# An Age-Structured Model of HIV Latent Infection with Two Transmission Routes: Analysis and Optimal Control

Authors: Chunyang Qin; Xia Wang; Libin Rong
Journal: Complexity
 (2020)

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## Abstract

In addition to direct virus infection of target cells, HIV can also be transferred from infected to uninfected cells (cell-to-cell transmission). These two routes might facilitate viral production and the establishment of the latent virus pool, which is considered as a major obstacle to HIV cure. We studied an HIV infection model including the two infection routes and the time since latent infection. The basic reproductive ratioR0 was derived. The existence, positivity, and boundedness of the solution are proved. We investigated the existence of steady states and their stability, which were shown to depend on R0. We established the global asymptotic dynamical behavior by proving the existence of the global compact attractor and uniform persistence of the system and by applying the method of Lyapunov functionals. In the end, we formulated and solved the optimal control problem for the age-structured model. The necessary condition for minimization of the viral level and the cost of drug treatment was obtained, and numerical simulations of various optimal control strategies were performed.


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## Body

## 1. Introduction

Human immunodeficiency virus (HIV) is a retrovirus that causes acquired immunodeficiency syndrome (AIDS) by infecting healthyCD4+ T cells, macrophages, and dendritic cells [1]. After infection, infected cells can generate new virus, which result in more infection and viral production. In order to investigate the within-host dynamics of HIV infection, many researchers have established different types of mathematical models (see, e.g., [2–11] and references cited therein). Most of these models focused on the infection of uninfected CD4+ T cells by cell-free virus particles. In addition to cell-free viral infection, target cells can also be infected by virus transmitted directly from infected cells (cell-to-cell transmission) [12–20]. When infected CD4+ T cells encounter uninfected CD4+ T cells and form viral synapses, HIV may spread by cell-to-cell transmission. The two routes of viral transmission may provide a synergistic way for virus infection [2, 21, 22]. The relative contributions of these two routes to viral production remain unclear. There is some evidence that HIV transmits more validly via cell-to-cell transmission than cell-free virus infection [3, 23–30].Models with age structure have been used in studying viral dynamics because many within-host processes, such as viral production by infected cells, may rely on the time after infection, i.e., the age of infection [31–41]. For example, Rong et al. [42] studied age of infection models with combination therapy including reverse transcriptase inhibitor (RTI), protease inhibitor (PI), and entry/fusion inhibitor. Wang et al. in [39] developed an HIV model with infection age and general nonlinear rates of viral infection in the infected n types of target cells to study the transmission dynamics of HIV.HIV latent infection of target cells remains to be a major barrier to viral clearance. To study the low viral persistence despite long-term antiretroviral therapy, mathematical models have been developed by incorporating cellular compartments or reservoirs such as latently infectedCD4+ T cells [43]. To explain the extremely slow decay of the latent viral pool and the viral persistence during therapy, Alshorman et al. [44] developed an HIV infection model involving latently infected cells with age structure. They showed that the model can generate a low viral load persistence and the extremely slow decay of the latent reservoir during prolonged therapy. In this work, we study an age-structured HIV model that includes two transmission routes to study virus dynamics. We will analyze the stability of the equilibria using the basic reproductive number. Because lifelong therapy is required for HIV-infected inhibitors, we formulate and solve the optimal control problem. The necessary condition for minimizing the viral level and the cost of drug treatment was obtained. Different optimal treatment strategies were evaluated by numerical investigations.


## 2. The Model

We introduce an HIV latent infection model with age structure and the two transmission routes. The model has four state variables: uninfectedCD4+ target cells Tt, age-structured latently infected cells, Lt,a, infected cells that can produce virus, It, and virus Vt. The model can be expressed as follows:(1)dTtdt=λ−dTTt−βTtVt−kTtIt,∂Lt,a∂t+∂Lt,a∂a=ρaLt,a−αa+δLaLt,a,Lt,0=fβTtVt+fkTtIt,dItdt=1−fβTtVt+kTtIt+∫0∞αaLt,ada−δIt,dVtdt=NδIt−cVt,T0=T0∈ℝ+,L0,⋅=L0⋅∈L+1,I0=I0∈ℝ+,V0=V0∈ℝ+.The model assumes thatλ and dT are the recruitment rate and death rate of uninfected CD4+ T cells, respectively. The parameter β is the infection rate of uninfected CD4+ T cells by free virus and k is the rate of transmission by infected cells. The constant f∈0,1 is the fraction of new infection that becomes latency. The rate of proliferation ρa and the rate of death δLa of latently infected cells are assumed to be functions of the age of infection. The activation rate αa also depends on the age because latently infected cells need to wait until they encounter their relevant antigen. The total number of productive infected cells It obtained by latently infected cell activation is given by ∫0∞αaLt,ada. δ represents the death rate of infected cells. N denotes the total number of virions released by one infected cell during its life cycle. The viral clearance rate is c. A schematic diagram of the full system is given in Figure 1.Figure 1
Schematic diagram of model (1).To investigate the dynamics of (1), we need the following assumptions.A1L0,0=L00=fβT0V0+kT0I0.A2ρ⋅,α⋅,δL⋅∈L+∞0,∞ are uniformly continuous, where L+∞0,∞ is the nonnegative cone of the Banach space L∞0,∞.The size of the latent reservoir remains relatively stable or declines extremely slowly during suppressive therapy [45]. Thus, the proliferation rate cannot exceed the sum of the activation rate and the death rate of latently infected cells. We have the following assumption as also used in [44].A3 Because the latent reservoir is relatively stable during therapy, we let(2)supaρa≤infaαa+δLa.The state space of (1),(3)X+=ℝ+×L+10,∞×ℝ+×ℝ+,is the nonnegative cone of the Banach space ℝ×L10,∞×ℝ×ℝ equipped with the norm(4)T,L,I,V=T+L1+I+V,for T,L,I,V∈ℝ×L10,∞×ℝ×ℝ.From Iannelli [46] and Magal [47], if the initial condition T0,L0,I0,V0∈X+ satisfies (A1), then system (1) has a unique continuous solution in X+. Therefore, we can define a solution semiflow Φ:ℝ+×X+⟶X+ of (1) by Φt,T0,L0,I0,V0=Tt,Lt,⋅,It,Vt.Let(5)F1t=Tt+∫0∞Lt,ada,F2t=Tt+It.It follows from (1) that(6)dF1tdt≤λ−dTT−∫0∞αa+δLa−ρaLt,ada=λ−dTT−∫0∞maLt,ada≤λ−μ1F1t,where ma=αa+δLa−ρa, μ1=mindT,m0,andm0=infa∈0,∞ma. This leads to limsupt⟶∞F1t≤λ/μ1.(7)dF2tdt≤λ−dTT+∫0∞αaLt,ada−δI≤λ+α∞λμ1−dTT−δI≤λ+α∞λμ1−μ2F2t,which means that limsupt⟶∞F2t≤λ/μ2+α∞λ/μ1μ2, where μ2=mindT,δ.Because(8)dVtdt=NδI−cV≤Nδλμ2+α∞λμ1μ2−cV,we obtain limsupt⟶∞Vt≤Nδ/cλ/μ2+α∞λ/μ1μ2.The set(9)Ω=T,L,I,V∈X+|T+L1≤λμ1,T+I≤λμ2+α∞λμ1μ2,V≤Nδcλμ2+α∞λμ1μ2is a positive invariant and attracting subset for system (1).


## 3. Analysis of Model (1)

### 3.1. The Existence of Equilibria and Basic Reproduction Number

System (1) always has an infection-free equilibrium E0=T0,0,0,0, where T0=λ/dT.Denote(10)ξ=∫0∞αaσada,σa=exp−∫0amsds,fora∈ℝ+,where ms=αs+δLs−ρs.Define(11)R0=NβλcdT+kλδdTfξ+1−f=NβλcdTfξ+1−f+kλδdTfξ+1−f=ΔR01+R02,where R0 is the basic reproductive ratio of model (1) and R01 is the portion from cell-free virus infection while R02 is from the other transmission route (see details in [40]).WhenR0 is greater than 1, the model has a positive equilibrium, given by E∗=T∗,L∗,I∗,V∗∈Ω, which satisfies(12)λ−dTT∗−βT∗V∗−kT∗I∗=0,ddaL∗a=−maL∗a,L∗0=fβT∗V∗+fkT∗I∗,1−fβT∗V∗+1−fkT∗I∗+ξL∗0−δI∗=0,NδI∗=cV∗.Straightforward calculation yields the infected equilibrium:(13)T∗=cδNδβfξ+1−f+kcfξ+1−f=λdTR0,L∗a=λR0−1fR0σa,I∗=cdTR0−1Nδβ+kc,V∗=NδdTR0−1Nδβ+kc.


### 3.2. Local Stability

LetE¯=T¯,L¯,I¯,V¯ be any equilibrium of (1). After linearizing (1) at E¯, we get the characteristic equation as follows:(14)ΔE¯=τ+dT+βV¯+kI¯0kT¯βT¯fβV¯+fkI¯−1fkT¯fβT¯−1−fβV¯+kI¯−ξττ+δ−1−fkT¯−1−fβT¯00−Nδτ+c=0,where τ denotes the eigenvalue of the characteristic equation (14) and ξτ=∫0∞αaσae−τada. Obviously, the equilibrium E¯ of (1) is LAS (i.e., locally asymptotically stable) if the real parts of the roots of equation (14) are all negative; if at least one eigenvalue of (14) has a positive real part, then the equilibrium E¯ of (1) is unstable.Theorem 1.
The infection-free equilibriumE0 of system (1) is LAS when R0<1 and it is unstable when R0>1.Proof.
The characteristic equation of equilibriumE0 of system (1) can be simplified as(15)ΔE0=τ+dTτ+δτ+ch1τ=0,where(16)h1τ=1−kT01−f+f∫0∞αaσae−τadaτ+δ−NδβT01−f+f∫0∞αaσae−τadaτ+δτ+c.
Therefore, the stability ofE0 is decided by the roots of the equation h1τ=0. When R0<1, we want to show that the real parts of all the roots of h1τ=0 are negative. It is proved by contradiction. Let τ0 be a root of h1τ=0 with nonnegative real part. It follows from equation (16) that(17)1=kT01−f+f∫0∞αaσae−aτ0daτ0+δ+NδβT01−f+f∫0∞αaσae−aτ0daτ0+δτ0+c≤kT01−f+fξδ+NδβT01−f+fξδc=R0,which leads to a contradiction since R0<1. Thus, we know that each root of equation (15) is negative or has negative real part. Therefore, infection-free equilibrium E0 is LAS when R0<1.
IfR0>1, we have h10=1−R0<0. From limτ⟶∞h1τ=1>0, the equation h1τ=0 has at least one positive real root, which means that the equilibrium E0 is unstable for R0>1.
Next, we show that whenR0>1, the positive equilibrium E∗ is LAS.Theorem 2.
WhenR0>1, the infected equilibrium E∗ of (1) is LAS.Proof.
Using the equilibrium equalitydTR0=dT+βV∗+kI∗ and equation (14), we have the characteristic equation at E∗:(18)ΔE∗=τ+dTR0τ+cτ+δ−τ+dTkT∗τ+c+NδβT∗⋅1−f+f∫0∞αaσae−τada=0.
We now claim that equation (18) has no root with a nonnegative real part. If it is not, let us assume that it has a root τ0 with Reτ0≥0. Equation (18) becomes the following by dividing τ0+dTR0τ0+cτ0+δ:(19)1=τ0+dTkT∗τ0+c+NδβT∗1−f+f∫0∞αaσae−τ0adaτ0+dTR0τ0+cτ0+δ.
Taking the modulus of the two sides of equation (19), we have(20)1<kT∗1−f+fξδ+NβT∗1−f+fξc=R0T0⋅T∗=1,which is a contradiction. Therefore, the equilibrium E∗ is LAS when R0>1.


## 3.1. The Existence of Equilibria and Basic Reproduction Number

System (1) always has an infection-free equilibrium E0=T0,0,0,0, where T0=λ/dT.Denote(10)ξ=∫0∞αaσada,σa=exp−∫0amsds,fora∈ℝ+,where ms=αs+δLs−ρs.Define(11)R0=NβλcdT+kλδdTfξ+1−f=NβλcdTfξ+1−f+kλδdTfξ+1−f=ΔR01+R02,where R0 is the basic reproductive ratio of model (1) and R01 is the portion from cell-free virus infection while R02 is from the other transmission route (see details in [40]).WhenR0 is greater than 1, the model has a positive equilibrium, given by E∗=T∗,L∗,I∗,V∗∈Ω, which satisfies(12)λ−dTT∗−βT∗V∗−kT∗I∗=0,ddaL∗a=−maL∗a,L∗0=fβT∗V∗+fkT∗I∗,1−fβT∗V∗+1−fkT∗I∗+ξL∗0−δI∗=0,NδI∗=cV∗.Straightforward calculation yields the infected equilibrium:(13)T∗=cδNδβfξ+1−f+kcfξ+1−f=λdTR0,L∗a=λR0−1fR0σa,I∗=cdTR0−1Nδβ+kc,V∗=NδdTR0−1Nδβ+kc.


## 3.2. Local Stability

LetE¯=T¯,L¯,I¯,V¯ be any equilibrium of (1). After linearizing (1) at E¯, we get the characteristic equation as follows:(14)ΔE¯=τ+dT+βV¯+kI¯0kT¯βT¯fβV¯+fkI¯−1fkT¯fβT¯−1−fβV¯+kI¯−ξττ+δ−1−fkT¯−1−fβT¯00−Nδτ+c=0,where τ denotes the eigenvalue of the characteristic equation (14) and ξτ=∫0∞αaσae−τada. Obviously, the equilibrium E¯ of (1) is LAS (i.e., locally asymptotically stable) if the real parts of the roots of equation (14) are all negative; if at least one eigenvalue of (14) has a positive real part, then the equilibrium E¯ of (1) is unstable.Theorem 1.
The infection-free equilibriumE0 of system (1) is LAS when R0<1 and it is unstable when R0>1.Proof.
The characteristic equation of equilibriumE0 of system (1) can be simplified as(15)ΔE0=τ+dTτ+δτ+ch1τ=0,where(16)h1τ=1−kT01−f+f∫0∞αaσae−τadaτ+δ−NδβT01−f+f∫0∞αaσae−τadaτ+δτ+c.
Therefore, the stability ofE0 is decided by the roots of the equation h1τ=0. When R0<1, we want to show that the real parts of all the roots of h1τ=0 are negative. It is proved by contradiction. Let τ0 be a root of h1τ=0 with nonnegative real part. It follows from equation (16) that(17)1=kT01−f+f∫0∞αaσae−aτ0daτ0+δ+NδβT01−f+f∫0∞αaσae−aτ0daτ0+δτ0+c≤kT01−f+fξδ+NδβT01−f+fξδc=R0,which leads to a contradiction since R0<1. Thus, we know that each root of equation (15) is negative or has negative real part. Therefore, infection-free equilibrium E0 is LAS when R0<1.
IfR0>1, we have h10=1−R0<0. From limτ⟶∞h1τ=1>0, the equation h1τ=0 has at least one positive real root, which means that the equilibrium E0 is unstable for R0>1.
Next, we show that whenR0>1, the positive equilibrium E∗ is LAS.Theorem 2.
WhenR0>1, the infected equilibrium E∗ of (1) is LAS.Proof.
Using the equilibrium equalitydTR0=dT+βV∗+kI∗ and equation (14), we have the characteristic equation at E∗:(18)ΔE∗=τ+dTR0τ+cτ+δ−τ+dTkT∗τ+c+NδβT∗⋅1−f+f∫0∞αaσae−τada=0.
We now claim that equation (18) has no root with a nonnegative real part. If it is not, let us assume that it has a root τ0 with Reτ0≥0. Equation (18) becomes the following by dividing τ0+dTR0τ0+cτ0+δ:(19)1=τ0+dTkT∗τ0+c+NδβT∗1−f+f∫0∞αaσae−τ0adaτ0+dTR0τ0+cτ0+δ.
Taking the modulus of the two sides of equation (19), we have(20)1<kT∗1−f+fξδ+NβT∗1−f+fξc=R0T0⋅T∗=1,which is a contradiction. Therefore, the equilibrium E∗ is LAS when R0>1.


## 4. Global Dynamics of (1)

In the present section, by using suitable Lyapunov functionals, we establish the global asymptotic dynamics of the equilibria of (1). The results acquired in the following indicate that the GAS (i.e., global asymptotic stability) is completely determined by the basic reproduction number. We start with the global stability of E0 by applying the method of Lyapunov functionals when R0<1 (see proof in Appendix A).Theorem 3.
The infection-free equilibriumE0 of (1) is GAS when R0<1.
To prove the equilibriumE∗ of (1) is GAS, we need some preparation.
Letρ:Ω⟶ℝ+ by(21)ρT,L,I,V=βTtVt+kTtIt+∫0∞αaLt,ada,for T,L,I,V∈Ω. Define(22)Ω0=T0,L0,I0,V0∈Ω|There ist0∈ℝ+ensuringρΦt0,x0>0.
Given anyx0=T0,L0,I0,V0∈Ω\Ω0, we have limt⟶∞Φt,x0=E0. From Lemma 3.2 of Hale [48] and Theorem 3.2 of Thieme [49], one can obtain the following results with standard arguments (also see Chen et al. [50]).Theorem 4.
SupposeR0>1. Then, the following statements are true. (i) There exists a global attractor A for the solution semiflow Φ of (1) in Ω0. (ii) System (1) is uniformly strongly ρ-persistent, that is, there exists ε0>0 (independent of initial values) such that(23)liminft⟶∞ρΦt,x>ε0,forx∈Ω0.
It can only contain points that pass through their total trajectories because the global attractorA is invariant. A total trajectory of Φ is a function ω:ℝ⟶X+ ensuring that Φs,ωt=ωs+t for all t∈ℝ and s∈ℝ+. For a total trajectory, Lt,a=Lt−a,0σa for all a∈ℝ+ and t∈ℝ. The alpha limit of a total trajectory ωt passing through ω0=ω0 is(24)αω0=∩t≤0∪s≤tωs¯⊆A∩Ω0.Corollary 1.
We assume thatR0>1. Then, there exists ε0>0 such that T,L,I,V≥ε0 for all t∈ℝ, where Tt,Lt,0,It,Vt is any total trajectory in A.Proof.
SinceΩ is attracting and invariant, there is t1∈ℝ+ ensuring that(25)It≤λ+α∞λ/μ1μ2,Vt≤Nδλ+α∞λ/μ1cμ2,fort≥t1.
Fort≥t1, from the first equation of system (1), we get(26)dTtdt≥λ−dT+βNδλ+α∞λ/μ1cμ2+kλ+α∞λ/μ1μ2Tt,which leads to(27)liminft⟶∞Tt≥λdT+βNδλ+α∞λ/μ1/cμ2+kλ+α∞λ/μ1/μ2=ΔεT.
By invariance,Tt≥εT for all t∈ℝ.
It follows from Theorem4 that there is ε¯>0 ensuring that(28)∫0∞αaLt,ada+βTtVt+kTtIt≥ε¯,fort∈ℝ.
This, combined with system (1), gives(29)dItdt≥ε¯−δIt,fort∈ℝ.
Therefore, we have(30)liminft⟶∞It≥ε¯δ=ΔεI,and thus It≥εI for t∈ℝ by invariance again.
It follows from the last equation of (1) that(31)liminft⟶∞Vt≥NδεIc=ΔεV.
Thus,(32)Lt,0=fβTV+fkTI≥fβεTεv+fεTεI=ΔεL,fort∈ℝ.
Lettingε0=minεT,εL,εI,εV, we complete the proof.
The following theorem establishes the GAS ofE∗ using the methods of Lyapunov functionals when R0>1. The proof is given in Appendix B.Theorem 5.
The infected equilibriumE∗ of system (1) is GAS when R0>1.


## 5. The Optimal Control Problem

To obtain the optimal treatment strategies, we study the following model with controls(33)dTtdt=λ−dTTt−1−ε1tβTtVt−1−ε1tkTtIt,∂Lt,a∂t+∂Lt,a∂a=ρaLt,a−αa+δLaLt,a,Lt,0=1−ε1tfβTtVt+1−ε1tfkTtIt,dItdt=1−f1−ε1tβTtVt+kTtIt+∫0∞αaLt,ada−δIt,dVtdt=1−ε2tNδIt−cVt,T0=T0∈ℝ+,L0,⋅=L0⋅∈L+1,I0=I0∈ℝ+,V0=V0∈ℝ+.One control termε1t is the effectiveness of RTIs, which block new infection. The other control term ε2t denotes the effectiveness of PIs, which reduce the number of infectious virions.We minimize the objective functional(34)Jε1,ε2=∫0tfr1ε12t+r2ε22t+qVtdt,where tf is the duration of the treatment and r1,r2,q are the positive weight coefficients to balance the control functions and the quantity of virus particles. The aim of this section is to minimize the objective functional defined in equation (34) by decreasing the viral load and the cost of drug treatment. The two control functions ε1t and ε2t represent the efficacy of RTIs and PIs satisfying 0≤εi,min≤εit≤εi,max<1,i=1,2.Problem 1.
We seek optimal controlsε1∗ and ε2∗ such that(35)Jε1∗,ε2∗=minJε1,ε2|ε1tandε2tare Lebesque‐integrable on0,tfwith values inUi=εi,min,εi,max,i=1,2,subject to system (33) with the boundary condition(36)Lt,0=f1−ε1tβTtVt+f1−ε1tkTtIt,and the initial values(37)T0=T0,L0,a=L0a,I0=I0,V0=V0.
We noticed that the existence of optimal control functions to Problem1 can be obtained by the methods used in [51, 52].
By the discretization-differential approach [52], we divide Lt,a into a number of classes from age 0 to amax, changing the age-structure model (33) to ODEs. We differentiate the ODEs to get an adjoint (costate) system. Let ajj=0n be a partition of 0,amax with a0=0,aj−1≤aj,Δaj=aj−aj−1j=1,2,…,n, and an=amax. The discrete system is(38)dTtdt=λ−dTTt−1−ε1tβTtVt−1−ε1tkTtIt,dLdtt,aj=−Lt,aj−Lt,aj−1Δaj−majLt,aj,dItdt=1−f1−ε1tβTtVt+kTtIt+∑j=1nαajLt,ajΔaj−δIt,dVtdt=1−ε2tNδIt−cVt,Lt,0=1−ε1tfβTtVt+kTtIt,L0,aj=L0aj.
LetLj=Lt,aj and X=T,L1,L2,…,Ln,I,VT. The constraint equations are given by(39)X˙=TL1L2⋯LnIV′=λ−dTT−1−ε1βTV+kTI−L1−1−ε1fβTV+kTIΔa1−ma1L1−L2−L1Δa2−ma2L2⋯−Ln−Ln−1Δan−manLn1−f1−ε1βTV+kTI+∑j=1nαajLjΔaj−δI1−ε2NδI−cV.
In the following, using the Pontryagin maximum principle [53], we will obtain the optimal control functions ε1∗ and ε2∗. To achieve this, we introduce a Lagrange multiplier or adjoint variable Y=ωt,λ1t,λ2t,…,λnt,ω1t,ω2tT and define the Lagrangian ℒ as follows:(40)ℒ=r1ε12+r2ε22+qV+ωλ−dTT−1−ε1tβTV+kTI+λ1−L1Δa1+1−ε1fβTV+kTIΔa1−ma1L1+∑j=2nλj−Lj−Lj−1Δaj−majLj+ω11−f1−ε1βTV+kTI+∑j=1nαajLjΔaj−δI+ω21−ε2NδI−cV−ξ1ε1−ε1,min−η1ε1,max−ε1−ξ2ε2−ε2,min−η2ε2,max−ε2,where ξit≥0,ηit≥0i=1,2 are the penalty multipliers satisfying ξitεi−εi,min=ηitεi,max−εii=1,2 at εi=εi∗ and ε1∗,ε2∗ are the optimal controls.Theorem 6.
Given optimal controls and solutions with respect to the corresponding constraint equation (1) that minimize the objective functional (34), there exists an adjoint variable Y=ωt,λ1t,λ2t,…,λnt,ω1t,ω2tT satisfying(41)Y˙=ωλ1λ2⋯λnω1ω2′=−−dTω−1−ε1βV+kIω−fλ1Δa1−1−fω1−λ1Δa1−ma1λ1+λ2Δa2+αa1Δa1ω1−λ2Δa2−ma2λ2+λ3Δa3+αa2Δa2ω1⋅⋅⋅−λnΔan−manλn+αanΔanω1−1−ε1kTω−fλ1Δa1−1−fω1−δω1+1−ε2Nδω2q−1−ε1βTω−fλ1Δa1−1−fω1−cω2,with the terminal conditions Ytf=0,0,0,…,0,0,0T. Furthermore, the optimal control functions ε1∗ and ε2∗ are given by(42)ε1∗=maxε1,min,minε1,max,−ωt+f/Δa1λ1t+1−fω1t2r1βTtVt+kTtIt,ε2∗=maxε2,min,minε2,max,NδItω2t2r2.Proof.
According to the necessary conditions of the fixed points ofℒ, the adjoint equations can be obtained by taking Y˙=∂ℒ/∂X with the terminal conditions Ytf=0,0,0,…,0,0,0T. The optimal controls ε1∗ and ε2∗ can be solved from the necessary optimal conditions ∂ℒ/∂ε1=0 and ∂ℒ/∂ε2=0, respectively. That is,(43)∂ℒ∂ε1=ωt−fΔa1λ1t−1−fω1tβTtVt+kTtIt+2r1ε1t−ξ1t+η1t=0,∂ℒ∂ε2=2r2ε2t−NδItω2t−ξ2t+η2t=0.
Therefore, one can solve the optimal controls as follows:(44)ε1∗=12r1−ωt+fΔa1λ1t+1−fω1tβTtVt+kTtIt+ξ1t−η1t,ε2∗=12r2NδItω2t+ξ2t−η2t.
Next, in order to derive a detailed expression for the optimal controlsε1∗ and ε2∗ without ξi,ηi, we will study all possible cases for the optimal controls.Case I. If we consider the set t|ε1,min<ε1∗t<ε1,max,ξ1t=η1t=0, then we obtain(45)ε1∗=12r1fΔa1λ1+1−fω1−ωβTV+kTI.Case II. If we consider the set t|ε1∗t=ε1,max,ξ1t=0, then we obtain(46)ε1,max=ε1∗=12r1fΔa1λ1+1−fω1−ωβTV+kTI−η1≤12r1fΔa1λ1+1−fω1−ωβTV+kTI.Case III. If we consider the set t|ε1∗t=ε1,min,η1t=0, then we obtain(47)ε1,min=ε1∗=12r1fΔa1λ1+1−fω1−ωβTV+kTI+ξ1≥12r1fλ1Δa1+1−fω1−ωβTV+kTI.
Thus, we have the optimal controlsε1∗ and ε2∗ in the forms of (42) by combining these three cases.


## 6. Numerical Simulations

### 6.1. Dynamics of the Latently InfectedCD4+ T Cells and Viral Load

We choose parameters according to a previous study [54]: λ=104 cells ml−1day−1, dT=0.01day−1, β=2.4×10−8 virion ml−1day−1, k=2.4×10−8 cell ml−1day−1, f=0.0001, δ=1day−1, N=2000cell−1day−1, and c=23day−1. Numerical simulation of model (1) with initial values T0,L0a,I0,V0=106,0,0,10−3 generates an infected equilibrium, which will be used as the initial value for model (1) under treatment.We suppose that the activation rateαa of latently infected CD4+ T cells satisfies the exponential decay function. The function is chosen as follows:(48)αa=a∗−amine−ua+amin,where amin and a∗ denote the minimum activation rate and initial activation rate of Lt,a, respectively. The parameter u represents the decay parameter determining how fast α decreases to its minimum value. Because these parameter values remain unknown, we choose a∗=0.05day−1, u=0.01day−1, and amin=0.01day−1 according to ref. [44] as default values in the following simulation.Using these parameter values, we findR0=2.1108>1 and system (33) has the unique positive equilibrium E∗=T∗,L∗,I∗,V∗, where T∗=4.7375×105 cells ml−1, L∗=42.5322 cells ml−1, I∗=5.2622×103 cells ml−1, and V∗=4.5758×105 RNA copies ml−1. Figures 2(a)–2(d) show the dynamics of uninfected CD4+ T cells, the latent reservoir, productively infected CD4+ T cells, and viral load of the model without treatment, respectively. To describe the stability of the latently infected CD4+ T cells, we choose γ=ρa−δLa=0.5amin. Thus, ma=αa−γ. In addition, we choose n=100 and amax=10 such that Δaj=ai−aj−1=0.1, where j=1,2,…,n. Thus, in the numerical simulation of the latent reservoir, age a is divided into 100 equal intervals. The dynamical behaviors of latently infected CD4+ T cells with different ages of infection are shown in Figure 2(b).Figure 2
Predicted dynamics of uninfectedCD4+ T cells, latently infected CD4+ T cells, productively infected CD4+ T cells, and viral load of the model before therapy. Using the parameter values given in the text, we have R0=2.1108>1, which ensures that the infected equilibrium E∗ is GAS.
(a)(b)(c)(d)The aim of the present section is to analyze the latently infected T cell and virus dynamics after treatment. We use an overall drug effectiveness of the combination of RTs and PIs, i.e.,ε=1−ε11−ε2 [44]. For comparison, we show the dynamics of model (1) before treatment in Figure 3(a). We find that both the virus and latently infected cells converge to positive equilibria. In Figure 3(b), we show the dynamics under therapy by using ε=0.99 and ρa−δLa=0.1amin.Figure 3
Dynamics of the virus and latent reservoir predicted by model (33). (a) The latently infected CD4+ T cells and viral load reach an infected equilibrium of model (33) without treatment and ρa−δLa=0.5amin, where amin=0.01day−1. (b) The latently infected CD4+ T cells and viral level are predicted to go extinct under treatment: ε=0.99 and ρa−δLa=0.1amin. (c) The latently infected CD4+ T cells and viral load persist at low levels under therapy: ε=0.99 and ρa−δLa=amin.
(a)(b)(c)Both the latently infectedCD4+ T cells and viral load are predicted to go extinct in this scenario. However, the decay of the latently infected CD4+ T cells and viral load is slow. In Figure 3(c), we use ε=0.99 but assume that ρa−δLa=amin. Both the latently infected CD4+ T cells and virus persist at a low level even under prolonged drug therapy.


### 6.2. Various Optimal Control Strategies

We performed numerical simulation of model (33) using the steady state of the model before treatment as the initial condition. When the optimal control is started at the infected equilibrium, we choose parameter q=1/V∗=4.5758×10−5 such that qV∗=1. The effectiveness of RTIs control ε1 and the effectiveness of PIs ε2 are used to optimize the objective function J (given by (34)). By studying the optimal treatment strategies, different control schemes are compared. We solve the optimality system numerically by applying the forward-backward sweep method [55] with Matlab. We choose the final time tf to be 1000 days. We investigate the following three cases.Case 1: the effectiveness of RTIsε1≠0 and the effectiveness of PIs ε2=0. In this case, we take the weight constant values r1=r2=54.5. In Figure 4, we obtain the optimal control diagram for ε1≠0 (solid lines) and ε2=0 (dashed lines). In Figure 5, we observe that the control strategy results in an increase in the level of uninfected cells and a decrease in the viral load.Case 2: the effectiveness of RTIsε1=0 and the effectiveness of PIs ε2≠0. We choose the weights r1=75.8 and r2=75.8. In Figure 6, we show the optimal control solution for ε1=0 (solid lines) and ε2≠0 (dashed lines). In Figure 7, we also see that the level of uninfected cells increases and viral load decreases despite oscillations.Case 3: the effectiveness of RTIsε1≠0 and the effectiveness of PIs ε2≠0.We choose different weight constants to investigate the optimal control in this case. In Figure8, we show the optimal control by choosing r1=r2=264 and fixing the other parameters as in Case 1. In Figure 9, we show the changes of uninfected cells and viral load under this optimal control. Uninfected cells increase and the viral load decreases under treatment. Using other combinations of the weight constants, we obtain the similar prediction. They are shown in Figures 10 and 11 for r1=470>r2=220 and in Figures 12 and 13 for r1=152<r2=496.Figure 4
Optimal control strategy withε1≠0 (solid lines) and ε2=0 (dashed lines).Figure 5
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1≠0 and ε2=0 (solid lines: without control; dashed lines: with control).
(a)(b)Figure 6
Optimal control strategy withε1=0 (solid lines) and ε2≠0 (dashed lines).Figure 7
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1=0 and ε2≠0 (solid lines: without control; dashed lines: with control).
(a)(b)Figure 8
Optimal control strategy withε1≠0 (solid lines) and ε2≠0 (dashed lines), where r1=r2=264.Figure 9
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1≠0 and ε2≠0. Solid lines are without control and dashed lines are with control, in which r1=r2=264.
(a)(b)Figure 10
Optimal control strategy withε1≠0 (solid lines) and ε2≠0 (dashed lines), where r1=470>r2=220.Figure 11
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1≠0 and ε2≠0. Solid lines are without control and dashed lines are with control, in which r1=470>r2=220.
(a)(b)Figure 12
Optimal control strategy withε1≠0 (solid lines) and ε2≠0 (dashed lines), where r1=152<r2=496.Figure 13
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1≠0 and ε2≠0. Solid lines are without control and dashed lines are with control, in which r1=152<r2=496.
(a)(b)


## 6.1. Dynamics of the Latently InfectedCD4+ T Cells and Viral Load

We choose parameters according to a previous study [54]: λ=104 cells ml−1day−1, dT=0.01day−1, β=2.4×10−8 virion ml−1day−1, k=2.4×10−8 cell ml−1day−1, f=0.0001, δ=1day−1, N=2000cell−1day−1, and c=23day−1. Numerical simulation of model (1) with initial values T0,L0a,I0,V0=106,0,0,10−3 generates an infected equilibrium, which will be used as the initial value for model (1) under treatment.We suppose that the activation rateαa of latently infected CD4+ T cells satisfies the exponential decay function. The function is chosen as follows:(48)αa=a∗−amine−ua+amin,where amin and a∗ denote the minimum activation rate and initial activation rate of Lt,a, respectively. The parameter u represents the decay parameter determining how fast α decreases to its minimum value. Because these parameter values remain unknown, we choose a∗=0.05day−1, u=0.01day−1, and amin=0.01day−1 according to ref. [44] as default values in the following simulation.Using these parameter values, we findR0=2.1108>1 and system (33) has the unique positive equilibrium E∗=T∗,L∗,I∗,V∗, where T∗=4.7375×105 cells ml−1, L∗=42.5322 cells ml−1, I∗=5.2622×103 cells ml−1, and V∗=4.5758×105 RNA copies ml−1. Figures 2(a)–2(d) show the dynamics of uninfected CD4+ T cells, the latent reservoir, productively infected CD4+ T cells, and viral load of the model without treatment, respectively. To describe the stability of the latently infected CD4+ T cells, we choose γ=ρa−δLa=0.5amin. Thus, ma=αa−γ. In addition, we choose n=100 and amax=10 such that Δaj=ai−aj−1=0.1, where j=1,2,…,n. Thus, in the numerical simulation of the latent reservoir, age a is divided into 100 equal intervals. The dynamical behaviors of latently infected CD4+ T cells with different ages of infection are shown in Figure 2(b).Figure 2
Predicted dynamics of uninfectedCD4+ T cells, latently infected CD4+ T cells, productively infected CD4+ T cells, and viral load of the model before therapy. Using the parameter values given in the text, we have R0=2.1108>1, which ensures that the infected equilibrium E∗ is GAS.
(a)(b)(c)(d)The aim of the present section is to analyze the latently infected T cell and virus dynamics after treatment. We use an overall drug effectiveness of the combination of RTs and PIs, i.e.,ε=1−ε11−ε2 [44]. For comparison, we show the dynamics of model (1) before treatment in Figure 3(a). We find that both the virus and latently infected cells converge to positive equilibria. In Figure 3(b), we show the dynamics under therapy by using ε=0.99 and ρa−δLa=0.1amin.Figure 3
Dynamics of the virus and latent reservoir predicted by model (33). (a) The latently infected CD4+ T cells and viral load reach an infected equilibrium of model (33) without treatment and ρa−δLa=0.5amin, where amin=0.01day−1. (b) The latently infected CD4+ T cells and viral level are predicted to go extinct under treatment: ε=0.99 and ρa−δLa=0.1amin. (c) The latently infected CD4+ T cells and viral load persist at low levels under therapy: ε=0.99 and ρa−δLa=amin.
(a)(b)(c)Both the latently infectedCD4+ T cells and viral load are predicted to go extinct in this scenario. However, the decay of the latently infected CD4+ T cells and viral load is slow. In Figure 3(c), we use ε=0.99 but assume that ρa−δLa=amin. Both the latently infected CD4+ T cells and virus persist at a low level even under prolonged drug therapy.


## 6.2. Various Optimal Control Strategies

We performed numerical simulation of model (33) using the steady state of the model before treatment as the initial condition. When the optimal control is started at the infected equilibrium, we choose parameter q=1/V∗=4.5758×10−5 such that qV∗=1. The effectiveness of RTIs control ε1 and the effectiveness of PIs ε2 are used to optimize the objective function J (given by (34)). By studying the optimal treatment strategies, different control schemes are compared. We solve the optimality system numerically by applying the forward-backward sweep method [55] with Matlab. We choose the final time tf to be 1000 days. We investigate the following three cases.Case 1: the effectiveness of RTIsε1≠0 and the effectiveness of PIs ε2=0. In this case, we take the weight constant values r1=r2=54.5. In Figure 4, we obtain the optimal control diagram for ε1≠0 (solid lines) and ε2=0 (dashed lines). In Figure 5, we observe that the control strategy results in an increase in the level of uninfected cells and a decrease in the viral load.Case 2: the effectiveness of RTIsε1=0 and the effectiveness of PIs ε2≠0. We choose the weights r1=75.8 and r2=75.8. In Figure 6, we show the optimal control solution for ε1=0 (solid lines) and ε2≠0 (dashed lines). In Figure 7, we also see that the level of uninfected cells increases and viral load decreases despite oscillations.Case 3: the effectiveness of RTIsε1≠0 and the effectiveness of PIs ε2≠0.We choose different weight constants to investigate the optimal control in this case. In Figure8, we show the optimal control by choosing r1=r2=264 and fixing the other parameters as in Case 1. In Figure 9, we show the changes of uninfected cells and viral load under this optimal control. Uninfected cells increase and the viral load decreases under treatment. Using other combinations of the weight constants, we obtain the similar prediction. They are shown in Figures 10 and 11 for r1=470>r2=220 and in Figures 12 and 13 for r1=152<r2=496.Figure 4
Optimal control strategy withε1≠0 (solid lines) and ε2=0 (dashed lines).Figure 5
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1≠0 and ε2=0 (solid lines: without control; dashed lines: with control).
(a)(b)Figure 6
Optimal control strategy withε1=0 (solid lines) and ε2≠0 (dashed lines).Figure 7
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1=0 and ε2≠0 (solid lines: without control; dashed lines: with control).
(a)(b)Figure 8
Optimal control strategy withε1≠0 (solid lines) and ε2≠0 (dashed lines), where r1=r2=264.Figure 9
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1≠0 and ε2≠0. Solid lines are without control and dashed lines are with control, in which r1=r2=264.
(a)(b)Figure 10
Optimal control strategy withε1≠0 (solid lines) and ε2≠0 (dashed lines), where r1=470>r2=220.Figure 11
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1≠0 and ε2≠0. Solid lines are without control and dashed lines are with control, in which r1=470>r2=220.
(a)(b)Figure 12
Optimal control strategy withε1≠0 (solid lines) and ε2≠0 (dashed lines), where r1=152<r2=496.Figure 13
The level of (a) uninfected cells and (b) viral load corresponding to the optimal control strategy withε1≠0 and ε2≠0. Solid lines are without control and dashed lines are with control, in which r1=152<r2=496.
(a)(b)


## 7. Conclusion

HIV infection is still a serious public health problem in the world. More than 700000 people died of HIV-related causes in 2018 [56]. In addition to cell-free virus infection, HIV can also be transmitted directly from infected cells to uninfected cells. Although there is no cure for HIV infection at present, the antiretroviral drugs are effective and can control the viral level below the detection limit, largely preventing the spread of the disease [56]. HIV can remain at a low level despite long-term therapy. The latently infected CD4+ T cells are considered to be a major barrier to viral eradication. In this paper, we developed and established an age-structured model of HIV latent infection with both cell-to-cell infection and cell-to-free transmission. We showed that the solution of system (1) is positive and ultimately bounded. We derived the basic reproduction number R0, which is an important threshold value to decide the global dynamics of the model. According to Theorems 1 and 3, the infection-free equilibrium E0 is GAS when R0<1, which means that HIV infection is predicted to be eliminated. To obtain the global stability of the infected equilibrium E∗, we proved the uniform persistence of system (1), followed by the global threshold dynamics using the approach of Lyapunov functionals when R0>1. Numerical simulations were performed to illustrate the asymptotic behavior of the solution (Figure 2). We found that the balance between the activation rate and the net generation rate of latently infected CD4+ T cells plays a critical role in generating the slow decay of the latent reservoir (see Figure 3) and the persistence of low viral load during treatment.There are various classes of antiretroviral drugs targeting different stages of viral infection and replication. In order to achieve the optimal treatment strategies and the necessary conditions that minimize the cost of drug treatment and the viral level, we have formulated an optimal control problem using the age-structured model (33). We obtained the optimal control solutions ε1∗ and ε2∗ using the Pontryagin maximum principle [53]. In numerical simulations, we chose different weight constants and explored different treatment strategies (e.g., treatment with a single class of drugs or a combination of different classes of drugs). In any case, the optimal control strategy resulted in a decrease in the viral load and an increase in the level of uninfected cells.



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