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

In addition to direct virus infection of target cells, HIV can also be transferred from infected to uninfected cells (cell-to-cell transmission). *ese 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. *e basic reproductive ratio R0 was derived. *e 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. *e 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.

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][32][33][34][35][36][37][38][39][40][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 infected CD4 + 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. ey 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 agestructured 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. e necessary condition for minimizing the viral level and the cost of drug treatment was obtained. Different optimal treatment strategies were evaluated by numerical investigations.

The Model
We introduce an HIV latent infection model with age structure and the two transmission routes. e model has four state variables: uninfected CD4 + target cells T(t), agestructured latently infected cells, L(t, a), infected cells that can produce virus, I(t), and virus V(t). e model can be expressed as follows: e model assumes that λ and d T are the recruitment rate and death rate of uninfected CD4 + T cells, respectively. e parameter β is the infection rate of uninfected CD4 + T cells by free virus and k is the rate of transmission by infected cells. e constant f ∈ (0, 1) is the fraction of new infection that becomes latency. e rate of proliferation ρ(a) and the rate of death δ L (a) of latently infected cells are assumed to be functions of the age of infection. e activation rate α(a) also depends on the age because latently infected cells need to wait until they encounter their relevant antigen. e total number of productive infected cells I(t) obtained by latently infected cell activation is given by ∞ 0 α(a)L(t, a)da. δ represents the death rate of infected cells. N denotes the total number of virions released by one infected cell during its life cycle. e viral clearance rate is c. A schematic diagram of the full system is given in Figure 1.
To investigate the dynamics of (1), we need the following assumptions.
e size of the latent reservoir remains relatively stable or declines extremely slowly during suppressive therapy [45]. us, 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]. (2) e state space of (1), 2 Complexity is the nonnegative cone of the Banach space From Iannelli [46] and Magal [47], if the initial condition (T 0 , L 0 , I 0 , V 0 ) ∈ X + satisfies (A 1 ), then system (1) has a unique continuous solution in X + . erefore, we can define a solution semiflow Φ: Let It follows from (1) that where is a positive invariant and attracting subset for system (1).

Analysis of Model (1)
3.1. e Existence of Equilibria and Basic Reproduction Number. System (1) always has an infection-free equilib- where m(s) � α(s) + δ L (s) − ρ(s). Define where R 0 is the basic reproductive ratio of model (1) and R 01 is the portion from cell-free virus infection while R 02 is from the other transmission route (see details in [40]). When R 0 is greater than 1, the model has a positive equilibrium, given by E * � (T * , L * , I * , V * ) ∈ Ω, which satisfies Figure 1: Schematic diagram of model (1).
Straightforward calculation yields the infected equilibrium: 3.2. Local Stability. Let E � (T, L, I, V) be any equilibrium of (1). After linearizing (1) at E, we get the characteristic equation as follows: where τ denotes the eigenvalue of the characteristic equation (14) and ξ(τ) � ∞ 0 α(a)σ(a)e − τa da. 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.
e infection-free equilibrium E 0 of system (1) is LAS when R 0 < 1 and it is unstable when R 0 > 1.

Proof.
e characteristic equation of equilibrium E 0 of system (1) can be simplified as where erefore, the stability of E 0 is decided by the roots of the equation h 1 (τ) � 0. When R 0 < 1, we want to show that the real parts of all the roots of h 1 (τ) � 0 are negative. It is proved by contradiction. Let τ 0 be a root of h 1 (τ) � 0 with nonnegative real part. It follows from equation (16) that which leads to a contradiction since R 0 < 1. us, we know that each root of equation (15) is negative or has negative real part. erefore, infection-free equilibrium E 0 is LAS when R 0 < 1.
Next, we show that when R 0 > 1, the positive equilibrium E * is LAS.
Proof. Using the equilibrium equality d T R 0 � d T + βV * + kI * and equation (14), we have the characteristic equation at E * : 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 τ 0 with Re(τ 0 ) ≥ 0. Equation (18) becomes the following by dividing Taking the modulus of the two sides of equation (19), we have which is a contradiction. erefore, the equilibrium E * is LAS when R 0 > 1.

Global Dynamics of (1)
In the present section, by using suitable Lyapunov functionals, we establish the global asymptotic dynamics of the equilibria of (1). e 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 E 0 by applying the method of Lyapunov functionals when R 0 < 1 (see proof in Appendix A).
To prove the equilibrium E * of (1) is GAS, we need some preparation.
It can only contain points that pass through their total trajectories because the global attractor A is invariant. A total trajectory of Φ is a function ω: R ⟶ X + ensuring that Φ(s, ω(t)) � ω(s + t) for all t ∈ R and s ∈ R + . For a total trajectory, L(t, a) � L(t − a, 0)σ(a) for all a ∈ R + and t ∈ R. e alpha limit of a total trajectory ω(t) passing through Proof. Since Ω is attracting and invariant, there is t 1 ∈ R + ensuring that For t ≥ t 1 , from the first equation of system (1), we get It follows from eorem 4 that there is ε > 0 ensuring that is, combined with system (1), gives erefore, we have and thus I(t) ≥ ε I for t ∈ R by invariance again. It follows from the last equation of (1) that us, Letting ε 0 � min ε T , ε L , ε I , ε V , we complete the proof. e following theorem establishes the GAS of E * using the methods of Lyapunov functionals when R 0 > 1. e proof is given in Appendix B. □ Theorem 5.
e infected equilibrium E * of system (1) is GAS when R 0 > 1.

The Optimal Control Problem
To obtain the optimal treatment strategies, we study the following model with controls One control term ε 1 (t) is the effectiveness of RTIs, which block new infection. e other control term ε 2 (t) denotes the effectiveness of PIs, which reduce the number of infectious virions. 6 Complexity We minimize the objective functional where t f is the duration of the treatment and r 1 , r 2 , q are the positive weight coefficients to balance the control functions and the quantity of virus particles. e 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. e two control functions ε 1 (t) and ε 2 (t) represent the efficacy of RTIs and Problem 1. We seek optimal controls ε * 1 and ε * 2 such that subject to system (33) with the boundary condition and the initial values We noticed that the existence of optimal control functions to Problem 1 can be obtained by the methods used in [51,52].
By the discretization-differential approach [52], we divide L(t, a) into a number of classes from age 0 to a max , changing the age-structure model (33) to ODEs. We differentiate the ODEs to get an adjoint (costate) system. Let and a n � a max . e discrete system is Let L j � L(t, a j ) and X � (T, L 1 , L 2 , . . . , L n , I, V) T . e constraint equations are given by

Complexity
Proof. According to the necessary conditions of the fixed points of L, the adjoint equations can be obtained by taking _ Y � zL/zX with the terminal conditions Y(t f ) � (0, 0, 0, . . . , 0, 0, 0) T . e optimal controls ε * 1 and ε * 2 can be solved from the necessary optimal conditions zL/zε 1 � 0 and zL/zε 2 � 0, respectively. at is, erefore, one can solve the optimal controls as follows: Next, in order to derive a detailed expression for the optimal controls ε * 1 and ε * 2 without ξ i , η i , we will study all possible cases for the optimal controls.

Case
I. If we consider the set t|ε 1, min < ε * 1 (t) < ε 1, max , ξ 1 (t) � η 1 (t) � 0, then we obtain  Figure 2: Predicted dynamics of uninfected CD4 + 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 R 0 � 2.1108 > 1, which ensures that the infected equilibrium E * is GAS.

Dynamics of the Latently Infected CD4 + T Cells and Viral
Load. We choose parameters according to a previous study [54]: generates an infected equilibrium, which will be used as the initial value for model (1) under treatment.
We suppose that the activation rate α(a) of latently infected CD4 + T cells satisfies the exponential decay function. e function is chosen as follows: where a min and a * denote the minimum activation rate and initial activation rate of L(t, a), respectively. e parameter u represents the decay parameter determining how fast α decreases to its minimum value. Because these parameter values remain unknown, we choose a * � 0.05 day − 1 , u � 0.01 day − 1 , and a min � 0.01 day − 1 according to ref. [44] as default values in the following simulation. Using these parameter values, we find R 0 � 2.1108 > 1 and system (33)  addition, we choose n � 100 and a max � 10 such that 1, 2, . . . , n). us, in the numerical simulation of the latent reservoir, age a is divided into 100 equal intervals. e dynamical behaviors of latently infected CD4 + T cells with different ages of infection are shown in Figure 2(b). e 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 − ε 1 )(1 − ε 2 ) [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 ρ(a) − δ L (a) � 0.1a min .
Both the latently infected CD4 + 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 ρ(a) − δ L (a) � a min . Both the latently infected CD4 + T cells and virus persist at a low level even under prolonged drug therapy.

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. e effectiveness of RTIs control ε 1 and the effectiveness of PIs ε 2 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 forwardbackward sweep method [55] with Matlab. We choose the final time t f to be 1000 days. We investigate the following three cases.
Case 1: the effectiveness of RTIs ε 1 ≠ 0 and the effectiveness of PIs ε 2 � 0. In this case, we take the weight constant values r 1 � r 2 � 54.5. In Figure 4, we obtain the optimal control diagram for ε 1 ≠ 0 (solid lines) and ε 2 � 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 RTIs ε 1 � 0 and the effectiveness of PIs ε 2 ≠ 0. We choose the weights r 1 � 75.8 and r 2 � 75.8. In Figure 6, we show the optimal control solution for ε 1 � 0 (solid lines) and ε 2 ≠ 0 (dashed  lines). In Figure 7, we also see that the level of uninfected cells increases and viral load decreases despite oscillations.
We choose different weight constants to investigate the optimal control in this case. In Figure 8, we show the optimal control by choosing r 1 � r 2 � 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. ey are shown in Figures 10 and 11 for r 1 � 470 > r 2 � 220 and in Figures 12  and 13 for r 1 � 152 < r 2 � 496.

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. e 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 R 0 , which is an important threshold value to decide the global dynamics of the model. According to eorems 1 and 3, the infection-free equilibrium E 0 is GAS when R 0 < 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 R 0 > 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. ere 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 ε * 1 and ε * 2 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.

A. Proof of Theorem 3
In this section, we prove eorem 3. We define a Lyapunov functional Calculating the derivative of Q(t) along the solution of system (1), we have Since λ � d T T 0 and L(t, 0) � fβTV + fkTI, we have

Complexity
Note that erefore, it follows from (A.4) that us, dQ(t)/dt < 0 when R 0 is less than 1. Furthermore, dQ(t)/dt � 0 if and only if T(t) is exactly T 0 and V(t) is 0. It is easy to see that E 0 is the largest invariant set in erefore, E 0 is globally attractive. is, combined with eorem 1, implies that E 0 is globally asymptotically stable.

B. Proof of Theorem 5
From eorem 2, it suffices to show that A � E * { }. To construct a Lyapunov functional, we define g(x) � x − 1 − ln x, x > 0. Note that g(x) ≥ 0 for all x > 0 and g(x) � 0 if and only if x � 1.