding of diseases and various drug therapy strategies. Most of the mathematical models of viral infection presented in the literature did not differentiate between the short-lived infected cells and chronically infected cells. The chronically infected cells produce much smaller amounts of virus than the short-lived infected cells and die at a much slower rate [18]. The virus dynamics model with chronically infected cells and under the effect of antiviral drug therapy was introduced in [18] as
(1)T˙=λ-dT-(1-ɛ)kTV,T˙*=(1-α)(1-ɛ)kTV-δT*,C˙*=α(1-ɛ)kTV-aC*,V˙=NTδT*+NCaC*-cV,
where T, T*, C*, and V are the concentration of the uninfected cells, short-lived infected cells, chronically infected cells, and free virus particles, respectively. The constant λ is the rate at which new uninfected cells are generated and d is the natural death rate constant of uninfected cells. k is the infection rate constant. The fractions (1-α) and α with 0<α<1 are the probabilities that, upon infection, an uninfected cell will become either short-lived infected or chronically infected. δ and a are the death rate constants of the short-lived infected cells and chronically infected cells, respectively. NT and NC are the average number of virions produced in the lifetime of the short-lived infected and chronically infected cells, respectively. The chronically infected cells produce much smaller amounts of virus than the short-lived infected cells and die at a much slower rate (i.e., NT>NC and δ>a). The free viruses are cleared with rate constant c. The drug efficacy is denoted by ɛ and 0≤ɛ≤1.
It is observed that the basic and global properties of model (1) are not studied in the literature. Moreover, model (1) did not take into consideration the immune response. During viral infections, the host immune system reacts with antigen-specific immune response. The immune system is described as having two “arms”: the cellular arm, which depends on T cells to mediate attacks on virally infected or cancerous cells, and the humoral arm, which depends on B cells. The B cell is a type of blood cell which belongs to a group of white blood cells (WBCs) called lymphocytes. WBCs protect the body from infection. The main job of B cells is to fight infection. B cells get activated when an infection occurs and they produce molecules called antibodies that attach to the surface of the infectious agent. These antibodies either kill the infection causing organism or make it prone to attack by other WBCs. They play a major role in the immune system, which guards the body against infection. Virus infection models with antibody immune response have been analyzed by many researchers (see [19–28]). However, in all of these works, the chronically infected cells have been neglected.
In this paper, we propose two virus infection models with antibody immune response and chronically infected cells. In the first model, bilinear incidence rate which is based on the law of mass-action is considered. The second model generalizes the first one where the incidence rate is given by a saturation functional response. The global stability of all equilibria of the models is established using the method of Lyapunov function. We prove that the global dynamics of the models are determined by two threshold parameters R0 and R1. If R0≤1, then the infection-free equilibrium is globally asymptotically stable (GAS), if R1≤1<R0, then the infected equilibrium without antibody immune response exists and it is GAS, and if R1>1 then the infected equilibrium with antibody immune response exists and it is GAS.
2. Model with Bilinear Incidence Rate
In this section we propose a viral dynamics model with antibody immune response, taking into consideration the chronically infected cells. Based on the mass-action principle, we assume that the incidence rate of infection is bilinear; that is, the infection rate per virus and per uninfected cell is constant:
(2)T˙=λ-dT-(1-ɛ)kTV,(3)T˙*=(1-α)(1-ɛ)kTV-δT*,(4)C˙*=α(1-ɛ)kTV-aC*,(5)V˙=NTδT*+NCaC*-cV-rVZ,(6)Z˙=gVZ-μZ,
where Z is the concentration of antibody immune cells. The viruses are attacked by the antibodies with rate rVZ. The antibody immune cells are proliferated at rate gVZ and die at rate μZ. All the other variables and parameters of the model have the same meanings as given in (1).
2.1. Positive Invariance
We note that model (2)–(6) are biologically acceptable in the sense that no population goes negative. It is straightforward to check the positive invariance of the nonnegative orthant ℝ+5 by model (2)–(6) (see, e.g., [6]). In the following, we show the boundedness of the solution of model (2)–(6).
Proposition 1.
There exist positive numbers Li, i=1,2,3, such that the compact set
(7)Ω={(T,T*,C*,V,Z)∈ℝ+4:0≤T,T*,C*≤L1,r0≤V≤L2,0≤Z≤L3{(T,T*,C*,V,Z)∈ℝ+4:0≤T,T*,C*≤L1,}
is positively invariant.
Proof.
To show the boundedness of the solutions we let G1(t)=T(t)+T*(t)+C*(t); then
(8)G˙1(t)=λ-dT(t)-(1-ɛ)kT(t)V(t)+(1-α)(1-ɛ)kT(t)V(t)-δT*+α(1-ɛ)kT(t)V(t)-aC*(t)≤λ-s1G1(t),
where s1=min{d,a,δ}. Hence G1(t)≤L1, if G1(0)≤L1 where L1=λ/s1. Since T(t)>0, T*(t)≥0, and C*(t)≥0, then 0≤T(t), T*(t), C*(t)≤L1 if 0≤T(0)+T*(0)+C*(0)≤L1. Let G2(t)=V(t)+(r/g)Z(t); then
(9)G˙2(t)=NTδT*(t)+NCaC*(t)-cV(t)-rμgZ(t)≤(NTδ+NCa)L1-s2(V(t)+rgZ(t))=(NTδ+NCa)L1-s2G2(t),
where s2=min{c,μ}. Hence G2(t)≤L2, if G2(0)≤L2, where L2=(NTδ+NCa)L1/s2. Since V(t)≥0 and Z(t)≥0 then 0≤V(t)≤L2 and 0≤Z(t)≤L3 if 0≤V(0)+(r/g)Z(0)≤L2, where L3=gL2/r.
2.2. Equilibria
System (2)–(6) always admits an infection-free equilibrium E0=(T0,0,0,0,0), where T0=λ/d. In addition to E0, the system can have an infected equilibrium without antibody immune response E1(T1,T1*,C1*,V1,0) and an infected equilibrium with antibody immune response E2(T2,T2*,C2*,V2,Z2) where
(10)T1=c(1-ɛ)k[(1-α)NT+αNC],T1*=(1-α)λ{(1-ɛ)kT0[(1-α)NT+αNC]-c}δ(1-ɛ)kT0[(1-α)NT+αNC],C1*=αλ{(1-ɛ)kT0[(1-α)NT+αNC]-c}a(1-ɛ)kT0[(1-α)NT+αNC],V1=d{(1-ɛ)kT0[(1-α)NT+αNC]-c}(1-ɛ)kc,T2=λggd+(1-ɛ)kμ,T2*=(1-α)(1-ɛ)kλμδ(dg+(1-ɛ)kμ),C2*=α(1-ɛ)kλμa(dg+(1-ɛ)kμ),V2=μg,Z2=cr(dg(1-ɛ)kT0[(1-α)NT+αNC]c(dg+(1-ɛ)kμ)-1).
We discuss the local stability of the infection-free equilibrium E0. At the infection-free equilibrium E0(T0,0,0,0,0), the system has the Jacobian matrix given by
(11)JE0=[-d00-(1-ɛ)kT000-δ0(1-α)(1-ɛ)kT0000-aα(1-ɛ)kT000δNTaNC-c00000-μ].
The characteristic equation of the Jacobian matrix evaluated at E0 is
(12)(s+d)(s+μ)(s3+a1s2+a2s+a3)=0,
where
(13)a1=a+c+δ,a2=ac+aδ+cδ-(1-α)(1-ɛ)kT0NTδ-α(1-ɛ)kT0NCa,a3=acδ(1-(1-ɛ)kT0[(1-α)NT+αNC]c).
We observe that (12) has two negative eigenvalues s1=-d and s2=-μ. By the Routh-Hurwitz criterion, the remaining three eigenvalues of (12) have negative real parts if a1>0, a3>0, and a1a2-a3>0. We have a1>0 and if (1-ɛ)kT0[(1-α)NT+αNC]/c<1, then a3>0 and
(14)a1a2-a3=aδ2+a2δ+2acδ+a(a+c)[c-α(1-ɛ)kT0NC]+δ(δ+c)[c-(1-α)(1-ɛ)kT0NT]>0.
Now we define the basic reproduction number for system (2)–(6) as
(15)R0=(1-ɛ)kT0[(1-α)NT+αNC]c.
It follows that the equilibria E1 and E2 can be written as
(16)T1=T0R0,T1*=(1-α)λδ(R0-1)R0,C1*=αλa(R0-1)R0,V1=d(1-ɛ)k(R0-1),T2=λggd+(1-ɛ)kμ,T2*=(1-α)(1-ɛ)kλμδ(dg+(1-ɛ)kμ),C2*=α(1-ɛ)kλμa(dg+(1-ɛ)kμ),V2=μg,Z2=cr(dgR0dg+(1-ɛ)kμ-1).
We note that T1, T1*, C1*, and V1 are positive when R0>1 and that Z2>0 when dgR0/(dg+(1-ɛ)kμ)>1. Now we define another threshold parameter R1 as
(17)R1=R01+((1-ɛ)kμ/dg).
Clearly R1<R0.
From (16) we have the following statements:
if R0≤1, then there exists only positive equilibrium E0;
if R1≤1<R0, then there exist two positive equilibria E0 and E1;
if R1>1, then there exist three positive equilibria E0, E1, and E2.
2.3. Global Stability Analysis
In this section, we study the global stability of all the equilibria of system (2)–(6) employing the method of Lyapunov function.
Theorem 2.
For system (2)–(6), if R0≤1, then E0 is GAS.
Proof.
Define a Lyapunov function U0 as follows:
(18)U0=T0(TT0-1-ln(TT0))+η1T*+η2C*+η3V+η4Z,
where ηi, i=1,…,4, are positive constants to be determined below. Calculating the derivative of U0 along the solutions of the system (2)–(6) and applying λ=T0d, we obtain
(19)dU0dt=(1-T0T)(λ-dT-(1-ɛ)kTV)+η1((1-α)(1-ɛ)kTV-δT*)+η2(α(1-ɛ)kTV-aC*)+η3(NTδT*+NCaC*-cV-rVZ)+η4(gVZ-μZ).
Let ηi, i=1,…,4, be chosen such as
(20)(1-α)η1+αη2=1,η1-NTη3=0,η2-NCη3=0,gη4-rη3=0.
The solution of (20) is given by
(21)η1=NT(1-α)NT+αNC,η2=NC(1-α)NT+αNC,η3=1(1-α)NT+αNC,η4=rg[(1-α)NT+αNC].
The values of ηi, i=1,…,4, given by (21) will be used throughout the paper. Then
(22)dU0dt=(1-T0T)(λ-dT)+(1-ɛ)kT0V-η3cV-η4μZ=-d(T-T0)2T+η3c(R0-1)V-η4μZ.
If R0≤1 then dU0/dt≤0 for all T,V,Z>0. Thus the solutions of system (2)–(6) limit to M, the largest invariant subset of {dU0/dt=0}. Clearly, it follows from (22) that dU0/dt=0 if and only if T=T0, 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 (5) we derive that
(23)0=V˙=NTδT*+NCaC*.
Since T*, C*≥0, then T*=C*=0. Hence dU0/dt=0 if and only if T=T0, T*=0, C*=0, V=0, and Z=0. It follows from LaSalle's invariance principle that the infection-free equilibrium E0 is GAS when R0≤1.
Theorem 3.
For system (2)–(6), if R1≤1<R0, then E1 is GAS.
Proof.
Define the following Lyapunov function:
(24)U1=T1(TT1-1-ln(TT1))+η1T1*(T*T1*-1-ln(T*T1*))+η2C1*(C*C1*-1-ln(C*C1*))+η3V1(VV1-1-ln(VV1))+η4Z.
The time derivative of U1 along the trajectories of (2)–(6) is given by
(25)dU1dt=(1-T1T)(λ-dT-(1-ɛ)kTV)+η1(1-T1*T*)((1-α)(1-ɛ)kTV-δT*)+η2(1-C1*C*)(α(1-ɛ)kTV-aC*)+η3(1-V1V)(NTδT*+NCaC*-cV-rVZ)+η4(gVZ-μZ).
Applying λ=dT1+(1-ɛ)kT1V1 we get
(26)dU1dt=(1-T1T)(dT1-dT)+(1-ɛ)kT1V1(1-T1T)+(1-ɛ)kT1V-η1(1-α)(1-ɛ)kTVT1*T*+η1δT1*-η2α(1-ɛ)kTVC1*C*+η2aC1*-δη1V1T*V-aη2V1C*V-cη3V+cη3V1+rη3V1Z-μη4Z.
Using the following equilibrium conditions for E1,
(27)(1-α)(1-ɛ)kT1V1=δT1*,α(1-ɛ)kT1V1=aC1*,cV1=NTδT1*+NCaC1*,
then we have (1-ɛ)kT1V1=η1δT1*+η2aC1* and
(28)dU1dt=-d(T-T1)2T+η1δT1*(1-T1T)+η2aC1*(1-T1T)-η1δT1*TVT1*T1V1T*+η1δT1*-η2aC1*TVC1*T1V1C*+η2aC1*-η1δT1*V1T*VT1*-η2aC1*V1C*VC1*+η1δT1*+η2aC1*+rη3(V1-μg)Z=-d(T-T1)2T+η1δT1*[3-T1T-T1*TVT*T1V1-V1T*VT1*]+η2aC1*[3-T1T-C1*TVC*T1V1-C*V1C1*V]+rη3(dg+(1-ɛ)kμg(1-ɛ)k)(R1-1)Z.
We have that if R0>1, then T1,T1*,C1*,V1>0. Since the arithmetical mean is greater than or equal to the geometrical mean, then if R1≤1 then dU1/dt≤0 for all T,T*,C*,V,Z>0. It can be seen that dU1/dt=0 if and only if T=T1, T*=T1*, C*=C1*, V=V1, and Z=0. LaSalle's invariance principle implies global stability of E1.
Theorem 4.
For system (2)–(6), if R0≤1, then E0 is GAS.
Proof.
We consider a Lyapunov function
(29)U2=T2(TT2-1-ln(TT2))+η1T2*(T*T2*-1-ln(T*T2*))+η2C2*(C*C2*-1-ln(C*C2*))+η3V2(VV2-1-ln(VV2))+η4Z2(ZZ2-1-ln(ZZ2)).
Further, function U2 along the trajectories of system (2)–(6) satisfies
(30)dU2dt=(1-T2T)(λ-dT-(1-ɛ)kTV)+η1(1-T2*T*)((1-α)(1-ɛ)kTV-δT*)+η2(1-C2*C*)(α(1-ɛ)kTV-aC*)+η3(1-V2V)(NTδT*+NCaC*-cV-rVZ)+η4(1-Z2Z)(gVZ-μZ).
Using the following equilibrium conditions for E2,
(31)λ=dT2+(1-ɛ)kT2V2,(1-α)(1-ɛ)kT2V2=δT2*,α(1-ɛ)kT2V2=aC2*,cV2+rV2Z2=NTδT2*+NCaC2*,
we get
(32)dU2dt=-d(T-T2)2T+(1-ɛ)kT2V2(1-T2T)+(1-ɛ)kT2V-η1(1-α)(1-ɛ)kTVT2*T*+δη1T2*-η2α(1-ɛ)kTVC2*C*+aη2C2*-δη1V2T*V-aη2V2C*V-cη3V+cη3V2+rη4V2Z-rη4Z2V+μη4Z2-μη4Z=-d(T-T2)2T+η1δT2*(1-T2T)+η2aC2*(1-T2T)-η1δT2*TVT2*T2V2T*+η1δT2*-η2aC2*TVC2*T2V2C*+η2aC2*-η1δT2*V2T*VT2*-η2aC2*V2C*VC2*+η1δT2*+η2aC2*=-d(T-T2)2T+η1δT2*[3-T2T-T2*TVT*T2V2-V2T*VT2*]+η2aC2*[3-T2T-C2*TVC*T2V2-C*V2C2*V].
Thus, if R1>1, then T2,T2*,C2*,V2 and Z2>0. Since the arithmetical mean is greater than or equal to the geometrical mean, then dU2/dt≤0. It can be seen that dU2/dt=0 if and only if T=T2, T*=T2*, C*=C2*, and V=V2. From (5), if V=V2, then V˙=0 and 0=NTδT2*+NCaC2*-cV-rV2Z=0, so Z=Z2 and hence dU2/dt is equal to zero at E2. So, the global stability of the equilibrium E2 follows from LaSalle's invariance principle.
3. Model with Saturation Incidence Rate
In model (2)–(6), the infection process is characterized by bilinear incidence rate (1-ɛ)kxv. However, there are a number of reasons why this bilinear incidence can be insufficient to describe infection process in detail (see, e.g., [29–31]). For example, a less than linear response in v could occur when the concentration of viruses becomes higher, where the infectious fraction is high so that exposure is very likely [29]. Experiments reported in [32] strongly suggested that the infection rate of microparasitic infections is an increasing function of the parasite dose and is usually sigmoidal in shape (see, e.g., [33]). In [33], to place the model on more sound biological grounds, Regoes et al. replaced the mass-action infection rate with a dose-dependent infection rates. In this section, the incidence rate is given by a saturation functional response:
(33)T˙=λ-dT-(1-ɛ)kTV1+βV,(34)T˙*=(1-α)(1-ɛ)kTV1+βV-δT*,(35)C˙*=α(1-ɛ)kTV1+βV-aC*,(36)V˙=NTδT*+NCaC*-cV-rVZ,(37)Z˙=gVZ-μZ,
where β>0 is a constant, which represents the saturation infection rate constant.
All the variables and parameters have the same meanings as given in model (2)–(6).
3.1. Equilibria
Similar to the previous section, we can define two threshold parameters R0 and R1 for system (33)–(37) as
(38)R0=(1-ɛ)kT0[(1-α)NT+αNC]c,R1=R01+(dβμ+(1-ɛ)kμ/dg).
Clearly R1<R0. It is clear that system (33)–(37) has an infection-free equilibrium E0=(T0,0,0,0,0), where T0=λ/d. In addition to E0, the system can have an infected equilibrium without antibody immune response E1(T1,T1*,C1*,V1,0), where
(39)T1=βλ[(1-α)NT+αNC]+c((1-ɛ)k+dβ)[(1-α)NT+αNC],T1*=(1-α)cdδ((1-ɛ)k+dβ)[(1-α)NT+αNC](R0-1),C1*=αcda((1-ɛ)k+dβ)[(1-α)NT+αNC](R0-1),V1=d(1-ɛ)k+dβ(R0-1),
and infected equilibrium with antibody immune response E2(T2,T2*,C2*,V2,Z2), where
(40)T2=λ(g+βμ)gd+(1-ɛ)kμ+dβμ,T2*=(1-α)(1-ɛ)kλμδ(dg+(1-ɛ)kμ+dβμ),C2*=α(1-ɛ)kλμa(dg+(1-ɛ)kμ+dβμ),V2=μg,Z2=cr(R1-1).
It is clear from (39) and (40) that
if R0≤1, then there exists only positive equilibrium E0;
if R1≤1<R0, then there exist two positive equilibria E0 and E1;
if R1>1, then there exist three positive equilibria E0, E1, and E2.
3.2. Global Stability Analysis
In this section, we study the global stability of all the equilibria of system (33)–(37) employing the method of Lyapunov function and LaSalle's invariance principle.
Theorem 5.
For system (33)–(37), if R0≤1, then E0 is GAS.
Proof.
Define a Lyapunov function U0 as follows:
(41)U0=T0(TT0-1-ln(TT0))+η1T*+η2C*+η3V+η4Z.
Calculating the derivative of U0 along the solutions of system (33)–(37) and applying λ=T0d, we obtain
(42)dU0dt=(1-T0T)(λ-dT-(1-ɛ)kTV1+βV)+η1((1-α)(1-ɛ)kTV1+βV-δT*)+η2(α(1-ɛ)kTV1+βV-aC*)+η3(NTδT*+NCaC*-cV-rVZ)+η4(gVZ-μZ)=(1-T0T)(λ-dT)+(1-ɛ)kT0V1+βV-cη3V-μη4Z=-[d(T-T0)2T+η3cβR0V2(1+βV)+μη4Z]+cη3(R0-1)V.
Similar to the proof of Theorem 2, one can easily show that E0 is GAS when R0≤1.
Theorem 6.
For system (33)–(37), if R1≤1<R0, then E1 is GAS.
Proof.
Construct a Lyapunov function as follows:
(43)U1=T1(TT1-1-ln(TT1))+η1T1*(T*T1*-1-ln(T*T1*))+η2C1*(C*C1*-1-ln(C*C1*))+η3V1(VV1-1-ln(VV1))+η4Z.
The derivative of U1 along the trajectories of system (33)–(37) is given by
(44)dU1dt=(1-T1T)(λ-dT-(1-ɛ)kTV1+βV)+η1(1-T1*T*)((1-α)(1-ɛ)kTV1+βV-δT*)+η2(1-C1*C*)(α(1-ɛ)kTV1+βV-aC*)+η3(1-V1V)(NTδT*+NCaC*-cV-rVZ)+η4(gVZ-μZ).
Applying λ=dT1+((1-ɛ)kT1V1/(1+βV1)) we get
(45)dU1dt=(1-T1T)(dT1-dT)+(1-ɛ)kT1V11+βV1(1-T1T)+(1-ɛ)kT1V1+βV-η1(1-α)(1-ɛ)kTV1+βVT1*T*+η1δT1*-η2α(1-ɛ)kTV1+βVC1*C*+η2aC1*-η1δV1T*V-η2aV1C*V-cη3V+cη3V1+rη3V1Z-μη4Z.
Using the following equilibrium conditions for E1,
(46)(1-α)(1-ɛ)kT1V11+βV1=δT1*,α(1-ɛ)kT1V11+βV1=aC1*,cV1=NTδT1*+NCaC1*,
we get
(47)dU1dt=-d(T-T1)2T+η1δT1*(1-T1T)+η2aC1*(1-T1T)+(1-ɛ)kT1V11+βV1[V(1+βV1)V1(1+βV)-VV1]-η1δT1*TVT1*(1+βV1)T1V1T*(1+βV)+η1δT1*-η2aC1*TVC1*(1+βV1)T1V1C*(1+βV)+η2aC1*-η1δT1*V1T*VT1*-η2aC1*V1C*VC1*+η1δT1*+η2aC1*+rη3(V1-μg)Z=-d(T-T1)2T+(1-ɛ)kT1V11+βV1[-1+V(1+βV1)V1(1+βV)-VV1+1+βV1+βV1]+η1δT1*[4-T1T-TVT1*(1+βV1)T1V1T*(1+βV)-V1T*VT1*+η1δT1*w-1+βV1+βV1TVC1*(1+βV1)T1V1C*(1+βV)]+η2aC1*[4-T1T-TVC1*(1+βV1)T1V1C*(1+βV)-C*V1C1*V+η2aC1*w-1+βV1+βV1TVC1*(1+βV1)T1V1C*(1+βV)]+rη3(V1-μg)Z=-d(T-T1)2T-(1-ɛ)kT1V11+βV1[β(V-V1)2V1(1+βV)(1+βV1)]+η1δT1*[4-T1T-TVT1*(1+βV1)T1V1T*(1+βV)-V1T*VT1*+η1δT1*w-1+βV1+βV1TVC1*(1+βV1)T1V1C*(1+βV)]+η2aC1*[4-T1T-TVC1*(1+βV1)T1V1C*(1+βV)-C*V1C1*V+η2aC1*w-1+βV1+βV1TVC1*(1+βV1)T1V1C*(1+βV)]+rη3(dg+(1-ɛ)kμ+dβμg(1-ɛ)k+dgβ)(R1-1)Z.
We have that if R1≤1<R0, then dU1/dt≤0 where equality occurs at E1. LaSalle's invariance principle implies global stability of E1.
Theorem 7.
For system (33)–(37), if R1>1, then E2 is GAS.
Proof.
We consider a Lyapunov function as follows:
(48)U2=T2(TT2-1-ln(TT2))+η1T2*(T*T2*-1-ln(T*T2*))+η2C2*(C*C2*-1-ln(C*C2*))+η3V2(VV2-1-ln(VV2))+η4Z2(ZZ2-1-ln(ZZ2)).
Further, function U2 along the trajectories of system (33)–(37) satisfies
(49)dU2dt=(1-T2T)(λ-dT-(1-ɛ)kTV1+βV)+η1(1-T2*T*)((1-α)(1-ɛ)kTV1+βV-δT*)+η2(1-C2*C*)(α(1-ɛ)kTV1+βV-aC*)+η3(1-V2V)(NTδT*+NCaC*-cV-rVZ)+η4(1-Z2Z)(gVZ-μZ).
Using the following equilibrium conditions for E2,
(50)λ=dT2+(1-ɛ)kT2V21+βV2,δT2*=(1-α)(1-ɛ)kT2V21+βV2,aC2*=α(1-ɛ)kT2V21+βV2,cV2+rV2Z2=NTδT2*+NCaC2*,
we get
(51)dU2dt=-d(T-T2)2T+(1-ɛ)kT2V21+βV2(1-T2T)+(1-ɛ)kT2V1+βV-η1(1-α)(1-ɛ)kTV1+βVT2*T*+η1δT2*-η2α(1-ɛ)kTV1+βVC2*C*+η2aC2*-η1δV2T*V-η2aV2C*V-η3cV+η3cV2+η3rV2Z-η4gZ2V+μη4Z2-μη4Z=-d(T-T2)2T+η1δT2*(1-T2T)+η2aC2*(1-T2T)+(1-ɛ)kT2V21+βV2[V(1+βV2)V2(1+βV)-VV2]-η1δT2*TVT2*(1+βV2)T2V2T*(1+βV)+η1δT2*-η2aC2*TVC2*(1+βV2)T2V2C*(1+βV)+η2aC2*-η1δT2*V2T*VT2*-η2aC2*V2C*VC2*+η1δT2*+η2aC2*=-d(T-T2)2T-(1-ɛ)kT2V21+βV2[β(V-V2)2V2(1+βV)(1+βV2)]+η1δT2*[4-T2T-TVT2*(1+βV2)T2V2T*(1+βV)-V2T*VT2*+η1δT2*w-1+βV1+βV2TVT2*(1+βV2)T2V2T*(1+βV)]+η2aC2*[4-T2T-TVC2*(1+βV2)T2V2C*(1+βV)-C*V2C2*V+η1δT2*w-1+βV1+βV2TVC2*(1+βV2)T2V2C*(1+βV)].
Similar to the proof of Theorem 4, one can show that E2 is GAS.
4. Numerical Simulations
We now use simple numerical simulations to illustrate our theoretical results for the two models. In both models we will fix the following data: λ=10 mm−3 day−1,d=0.01 day−1, k=0.001 mm3 day−1, δ=0.5 day−1, α=0.5, a=0.1 day-1, c=3 day−1, NT=10, NC=5, r=0.01 mm3 day−1, and μ=0.1 day−1. The other parameters will be chosen below. All computations were carried out by MATLAB.
4.1. Model with Bilinear Incidence Rate
In this section, we perform simulation results for model (2)–(6) to check our theoretical results given in Theorems 2–4. We have the following cases.
R0≤1. We choose ɛ=0.63 and g=0.01 mm3 day−1. Using these data we compute R0=0.92 and R1=0.672. Figures 1, 2, 3, 4, and 5 show that the numerical results are consistent with Theorem 2. We can see that, the concentration of uninfected cells is increased and converges to its normal value λ/d=1000 mm−3, while the concentrations of short-lived infected cells, chronically infected cells, free viruses, and antibody immune cells are decaying and tend to zero.
R1≤1<R0. We take ɛ=0 and g=0.005 mm3 day−1. In this case, R0=2.5 and R1=0.833. Figures 1–5 show that the numerical results are consistent with Theorem 3. We can see that the trajectory of the system will tend to the infected equilibrium without antibody immune response E1(400,6,27.77,15,0). In this case, the infection becomes chronic but with no persistent antibody immune response.
R1>1. We choose ɛ=0 and g=0.01 mm3 day−1. Then we compute R0=2.5 and R1=1.25. From Figures 1–5 we can see that our simulation results are consistent with the theoretical results of Theorem 4. We observe that the trajectory of the system will tend to the infected equilibrium with antibody immune response E2(500.04,5,23.15,10,57.03). In this case, the infection becomes chronic but with persistent antibody immune response.
The evolution of uninfected cells for model (2)–(6).
The evolution of short-lived infected cells for model (2)–(6).
The evolution of chronically infected cells for model (2)–(6).
The evolution of free viruses for model (2)–(6).
The evolution of antibody immune cells for model (2)–(6).
We note that the values of the parameters g, r, and μ have no impact on the value of R0, since R0 is independent of those parameters. This fact seems to suggest that antibodies do not play a role in eliminating the viruses. From the definition of R1, we can see that R1 can be increased by increasing g or decreasing μ.
Figures 1 and 4 show that the presence of antibody immune response (i.e., R1>1) reduces the concentration of free viruses and increases the concentration of uninfected cells. This can be seen by comparing the virus and uninfected cell components in the equilibria E1 and E2 under the condition R1>1. For model (2)–(6), simple calculation shows that
(52)V1-V2=(dg+(1-ɛ)kμg(1-ɛ)k)(R1-1).
It follows that if R1>1, then V2<V1. From (2) and at any equilibrium point E¯(T¯,T¯*,C¯*,V¯,Z¯) we have
(53)T¯=λd+(1-ɛ)kV¯.
Clearly, T¯ is a decreasing function of V¯. This yields that if R1>1, then V2<V1 and T2>T1.
4.2. Model with Saturation Functional Response
In this section, we perform simulation results to check Theorems 5–7. The parameter β is chosen as α=0.2 mm3. We have the following cases.
R0≤1. We take ɛ=0.63 and g=0.01 mm3 day−1. Using these data, we compute R0=0.92 and R1=0.273. The simulation results of this case are shown in Figures 6, 7, 8, 9, and 10. We can see that the numerical results are consistent with Theorem 5. It is observed that the viruses will be cleared and the uninfected cells will return to their normal value.
R1≤1<R0. To satisfy this condition, we take ɛ=0 and g=0.005 mm3 day−1. This will give R0=2.5 and R1=0.833. Figures 6–10 show that the numerical results are consistent with Theorem 6. We see that the infected equilibrium E1(800,2,9.25,5,0) is GAS, and the infection becomes chronic but with no persistent antibody immune response.
R1>1. This condition is satisfied by choosing ɛ=0 and g=0.01 mm3 day−1. This yields R0=2.5 and R1=1.25. Figures 6–10 demonstrate the global stability of E2(832.58,1.67,7.71,3.34,74.55). Then, the infection becomes chronic but with persistent antibody immune response.
The evolution of uninfected cells for model (33)–(37).
The evolution of short-lived infected cells for model (33)–(37).
The evolution of chronically infected cells for model (33)–(37).
The evolution of free viruses for model (33)–(37).
The evolution of antibody immune cells for model (33)–(37).
From the definition of the parameter R0, we can see that the value of the saturation infection rate constant β has no impact on the value of R0. This means that saturation does not play a role in eliminating the virus. From the definition of R1, we can see that R1 can be increased by increasing g or decreasing μ and β.
Figures 6 and 9 show that if R1>1 the antibody immune response reduces the concentration of free viruses and increases the concentration of uninfected cells. For model (33)–(37), simple calculation shows that
(54)V1-V2=(dg+(1-ɛ)kμ+dβμg(1-ɛ)k+dgβ)(R1-1).
As a result, if R1>1, then V2<V1. From (33) and at any equilibrium point E¯(T¯,T¯*,C¯*,V¯,Z¯) we have
(55)T¯=(1+βV¯)λd+(1-ɛ)kV¯+dV¯β,dT¯dV¯=-(1-ɛ)kλ(d+(1-ɛ)kV¯+dV¯β)2.
Then, T¯ is a decreasing function of V¯. It follows that if R1>1 then V2<V1 and T2>T1.
5. Conclusions
In this paper, we have proposed two virus infection models with antibody immune response taking into account the chronically infected cells. In the first model we have assumed that the incidence rate of infection is bilinear while in the second model the incidence rate is given by saturation functional response. We have shown that the dynamics of the models are fully determined by two threshold parameters R0 and R1. The parameter R0 determines whether a chronic infection can be established while R1 determines whether a persistent antibody response can be established. By constructing Lyapunov function and using LaSalle's invariance principle, we have investigated the global stability of all equilibria of the two models. We have proven that if R0≤1 then the infection-free equilibrium E0 is GAS, and the viruses are cleared. If R1≤1<R0, then the infected equilibrium without antibody immune response E1 exists and it is GAS, and the infection becomes chronic but with no persistent antibody immune response. If R1>1, then the infected equilibrium with antibody immune response E2 exists and it is GAS, and the infection is chronic with persistent antibody immune response. Numerical simulations have been performed for the two models. Our simulation results confirm the analytic results given in Theorems 2–7.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 130-078-D1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are also grateful to Professor Malay Banerjee and to the anonymous reviewers for constructive suggestions and valuable comments, which improve the quality of the paper.
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