Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 828509, 19 pagesdoi:10.1155/2011/828509

Research Article

An Efficient Therapy Strategy under a Novel HIV Model

Chunming Zhang,¹ Xiaofan Yang,¹ Wanping Liu,¹ and Lu-Xing Yang²

¹College of Computer Science, Chongqing University, Chongqing 400044, China²College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

Received 20 April 2011; Revised 19 August 2011; Accepted 19 August 2011

Academic Editor: Antonia Vecchio

Copyright © 2011 Chunming Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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: cells (uninfected T cells), cells (early-stage infected T cells), and cells (late-stage infected T cells). Let , and denote the numbers of cells, cells, and cells, respectively. Likewise, M cells are classified into three categories: cells (uninfected M cells), cells (early-stage infected M cells), and cells (late-stage infected M cells). Let , , and denote the numbers of cells, cells and cells, respectively. Besides, let denote the number of free viruses. Sedaghat et al. [13] made the following reasonable assumptions.(A₁) cells are produced with constant rate . cells are produced with constant rate . (A₂) cells become cells with constant rate . cells become cells with constant rate . (A₃) cells become cells with constant rate . cells become cells with constant rate . (A₄)These cells die with constant rates , , , , , and respectively. (A₅)Free viruses () are cleared at a rate , produced by cells with a burst size of , and produced by cells with a burst size of , respectively.

Figure 1: The HIV model.

Under these assumptions, Sedaghat et al. [13] deduced the following system of ordinary differential equations: 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.

2. Presentation of a New Model

For our purpose, let us introduce the following notations (see Figure 2): :the dosage of RTI at time , which is assumed to take values in the interval ;:the dosage of PI at time , which is assumed to take values in ;:the capability of preventing cells from becoming cells with per unit dosage of RTI;:the capability of preventing cells from becoming cells with per unit dosage of RTI;:the capability of preventing cells from producing viruses with per unit dosage of PI;:the capability of preventing cells from producing viruses with per unit dosage of PI.

Figure 2: The HIV model with therapy strategy.

Next, let us consider the following assumptions.(A₆)Due to the effect of RTIs, cells become cells with rate , and cells become cells with rate , where and are constants. (A₇)Due to the effect of PIs, Free viruses () are produced by and cells with a burst size of and , respectively, where and are constants.

Under assumptions (A₁)–(A₇), we can derive the following system of ordinary differential equations:

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 represent the benefit of per cell and per cell, respectively, and represent the cost of per unit RTI and per unit PI, respectively. Our goal is to obtain an optimal control pair such that where is the admissible control set defined by

3. Existence of an Optimal Control Pair

For our purpose, let us introduce the following four assumptions. (A₈)The set of control and corresponding state variables is nonempty.(A₉)The admissible control set is closed and convex.(A₁₀)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 with coefficients depending on time and state.(A₁₁)There exist positive constants and such that the integrand (denoted by ) of the objective functional (2.2) is concave and satisfies the condition .

In what follows, it is always assumed that assumptions (A₁)–(A₇) hold.

Theorem 3.1. Consider system (2.1) with initial conditions, and the objective functional (2.2). There exists such that

Proof. It suffices to verify the assumptions (A₈)–(A₁₁) 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 is obviously closed and convex, because both and 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 with coefficients depending on time and states. The fact that all state variables , , , , , , , and are bounded on , implies the rest of assumption (A₁₀).It is easy to see that is concave in . By setting , and , we can derive The proof is complete.

4. 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 and are given by

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

Here

where In addition, the in system (4.3) is

Next, adding the penalty term will give us the optimality condition where is an operator from to defined by where all are nonnegative penalty multipliers satisfying the following conditions:

According to the Pontryagin’s Maximum Principle, if the control and the corresponding state constitute an optimal pair, there exists an adjoint vector defined system (4.4) such that the function defined by (4.8) reaches its maximum on the set at the point . This gives the following result.

Theorem 4.1. Given an optimal control pair and a solution of the corresponding system, then there exist seven adjoint variables satisfying with the final conditions Furthermore, , where

Proof. According to the previous section, an optimal couple exists for maximizing the objective functional (2.2) subject to the system (2.1). Therefore, by Pontryagin’s Maximum Principle, there exists a vector satisfying That yields Through simple calculations, we derive system (4.11). The Pontryagin’s Maximum Principle gives the following necessary conditions to obtain the optimal pair : where . From (4.10) and (4.16), we have which implies On the other hand, which indicates Now from the constraint condition, the following three cases arise. Case 1. and . Then .Case 2. and . Then , which implies because .Case 3. and . Then , which leads to , owing to .Hence, we have . Similarly, we can get that .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 We substitute the expressions 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).

5. 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 at the points , where is the time step.

In the following, we define the state and adjoint variables , , , , , , , and the controls and in terms of nodal points , , , , , , , , , as the state and adjoint variables and the controls at initial time , while , , , , , , , , , as the state and adjoint variables and the controls at final time . 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 , by

We use the scheme developed by Gumel et al. [7] in the following way: Analogously, we have

By applying an analogous technology, we approximate the time derivative of the adjoint variables by their first-order backward-difference and we use the appropriated scheme as follows:

Hence, we can establish an algorithm to solve the optimality system and then to compute the optimal control pair by employing the GSS1 method (5.2)–(5.4) that we denote by IGSS1 method here (see Appendix B).

5.1. Numerical Results

By making some parameter value choices, computer simulation experiments are done to verify the effectiveness of our new model by comparing the disease progression before and after introducing the two optimal control variables , . 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.).

Table 1

Time (days)
BT
AT
BT
AT
BT
AT

0
1000
1000
1000
1000
1
1

2
980.2369
980.2012
926.3417
926.1494
15.33444
19.568136

4
955.4799
953.1210
856.9665
852.1262
283.2647
334.65888

6
839.4257
889.1900
754.0975
754.1006
4710.942
315.44809

8
270.1388
833.1982
372.5725
671.2911
38782.17
297.34008

10
11.92899
784.0453
41.26735
601.1371
99155.03
280.27154

20
0.511601
628.5318
1.021524
388.0657
255003.3
208.54835

30
0.379269
597.7372
0.756860
313.0676
335762.8
155.17956

40
0.334284
574.2037
0.666976
268.0361
376288.7
115.46816

50
0.322352
555.2309
0.643089
240.5969
387774.5
85.919158

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).

Figure 3: Uninfected T cells.

Figure 4: Uninfected M cells.

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).

Figure 5: Virus population before optimal controls.

Figure 6: Virus population after optimal controls.

Finally, the optimal controls , for drug administration are presented through Figures 7 and 8.

Figure 7: Optimal control variable .

Figure 8: Optimal control variable .

6. 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.

Appendices

A. The Theorem Used in Theorem 3.1

The equations where , with a nonempty open subset of and , are called a Cauchy problem or initial-value problem.

A solution to the Cauchy Problem is defined to be any pair in which is an open subinterval of containing is absolutely continuous, for all , and satisfies the above two equations at .

For with coordinates , define a norm on by

The following theorem applies the Lebesgue integral and the hypothesis is stated in terms of the rectangular subset of centered about (),

Theorem A.1 (see [19, p.182]). The Cauchy problem has a solution if for some centered about the restriction of to is continuous in for fixed , measurable in for fixed , and satisfies for some integrable over the interval .

B. An Algorithm Using the GSS1 Method

Algorithm B.1. Step 1. Step 2. for do Step 3. for , denote

It is easy to conclude that this algorithm takes execution time.

Acknowledgments

This work was financially supported by the Fundamental Research Funds for the Central Universities (no. CDJXS10181130), the New Century Excellent Talent Project of China (no. NCET-05-0759), and the National Natural Science Foundation of China (no. 10771227).

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