# Dynamic Analysis of an HIV Model Incorporating Cytotoxic T Lymphocytes and Vectored Immunoprophylaxis

Authors: Hongyan Chen; Jianfeng Luo
Journal: Journal of Mathematics
 (2022)

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## Abstract

The objective of this study is to investigate the effects of immune responses on HIV replication by using a novel HIV model that incorporates immune responses including cytotoxic T lymphocytes and antibodies. In this model, the cytotoxic lymphocyte cells are stimulated by infected T cells, and the antibodies are received continuously through vectored immunoprophylaxis. In the first step, we analyze the well-posedness of our proposed model. By obtaining the basic reproduction number, we also investigate the existence of equilibrium in three cases, including infection-free equilibrium, immune-free infection equilibrium, and immune-present infection equilibrium. As a result, we demonstrate our model can admit two immune-free infection equilibria, which are dependent on the basic reproduction number. Additionally, we study their local stability and find sufficient conditions for them. In particular, we show that immune-free infection equilibrium and immune-present infection equilibrium can become unstable from stable, and then a simple Hopf bifurcation can occur. Theoretical results about backward bifurcation and forward bifurcation are further derived. In addition, simulations reveal rich dynamic behaviors, such as backward bifurcation, forward bifurcation, and Hopf bifurcation. The rich dynamics of the proposed model demonstrate the importance and complexity of immune responses when fighting HIV replication.


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## Body

## 1. Introduction and Model

As the human immunodeficiency virus (HIV) emerged in the early 1980s, it quickly became a worldwide health issue, due to its infectiousness, mortality, and incurability. Biological studies have shown that HIV infects CD4+ T helper cells and attacks the immune system, causing the body to lose immunity. Humans are therefore susceptible to a variety of diseases, as well as malignant tumors. Therefore, revealing the HIV replication mechanism in vivo and predicting the impacts of interventions can help humans improve survival and reduce the therapy cost. Considering these two scenarios, mathematical models are imperative to reduce the cost and control virus replication. For example, the authors in [1–3] study COVID-19 and estimate the impacts of intervention strategies by mathematical models, which provides significant theoretical guidance for the process of human antiviral.It is clear that many scholars also studied HIV replication mechanism by mathematical models. Nowak and Bangham [4, 5] proposed a basic model to describe the variation of the virus in vivo:
(1)dTdt=Λ−δ1T−β1VT,dT∗dt=βVT−δ2T∗,dVdt=Nδ2T∗−δ3V,where T denotes the concentration of helper T cells, T∗ denotes the concentration of infected helper T cells, and V denotes the concentration of the virus. Λ is the production rate of new target cells. δ1 and δ2 are the death rates of uninfected cells and infected cells, respectively. β1 is the infection coefficient, and N is the burst coefficient of the virus when infected cells die. δ3 is the clearance rate of virus. The (1) model has been helpful in understanding HIV infection dynamics and developing specific treatment regimens [6, 7]. Based on this model, many mathematicians studied the dynamics of HIV infection in vivo [8–12].Up until now, there has not been an effective way to cure HIV in the world, which implies that controlling the spread of the virus in vivo is very essential for us. It is well known that antibodies, one main component of the immune response, which are produced by B lymphocytes are used to identify and neutralize free virus particles in the blood. It then becomes a promising and effective therapy to reduce the virus in the blood in clinic experiments. In 2011, Balazs et al. [13] carried out a vectored immunoprophylaxis experiment to bring new hope to eradicating HIV. The experiment showed that the humanized mice receiving vectored immunoprophylaxis appeared to be fully protected from HIV infection. Based on this experiment, the authors in [14] presented their new model which was different from the previous one:
(2)dTdt=Λ−δ1T−β1VT,dIdt=β1VT−δ2T∗,dVdt=Nδ2T∗−δ3V−γ2AV,dAdt=μ−δ5A−γVA.Here,A is the concentration of antibodies in vivo. μ represents the neutralizing antibodies produced at a constant rate after the injection. δ5 denotes the clearance rate of antibodies. γ2AV represents the loss rate of the virus under the attack of antibodies. The term γAV depicts the loss of antibody from the effect of antibodies’ involvement with the virus. Other parameters are the same with system (1).Another main component of the immune response is cytotoxic T lymphocytes (CTLs), which are the T cells that are capable of recognizing and killing infected cells, and they would not be infected by HIV. Some papers were devoted to investigate the HIV models under the impacts of CTLs [8, 15–17], which indicates the CTLs have a critical role on the viral infection. Moreover, many researchers have taken into account the effect of both CTLs’ response and antibodies [18–22] and the references therein. Specifically, in [20–22], the authors captured the main features of the complex interactions of HIV and immune responses and then formulated the impacts of immune responses as a term in the models instead of introducing a new variable. Their formulations make the mathematical analysis tractable. However, it cannot depict the dynamics of HIV under CTLs and antibodies in vivo exactly.Therefore, to study the dynamics of HIV replication in vivo under CTLs and antibodies, we introduce two new variables into (1) to propose our model, which incorporates the CTLs into model (2):
(3)dTdt=Λ−δ1T−β1VT,dT∗dt=β1VT−δ2T∗−γ1T∗C,dVdt=Nδ2T∗−δ3V−γ2AV−β1TV,dCdt=β2T∗C−δ4C,dAdt=μ−δ5A−γVA,where C denotes the concentration of CTL cells, γ1 represents the loss rate of infected cells under attack by CTLs, and δ4 is the death rate of the CTL cells. The last term β1VT in the third equation describes the loss rate of the virus because of entry into target cells. The term β2T∗C represents the increment of CTL cells stimulated by infected T cells. All the other parameters in (3) are the same as those in (2).We consider CTLs and antibodies as two major components of the immune response in our model (3), which describes HIV replication in vivo under the protection of the immune response. Compared to models that only consider CTLs or antibodies, the results are more informative. It can also be more accurate than those works in [20–22], where formulate the impacts of CTLs and antibodies as a term of the model. On the other hand, model (2) is proposed to investigate the viral dynamics for the introduction of vectored immunoprophylaxis antibodies in the experiment of [13]. As we mentioned before, our model (3) incorporates the CTLs into model (2), which can describe the dynamics of HIV under the CTLs in the vectored immunoprophylaxis experiment. It can provide the theoretical results for the impacts of CTLs in this experiment, which can be further extended to study the clinical possibilities of the experiment. Clearly, this model and its results cannot be derived from [18–22] and the references therein, where CTLs and antibodies are also considered in their models.The organization of this paper is as follows: In the next section, we study the model (3). We study the well-posedness of solutions in Section 2. In Section 3, we study the existence of equilibria and then present the stability analysis of infection-free equilibrium in Section 4. We conduct the numerical simulation of this model in Section 5. A brief summary and discussion are shown in the last section.


## 2. Well-Posedness of Solutions

For the sake of brevity, we offer the following scaling:(4)t¯=δ1t,T¯=β1δ1T,T¯∗=β2δ1T∗,V¯=β1δ1V,C¯=γ1δ1C,A¯=γ2δ1A,Λ¯=Λβ1δ12,β¯=β2β1,δ¯1=β2δ1,N¯=Nδ2β1δ1β2,δ¯2=δ3δ1,δ¯3=δ4δ1,μ¯=μγ2δ12,δ¯4=δ5δ1,γ¯=γβ1.Dropping the bars, model (3) becomes
(5)dTdt=Λ−T−VT,dT∗dt=βVT−δ1T∗−T∗C,dVdt=NT∗−δ2V−AV−VT,dCdt=T∗C−δ3C,dAdt=μ−δ4A−γAV.In this section, we show that all solutions of the system (5), for any given nonnegative initial conditions, are nonnegative.Theorem 1.
Any solution of system (5), Tt,T∗t,Vt,Ct,At, is nonnegative and ultimately bounded for t>0 with any provided nonnegative initial conditions.Proof.
. Firstly, we show thatTt is positive for t>0. Assume that t1>0 is the first time such that Tt1=0. Then, according to the first equation of system (5), we have dT/dtt=t1=Λ>0, which implies that Tt≥0 always holds for t>0 with T0>0. It is similar to show that Ct≥0 and At>0 always hold true for t>0 for any given positive initial conditions.To proveVt≥0 for t>0, we argue it by contradiction. Let t2>0 be the first time to make Vt2=0, then we have dV/dtt=t2=NT∗t2. Solving T∗t from the second equation of system (5) yields
(6)T∗t2=e−∫0t2δ1+CεdεT∗0+∫0t2βTθVθe∫0θδ1+Cεdεdθ,which means that T∗t2>0. Then we have dV/dtt=t2=NT∗t2>0 and then Vt≥0 for t>0. Furthermore, we can also get
(7)T∗t=e−∫0tδ1+CεdεT∗0+∫0tβTθVθe∫0θδ1+Cεdεdθ>0.It shows thatT∗t is nonnegative for t>0.Now we show that solutions of system (5) are ultimately uniformly bounded for t>0. Firstly, from system (5), we have
(8)dTtdt≤Λ−T,dAtdt≤μ−δ4At,which means that limsupt⟶∞Tt≤Λ and limsupt⟶∞At≤μ/δ4. Adding the former two equations yields
(9)dβTt+T∗tdt≤βΛ−d1βTt+T∗t,where d1=minβ,δ1, which means that limsupt⟶∞βTt+T∗t≤βΛ/d1. It follows that
(10)dVtdt≤NβΛd1−δ2Vt,and then limsupt⟶∞Vt≤NβΛ/d1δ2. Therefore, we obtain that
(11)dT∗t+Ctdt≤Nβ2Λ2d1δ2−d2T∗t+Ct,where d2=minδ1,δ3, which means that limsupt⟶∞T∗t+Ct≤Nβ2Λ2/d1d2δ2.Therefore, we get the following feasible region:(12)Γ=T,T∗,V,C,A∈ℝ+5:0<Tt≤Λ,0<βTt+T∗t≤βΛd1,0≤Vt≤NβΛd1δ2,0≤T∗t+Ct≤Nβ2Λ2d1d2δ2,0<At≤μδ4.It can be verified thatΓ is positively invariant with respect to system (5).


## 3. The Existence of Equilibria

It is easy to obtain that the infection-free equilibrium (IFE) isE0=Λ,0,0,0,μ/δ4. In virus dynamics, the basic reproduction number is a basic concept, which denotes the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [23]. Obviously, whether the basic reproduction number exceeds unit one is an important factor to determine the spread or extinction of the infection, biologically. Hence, it is necessary and reasonable for us to derive the basic reproduction number of model (5). By the method introduced in [23], we rewrite (5) as dx/dt=F−V, where x=T,T∗,V,C,AΤ.
(13)F=ΛβVTNT∗0μ,V=T+VTδ1T∗+T∗Cδ2V+AV+VT−T∗C+δ3Cδ4A+γAV.By computingR0=ρFL−1, where ρB denotes the spectral radius of a matrix B(14)F=0000000βΛ000N0000000000000,L=10T000δ100000δ2+Λ+μδ400000δ300000δ4,we can obtain that the basic reproduction number of virus in (5) is
(15)R0=NβΛδ1δ2+Λ+μ/δ4.


### 3.1. The Existence of the Immune-Free Infection Equilibrium

It is obvious that model (5) has an immune-free infection equilibrium E~=T~,T~∗,V~,0,A~. It is clear that the existence of this immune-free infection equilibrium is equivalent to study the existence of the infection equilibrium of the following system:
(16)dTdt=Λ−T−VT,dT∗dt=βVT−δ1T∗,dVdt=NT∗−δ2V−AV−VT,dAdt=μ−δ4A−γAV.Then, we study the existence of infection equilibrium of system (16). An infection equilibrium E~=T~,T~∗,V~,A~ satisfies
(17)Λ−T~−V~T~=0,βV~T~−δ1T~∗=0,NT~∗−δ2V~−A~V~−V~T~=0,μ−δ4A~−γA~V~=0.Obviously, we have(18)T~=Λ1+V~,T~∗=βV~δ1T~,A~=μδ4+γV~,where V~ satisfies the equation
(19)fV~=M1V~2+M2V~+M3=0,in which
(20)M1=δ2δ4γ>0,M2=δ4δ2δ4+μ−δ4γΛ+δ2R02−1−γμR02,M3=δ4Λ+δ2δ4+μ1−R02.FromM3, we know that system (16) has a unique infection equilibrium E~2=T2,T2∗,V2,A2, when R0>1, where
(21)V2=−M2+Δ2M1,T2=Λ1+V2,T2∗=βV2δ1T2,A2=μδ4+γV2,Δ=M22−4M1M3.As in the case ofR0<1, note that
(22)m1=γ2δ2+Λδ4+μ2>0,m2=−2δ2+Λδ4+μδ2+Λγ+μ−δ2δ4δ4γ,m3=γ−δ4δ2+Λγ−μ2+4Λγμδ42>0,R0c=−m2+m22−4m1m32m1.We define that(23)V1=−M2−Δ2M1,T1=Λ1+V1,T1∗=βV1δ1T1,A1=μδ4+γV1.Theorem 2.
System (16) has an infection equilibria E~2=T2,T2∗,V2,A2 if R0>1. Hence, system (5) admits an immune-free equilibria E2=T2,T2∗,V2,0,A2 if R0>1.Lemma 3.
Letμ+δ2δ4<μγ/δ4 and R0<1 hold. We have the following conclusions:
(1)
System (16) has two infection equilibria E~1=T1,T1∗,V1,A1 and E~2=T2,T2∗,V2,A2 when R0c<R0<1(2)
System (16) has a unique infection equilibrium E~1=E~2 when R0=R0c(3)
System (16) has no infection equilibrium when R0<R0cProof.
Clearly, (19) has no positive root when M2>0, two positive roots when M2<0 and Δ>0, has a positive root when M2<0 and Δ=0, and no positive root when Δ<0. So, we study the case that M2<0.
To determine the signs ofΔ in terms of the basic reproduction number, we denote R02 by R1 so that
(24)N=δ1δ2δ4+δ4Λ+μβΛδ4R1.
It is easy to obtain(25)Δ=m1R12+m2R1+m3,where mii=1,2,3 are given in (22). Since M2<0, we get
(26)R1>μ+δ2δ4+δ2γ+Λγδ4γδ2δ4+Λδ4+μ≔R2.
Suppose(27)μ+δ2δ4<μγδ4,then R2<1. Let hx=m1x2+m2x+m3; then we have
(28)hR2=m12R22+m2R2+m3=4δ2δ43δ2δ4+μ−μγδ4<0,h1=m1+m2+m3=γ−δ4μ−δ2δ422>0.
Ifμ+δ2δ4<μγ/δ4, we can get γ>δ4, and then
(29)Δ1=m22−4m1m3=16γ−δ4Λ+δ2δ4+μ2γ2δ2δ42μ>0.
It follows thathx=0 admits a root R0c, which is defined in (22), in interval (R2, 1). Clearly, we have Δ=0 when R0=R0c; Δ>0 when R0c<R0 and Δ<0 when 0<R0<R0c.
According to Lemma3, we can conclude the following theorem.Theorem 4.
Letμ+δ2δ4<μγ/δ4 and R0<1 hold.
(1)
System (5) has two immune-free equilibria E1=T1,T1∗,V1,0,A1 and E2=T2,T2∗,V2,0,A2 when R0c<R0<1(2)
System (5) has a unique immune-free equilibrium when R0=R0c(3)
System (5) has no immune-free equilibrium when R0<R0c


### 3.2. The Existence of the Immune-Present Infection Equilibrium of System (5)

An immune-present infection equilibriumE¯=T¯,T¯∗,V¯,C¯,A¯ satisfies
(30)Λ−T¯−V¯T¯=0,βV¯T¯−δ1T¯∗−T¯∗C¯=0,NT¯∗−δ2V¯−A¯V¯−V¯T¯=0,T¯∗C¯−δ3C¯=0,μ−δ4A¯−γA¯V¯=0.Direct computation leads to(31)T¯=Λ1+V¯,T¯∗=δ3,C¯=βΛV¯δ31+V¯−δ1,A¯=μδ4+γV¯,where V¯ satisfies the equation
(32)gV¯=a1V¯3+a2V¯2+a3V¯+a4,in which
(33)a1=δ2γ>0,a2=μ+δ2δ4+δ2γ+Λγ−Nδ3γ=δ2δ4γ2−R∗+δ4γΛ1−R∗+μγ1−R∗+δ22δ42+δ2γ2+Λγ2+μδ4δ4+γ,a3=μ+δ2δ4+δ4Λ−Nδ3γ−Nδ3δ4=μ+δ2δ4+δ4Λ1−R∗,a4=−Nδ3δ4<0,R∗=Nδ3δ4+γδ2δ4+μ+Λδ4.To ensure the existence of a positive equilibrium, we letβΛ−δ1δ3>0 and C¯>0, which is equivalent to
(34)V¯>δ1δ3βΛ−δ1δ3.By the mathematical analysis, we know thatgx=a1x3+a2x2+a3x+a4=0 must have a unique positive solution when a3<0R∗>1. When a3≥0R∗≤1, it is easy to get a2>0. Then gx=0 has also a unique positive solution when a3≥0. Therefore, we have the following theorem.Theorem 5.
System (5) has a unique immune-present infection equilibrium E3=T3,T3∗,V3,C3,A3 when V3>δ1δ3/βΛ−δ1δ3, where V3 is the positive solution of gV¯=0. Otherwise, system (5) has no immune-present infection equilibrium.


## 3.1. The Existence of the Immune-Free Infection Equilibrium

It is obvious that model (5) has an immune-free infection equilibrium E~=T~,T~∗,V~,0,A~. It is clear that the existence of this immune-free infection equilibrium is equivalent to study the existence of the infection equilibrium of the following system:
(16)dTdt=Λ−T−VT,dT∗dt=βVT−δ1T∗,dVdt=NT∗−δ2V−AV−VT,dAdt=μ−δ4A−γAV.Then, we study the existence of infection equilibrium of system (16). An infection equilibrium E~=T~,T~∗,V~,A~ satisfies
(17)Λ−T~−V~T~=0,βV~T~−δ1T~∗=0,NT~∗−δ2V~−A~V~−V~T~=0,μ−δ4A~−γA~V~=0.Obviously, we have(18)T~=Λ1+V~,T~∗=βV~δ1T~,A~=μδ4+γV~,where V~ satisfies the equation
(19)fV~=M1V~2+M2V~+M3=0,in which
(20)M1=δ2δ4γ>0,M2=δ4δ2δ4+μ−δ4γΛ+δ2R02−1−γμR02,M3=δ4Λ+δ2δ4+μ1−R02.FromM3, we know that system (16) has a unique infection equilibrium E~2=T2,T2∗,V2,A2, when R0>1, where
(21)V2=−M2+Δ2M1,T2=Λ1+V2,T2∗=βV2δ1T2,A2=μδ4+γV2,Δ=M22−4M1M3.As in the case ofR0<1, note that
(22)m1=γ2δ2+Λδ4+μ2>0,m2=−2δ2+Λδ4+μδ2+Λγ+μ−δ2δ4δ4γ,m3=γ−δ4δ2+Λγ−μ2+4Λγμδ42>0,R0c=−m2+m22−4m1m32m1.We define that(23)V1=−M2−Δ2M1,T1=Λ1+V1,T1∗=βV1δ1T1,A1=μδ4+γV1.Theorem 2.
System (16) has an infection equilibria E~2=T2,T2∗,V2,A2 if R0>1. Hence, system (5) admits an immune-free equilibria E2=T2,T2∗,V2,0,A2 if R0>1.Lemma 3.
Letμ+δ2δ4<μγ/δ4 and R0<1 hold. We have the following conclusions:
(1)
System (16) has two infection equilibria E~1=T1,T1∗,V1,A1 and E~2=T2,T2∗,V2,A2 when R0c<R0<1(2)
System (16) has a unique infection equilibrium E~1=E~2 when R0=R0c(3)
System (16) has no infection equilibrium when R0<R0cProof.
Clearly, (19) has no positive root when M2>0, two positive roots when M2<0 and Δ>0, has a positive root when M2<0 and Δ=0, and no positive root when Δ<0. So, we study the case that M2<0.
To determine the signs ofΔ in terms of the basic reproduction number, we denote R02 by R1 so that
(24)N=δ1δ2δ4+δ4Λ+μβΛδ4R1.
It is easy to obtain(25)Δ=m1R12+m2R1+m3,where mii=1,2,3 are given in (22). Since M2<0, we get
(26)R1>μ+δ2δ4+δ2γ+Λγδ4γδ2δ4+Λδ4+μ≔R2.
Suppose(27)μ+δ2δ4<μγδ4,then R2<1. Let hx=m1x2+m2x+m3; then we have
(28)hR2=m12R22+m2R2+m3=4δ2δ43δ2δ4+μ−μγδ4<0,h1=m1+m2+m3=γ−δ4μ−δ2δ422>0.
Ifμ+δ2δ4<μγ/δ4, we can get γ>δ4, and then
(29)Δ1=m22−4m1m3=16γ−δ4Λ+δ2δ4+μ2γ2δ2δ42μ>0.
It follows thathx=0 admits a root R0c, which is defined in (22), in interval (R2, 1). Clearly, we have Δ=0 when R0=R0c; Δ>0 when R0c<R0 and Δ<0 when 0<R0<R0c.
According to Lemma3, we can conclude the following theorem.Theorem 4.
Letμ+δ2δ4<μγ/δ4 and R0<1 hold.
(1)
System (5) has two immune-free equilibria E1=T1,T1∗,V1,0,A1 and E2=T2,T2∗,V2,0,A2 when R0c<R0<1(2)
System (5) has a unique immune-free equilibrium when R0=R0c(3)
System (5) has no immune-free equilibrium when R0<R0c


## 3.2. The Existence of the Immune-Present Infection Equilibrium of System (5)

An immune-present infection equilibriumE¯=T¯,T¯∗,V¯,C¯,A¯ satisfies
(30)Λ−T¯−V¯T¯=0,βV¯T¯−δ1T¯∗−T¯∗C¯=0,NT¯∗−δ2V¯−A¯V¯−V¯T¯=0,T¯∗C¯−δ3C¯=0,μ−δ4A¯−γA¯V¯=0.Direct computation leads to(31)T¯=Λ1+V¯,T¯∗=δ3,C¯=βΛV¯δ31+V¯−δ1,A¯=μδ4+γV¯,where V¯ satisfies the equation
(32)gV¯=a1V¯3+a2V¯2+a3V¯+a4,in which
(33)a1=δ2γ>0,a2=μ+δ2δ4+δ2γ+Λγ−Nδ3γ=δ2δ4γ2−R∗+δ4γΛ1−R∗+μγ1−R∗+δ22δ42+δ2γ2+Λγ2+μδ4δ4+γ,a3=μ+δ2δ4+δ4Λ−Nδ3γ−Nδ3δ4=μ+δ2δ4+δ4Λ1−R∗,a4=−Nδ3δ4<0,R∗=Nδ3δ4+γδ2δ4+μ+Λδ4.To ensure the existence of a positive equilibrium, we letβΛ−δ1δ3>0 and C¯>0, which is equivalent to
(34)V¯>δ1δ3βΛ−δ1δ3.By the mathematical analysis, we know thatgx=a1x3+a2x2+a3x+a4=0 must have a unique positive solution when a3<0R∗>1. When a3≥0R∗≤1, it is easy to get a2>0. Then gx=0 has also a unique positive solution when a3≥0. Therefore, we have the following theorem.Theorem 5.
System (5) has a unique immune-present infection equilibrium E3=T3,T3∗,V3,C3,A3 when V3>δ1δ3/βΛ−δ1δ3, where V3 is the positive solution of gV¯=0. Otherwise, system (5) has no immune-present infection equilibrium.


## 4. The Stability of Equilibra

The Jacobian matrix of (5) at an equilibrium E=T,T∗,V,C,A is
(35)JE=−V−10−T00Vβ−C−δ1βT−T∗0−VN−A−T−δ20−V0C0T∗−δ3000−γA0−Vγ−δ4.Theorem 6.
The infection-free equilibriumE0 is asymptotically stable if R0<1 and is unstable if R0>1. The immune-free equilibrium E1 is unstable when it exists.Proof.
The characteristic equation of the Jacobian matrix of (5) at E0, JE0, in λ is
(36)λ+1λ+δ3λ+δ4λ2+δ1+δ2+μδ4+Λλ+δ1δ2+Λ+μδ41−R02=0.
It follows that all the characteristic roots have negative real parts whenR0<1, and one characteristic root is positive when R0>1. Thus, we know that E0 is asymptotically stable when R0<1 and is unstable when R0>1.
The characteristic polynomial ofJE1 in λ is then
(37)F1λ=λ+δ3−T1∗λ4+c3λ3+c2λ2+c1λ+c0,where
(38)c3=δ1+1+γV1+δ2+A1+T1+δ4+1>0,c2=γV12+1+γδ1+δ2+γ1+T1+A1+δ4V1+δ1+δ2+A1+T11+δ4+δ4>0,c1=δ1δ2+δ4+A1+δ1+δ2γ+δ2+A1δ4−δ1γA1,c0=δ1V1δ4−γA1+δ2δ4+γV1.
We note that(39)c0=δ1V1δ4+γV12M1V1+M2−γδ4M2V1+2M3=−Δγδ4V1+1<0.
It then implies thatJE1 has at least a positive real eigenvalue, which means that E1 is unstable.
As for the stability ofE2, we can also obtain that the characteristic polynomial of JE2 in λ is then
(40)F2λ=λ+δ3−T2∗λ4+c^3λ3+c^2λ2+c^1λ+c^0,where
(41)c^3=δ1+1+γV2+δ2+A2+T2+δ4+1>0,c^2=γV22+1+γδ1+δ2+γ1+T2+A2+δ4V2+δ1+δ2+A2+T21+δ4+δ4>0,c^1=δ1δ2+δ4+A2+δ1+δ2γ+δ2+A2δ4−δ1γA2,c^0=δ1V2δ4−γA2+δ2δ4+γV2.Hence, according to Routh-Hurwitz criterion, we obtain the following theorem.Theorem 7.
When system (5) admits the immune-free equilibrium E2, it is locally stable when T2∗<δ3, c^1>0, c^0>0, c^3c^2>c^1, and c^1c^2c^3>c^12+c^32c^0.
Obviously, the characteristic polynomial ofJE3 in λ is then
(42)F3λ=λ5+c¯4λ4+c¯3λ3+c¯2λ2+c¯1λ+c¯0,where
(43)c¯4=δ1+1+γV3+δ2+A3+T3+δ4+1+C3>0,c¯3=1+γV3+1+δ3+δ4C3+γV32+δ1+δ2+δ4+A3+γ1+δ1+δ2+T3V3+δ1+δ2+A3+T31+δ4+δ4>0,c¯2=δ1+δ2+C3γV32+δ2+δ3+δ4+γ+δ3γ+A3V3C3+1+δ3δ4+1+δ2+A3+T3δ3C3+δ2+A3δ4+δ2+δ4+γ+A3−γA3δ1+δ2+T3−A3C3γV3+δ1+δ2+A3+T3δ4,c¯1=δ1δ2+δ2+δ3C3γV32+δ1+C3δ2+A3δ4+δ2+δ3A3+δ4+γ1+δ2+T3δ3C3V3+δ4+1+δ4δ2+A3+T3δ3C3>0,c¯0=δ3C3δ2γV32+δ2+A3δ4+δ2+T3γV3+δ1+A3+T3δ4>0.According to Routh-Hurwitz criterion, we can also obtain the following theorem.Theorem 8.
When system (5) admits the immune-present equilibrium E3, it is locally stable when c¯2>0, c¯3c¯4−c¯2>0, c¯2c¯3c¯4+c¯0c¯4−c¯22−c¯1c¯42>0, and c¯1c¯4c¯2c¯3−c¯1c¯4−c¯22+c¯02c¯1c¯4+c¯2c¯3−c¯32c¯4−c¯0>0.


## 5. Bifurcations and Numerical Simulation

In this section, we study bifurcations of model (3) on the basis of Section 3 and Section 4. When R0 crosses unit one, the IFE E0 loses stability, which results in a bifurcation where a curve of endemic equilibria emerges. The bifurcation is forward if the endemic curve occurs when R0 is slightly larger than one, and there is no positive equilibrium near the E0 for R0<1. In contrast, the bifurcation is backward if a positive equilibrium occurs when R0<1. Now, we derive the condition for the backward bifurcation from E0 of model (3). Notice that Theorem 4 implies that (24) is the existent condition of the backward bifurcation of model (5). By (4), we get an equivalent equation of (27):
(44)B1≔μγγ2δ52−δ3β1δ1−μβ1γ2δ1δ5>0.Thus, we show the proposition below.Proposition 9.
System (3) admits a backward bifurcation as R0 crosses unity if B1>0 and admits a forward bifurcation if B1<0.In this section, we implement numerical simulations on the basis of the MatCont package [24, 25] to testify the theoretical results above and explore more patterns of dynamical behaviors of model (3). We select Λ=100,δ1=0.008,δ2=0.08,δ3=3,δ4=0.05,δ5=0.02, β1=5×10−7, β2=5×10−5, γ1=0.0005, γ2=0.005, and μ=100 from [8, 14].Ifγ=10−5, we have B1=0.01075>0 and a backward bifurcation exists from Proposition 9. We derive a bifurcation figure shown in Figure 1. To further address the results of Figure 1 at length, we get evolutionary results, as presented by Figures 2 and 3.Figure 1
Two immune-free infection equilibria and an immune-present infection equilibrium coexist. The dashed (solid) curve represents an unstable (stable) equilibrium. LP is a limit point, H represents a Hopf point, and BP represents a branch point.Figure 2
(a) A numerical solution of model (3) from 4000,30,10000,20,50 tends to an infection-free equilibrium as time tends to infinity when R0=0.8. (b) A numerical solution of model (3) from 6140,635,16572,538,1 tends to an immune-free infection equilibrium as time tends to infinity when R0=0.8.
(a)(b)Figure 3
(a) A numerical solution of model (3) from 6140,635,16572,538,1 tends to an immune-present infection equilibrium as time tends to infinity when R0=1.25. (b) Numerical solutions of model (3) tend to the infection-free equilibrium as time tends to infinity with the different initial values when R0=0.7.
(a)(b)From Figure1, we find a Hopf bifurcation point (H) at R0=0.724, a fold bifurcation point (LP) at R0=0.6404, and a branch point (BP) at R0=0.819. It illustrates that two unstable immune-free infection equilibria coexist with a stable infection-free equilibrium when 0.6404<R0<0.724; a stable immune-free infection equilibrium, an unstable immune-free infection equilibrium, and a stable infection-free equilibrium coexist when 0.724<R0<0.819 (see Figure 2); and an unstable immune-free infection equilibrium coexists with a stable immune-present infection equilibrium when R0>0.819 (see Figure 3(a)). Further numerical simulations indicate that any solution tends to an infection-free equilibrium when R0<0.724 (see Figure 3(b)), which means that the subthreshold 0.724 of R0 is required to eliminate the infection. It is different to the classical backward bifurcation that needs R0<0.6404 to eliminate the infection.Whenγ=10−7, we have B1=−0.001625<0, and a forward bifurcation exists from Proposition 9. When we get a bifurcation figure shown in Figure 4, the detailed evolutionary processes are, correspondingly, illustrated in Figure 5.Figure 4
A immune-free infection equilibrium and an immune-present infection equilibrium coexist whenR0>1. The dashed (solid) curve represents an unstable (stable) equilibrium. BP represents a branch point.Figure 5
For two cases,R0=1.4 and R0=2.5, a numerical solution of model (3) from 4000,30,10000,20,50 tends to an immune-free infection equilibria and the immune-present infection equilibrium as time tends to infinity, respectively.
(a)(b)From Figure4, a branch point (BP) is observed at R0=1.979166. It demonstrates a stable infection-free equilibrium exists when R0<1; a stable immune-free infection equilibrium, and an unstable infection-free equilibrium coexist when 1<R0<1.979166 (see Figure 5(a)), and an unstable immune-free infection equilibrium coexists with a stable immune-present infection equilibrium when R0>1.979166 (see Figure 5(b)).Note that the qualitative difference between Figures1 and 4 results from the fact that the value of γ in Figure 1 is higher than it is in Figure 4. This indicates that a stronger loss of antibodies’ involvement with a virus can lead to a backward bifurcation, which supports the result of Proposition 9.


## 6. Conclusion and Discussion

In this paper, we focused on the modeling analysis of HIV epidemics, which incorporated immune systems based on CTLs and vectored immunoprophylaxis for controlling HIV replication. To address this role of immune response in detail, we integrated the immune response as a term into the model (3).Specifically, we gave the analytical condition of the existence of equilibria. We further studied the local stability of the existing equilibria. According to these theoretical results, we described the bifurcation figures of the given model. We noted that there exists a backward bifurcation at the infection-free equilibrium, thereby, demonstrating that driving the basic reproduction number (R0) below 1 is not enough to eliminate HIV. More significantly, we observed a Hopf point at R0=0.724, which indicates the classical strategy to drive R0 below a certain value (limit point) 0.6404 maybe unnecessary. Due to these findings, we noted that it is feasible to extinguish the infection even when R0 is below 0.724. Meanwhile, a forward bifurcation and an immune-present infection equilibrium exist in this model. Therefore, any strategy to drive R0 below 1 is adequate to clear the infection away, according to our results.Using numerical simulations and mathematical analyses of the proposed model, we can gain insights into the mechanisms of viral infection under the conditions of vectored immunoprophylaxis and immune response. With the introduction of the immune response, the model exhibits rich dynamical characteristics. As a result of the above findings, vectored immunoprophylaxis may be an effective therapy for HIV, and immune responses have a great influence on viral infection in vivo, which is overlooked in [14].



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*Source: 8043131-2022-09-27.xml*