An Efficient Therapy Strategy under a Novel HIV Model

By incorporating the chemotherapy into a previous model describing the interaction of the immune system with the human immunodeficiency virus HIV , this paper proposes a novel HIV virus spread model with control variables. Our goal is to maximize the number of healthy cells and, meanwhile, to minimize the cost of chemotherapy. In this context, the existence of an optimal control is proved. Experimental results show that, under this model, the spread of HIV virus can be controlled effectively.


Introduction
Numerous studies have been devoted to the description and understanding of the spread of infectious diseases especially, the acquired immunodeficiency syndrome AIDS 1-18 .Mathematical modeling of the human immunodeficiency virus HIV viral dynamics has offered many insights into the pathogenesis and treatment of HIV 1, 2, 4-10, 12-16, 18 .Consequently, many mathematical models have been developed to depict the relationships among HIV, etiological agent for AIDS and CD4 T lymphoblasts, which are the targets for the virus 13 .Some of these models investigate how to avoid an excessive use of drugs because it might be toxic to human body and, hence, cause damages 1, 4-6, 8-11, 14, 15, 17, 18 .Recently, Sedaghat et al. 13 proposed a model, which describes the law governing the transition of two populations of target cells, the T cells the abbreviation of the CD4 T lymphoblasts and the M cells say, macrophages, T cells in a lower state of activation, or another cell type , in the effect of free virus see Figure 1 .The T cells produce most of the plasma virus and are responsible for the first-phase decay, while the M cells are responsible for the second-phase decay.T cells are classified into three categories: T U cells uninfected T cells , T 1 cells early-stage infected T cells , and T 2 cells late-stage infected T cells .Let T U , T 1 and T 2 denote the numbers of T U cells, T 1 cells, and T 2 cells, respectively.Likewise, M cells are classified into three categories: M u cells uninfected M cells , M 1 cells early-stage infected M cells , and M 2 cells late-stage infected M cells .Let M u , M 1 , and M 2 denote the numbers of M U cells, M 1 cells and M 2 cells, respectively.Besides, let V denote the number of free viruses. 1.1 For a highly simplified version of this system, Sedaghat et al. 13 derived its analytic solution.
It is well known 5, 6, 8-11, 13, 15, 17 that there are mainly two categories of anti-HIV drugs: the reverse transcriptase inhibitors RTIs , which prevent new HIV infection by disrupting the conversion of viral RNA into DNA inside of T cells, and the protease inhibitors PIs , which reduce the number of virus particles produced by actively-infected T cells.
In consideration of this, this paper introduces a novel HIV model by incorporating the drug dosage into the above-mentioned model.Our goal is to maximize the number of healthy cells and, meanwhile, to minimize the cost of chemotherapy.In this context, the existence of an optimal control strategy is proved.Experimental results show that, under this model, the spread of HIV virus can be controlled effectively.

Presentation of a New Model
For our purpose, let us introduce the following notations see Figure 2 : u 1 t : the dosage of RTI at time t, which is assumed to take values in the interval 0, 1 ; u 2 t : the dosage of PI at time t, which is assumed to take values in 0, 1 ; The HIV model with therapy strategy.
Under assumptions A 1 -A 7 , we can derive the following system of ordinary differential equations:

2.1
Our target is to maximize the objective functional by increasing the number of healthy T and M cells and minimizing the cost based on the percentage effect of the chemotherapy given.For that purpose, we introduce the following objective functional where B 1 , B 2 represent the benefit of per T U cell and per M U cell, respectively, and A 1 , A 2 represent the cost of per unit RTI and per unit PI, respectively.Our goal is to obtain an optimal control pair u * 1 , u * 2 such that where U is the admissible control set defined by 2.4

Existence of an Optimal Control Pair
For our purpose, let us introduce the following four assumptions.
A 8 The set of control and corresponding state variables is nonempty.
A 9 The admissible control set U is closed and convex.
A 10 All the right hand sides of equations of system 2.1 are continuous, bounded above by a sum of bounded control and state, and can be written as a linear function of u with coefficients depending on time and state.
A 11 There exist positive constants c 1 , c 2 and β > 1 such that the integrand denoted by L y, u, t of the objective functional 2.2 is concave and satisfies the condition In what follows, it is always assumed that assumptions A 1 -A 7 hold.
Theorem 3.1.Consider system 2.1 with initial conditions, and the objective functional 2.2 .There exists Proof.It suffices to verify the assumptions A 8 -A 11 with respect to the seven ODEs of system 2.1 .
Since the coefficients involved in the system are bounded, and each state variable of the system is bounded on the finite time interval, it follows by a result see Appendix A from 19 we can obtain the existence to the solution of the system 2.1 .
The control set U U 1 × U 2 is obviously closed and convex, because both U 1 and U 2 are closed and convex sets.
By definition, each right hand side of the ODEs of system 2.1 is continuous and can be written as a linear function of u with coefficients depending on time and states.The fact that all state variables T U , T 1 , T 2 , M U , M 1 , M 2 , V , and U are bounded on t 0 , t 1 , implies the rest of assumption A 10 .
It is easy to see that L y, u, t is concave in U.By setting

3.2
The proof is complete.

Optimally Controlling Chemotherapy
In this section, we discuss the theorem related to the characterization of the optimal control.This result depends on the Pontryagin's Maximum Principle, which gives necessary conditions for the optimal control.First, we rewrite the system 2.1 in the following vector notation: where y t and A y, u, t are given by A y, u, t g 1 y, u, t , g 2 y, u, t , . . ., g 6 y, u, t , g 7 y, u, t T .

4.2
The Hamiltonian associated with our problem is where the adjoint vector λ t is defined by the adjoint equation

4.4
Here where In addition, the L y in system 4.3 is

4.7
Next, adding the penalty term will give us the optimality condition ξ y, u, λ, t H y, u, λ, t Γ u t ω t , 4.8 where Γ is an operator from Ê 2 to Ê 4 defined by where all ω ij , i, j 1, 2 are nonnegative penalty multipliers satisfying the following conditions: According to the Pontryagin's Maximum Principle, if the control u * t and the corresponding state y * t constitute an optimal pair, there exists an adjoint vector λ t defined system 4.4 such that the function ξ y, u, λ, t defined by 4.8 reaches its maximum on the set U at the point u * .This gives the following result.

4.11
with the final conditions

4.13
Proof.According to the previous section, an optimal couple y * t , u * t exists for maximizing the objective functional 2.2 subject to the system 2. The proof is complete.Now, the optimality system is given by incorporating the optimal control pair in the state system coupled with the adjoint system.Thus, we have

4.21
We substitute the expressions u * u * 1 , u * 2 in the above system.The uniqueness of the solution of the optimality system can be derived by a standard method refer to 6 for more details on the proof .

Numerical Algorithm and Results
The resolution of the optimal system is created improving the Gauss-Seidel-like implicit finite-difference method developed by 7 and denoted by GSS1 method.It consists on discretizing the interval t 0 , t 1 at the points t k kl t 0 k 0, 1, . . ., n , where l is the time step.
In the following, we define the state and adjoint variables 2 as the state and adjoint variables and the controls at initial time t 0 , while u n 2 as the state and adjoint variables and the controls at final time t 1 .As it is well known that the approximation of the time derivative by its first-order forward-difference is given, for the first state variable T U , by We use the scheme developed by Gumel et al. 7 in the following way: Analogously, we have  after introducing the two optimal control variables u * 1 t , u * 2 t .For the following parameters and initial values: The experimental results obtained are listed in Table 1 in which "before treatment" and "after treatment" are denoted by BT and AT, resp. .
For more clearness, it is better to present these comparative results by the following graphs.When the viruses attack the human body, uninfected T and M cells decrease see Figures 3 and 4 .
The viruses do not stop to proliferate and so its abundance dramatically increases see Figure 5 .However, after introducing the optimal controls, the situation changes.A few days later, the effect of chemotherapy starts to appear; which explains the growth of uninfected T and M cells and the diminishing of viruses see Figure 6 .
Finally, the optimal controls u * 1 t , u * 2 t for drug administration are presented through Figures 7 and 8

Conclusions
By incorporating the chemotherapy into a previous model describing the interaction of the immune system with the human immunodeficiency virus HIV , this paper has proposed a novel HIV virus spread model with control variables.Our goal is to maximize the number of healthy cells and, meanwhile, to minimize the cost of chemotherapy.In this context, the existence of an optimal control has been proved.Experimental results show that, under this model, the spread of HIV virus can be controlled effectively.
Our next work is to study other kinds of models, especially those with impulsive drug effect.The following theorem applies the Lebesgue integral and the hypothesis is stated in terms of the rectangular subset of R × R n centered about τ, ξ , for some m integrable over the interval τ − a, τ a .

B. An Algorithm Using the GSS1 Method
Algorithm B.1.

B.2
Step 3. for k 1, . . ., n, denote It is easy to conclude that this algorithm takes O n execution time.

3 Theorem A. 1
R a,b { t, x : |t − τ| ≤ a, |x − ξ| ≤ b}, a > 0, b > 0. A.see 19, p.182 .The Cauchy problem has a solution if for some R a,b ⊂ D centered about τ, ξ the restriction of f to R a,b is continuous in x for fixed t, measurable in t for fixed x, and satisfies f t, x ≤ m t , t, x ∈ R a,b , A.4 4These cells die with constant rates δ T U , δ T 1 , δ T 2 , δ M U , δ M 1 , and δ M 2 respectively.
Sedaghat et al. 13 made the following reasonable assumptions.A 1 T U cells are produced with constant rate θ T .M U cells are produced with constant rate θ M .A 2 T U cells become T 1 cells with constant rate β T .M U cells become M 1 cells with constant rate β M .A 3 T 1 cells become T 2 cells with constant rate k T .M 1 cells become M 2 cells with constant rate k M .A A 5 Free viruses V are cleared at a rate c, produced by T 2 cells with a burst size of N T , and produced by M 2 cells with a burst size of N M , respectively.Under these assumptions, Sedaghat et al. 13 deduced the following system of ordinary differential equations:

the capability of preventing T U cells from becoming T 1 cells with per unit dosage
Due to the effect of RTIs, T U cells become T 1 cells with rate β T 1 − u 1 t γ 1 , and M U cells become M 1 cells with rate β M 1 − u 1 t γ 2 , where γ 1 and γ 2 are constants.A 7 Due to the effect of PIs, Free viruses V are produced by T 2 and M 2 cells with a burst size of α 1 1 − u 2 t N T and α 2 1 − u 2 t N M , respectively, where α 1 and α 2 are constants.