Discrete Dynamics in Nature and Society

Volume 2011, Article ID 828509, 19 pages

http://dx.doi.org/10.1155/2011/828509

## An Efficient Therapy Strategy under a Novel HIV Model

^{1}College of Computer Science, Chongqing University, Chongqing 400044, China^{2}College 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.(A_{1}) cells are produced with constant rate . cells are produced with constant rate . (A_{2}) cells become cells with constant rate . cells become cells with constant rate . (A_{3}) cells become cells with constant rate . cells become cells with constant rate . (A_{4})These cells die with constant rates , , , , , and respectively. (A_{5})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.

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.

Next, let us consider the following assumptions.(A_{6})Due to the effect of RTIs, cells become cells with rate , and cells become cells with rate , where and are constants. (A_{7})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 (A_{1})–(A_{7}), 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. (A_{8})The set of control and corresponding state variables is nonempty.(A_{9})The admissible control set 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 with coefficients depending on time and state.(A_{11})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 (A_{1})–(A_{7}) 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 (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 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 (A_{10}).

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.).

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 , for drug administration are presented through Figures 7 and 8.

#### 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).

#### References

- B. M. Adams, H. T. Banks, M. Davidian et al., “HIV dynamics: modeling, data analysis, and optimal treatment protocols,”
*Journal of Computational and Applied Mathematics*, vol. 184, no. 1, pp. 10–49, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. M. Anderson, G. F. Medley, R. M. May, and A. M. Johnson, “A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS,”
*IMA Journal of Mathematics Applied in Medicine and Biology*, vol. 3, no. 4, pp. 229–263, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. Becker, “The uses of epidemic models,”
*Biometrics*, vol. 35, no. 1, pp. 295–305, 1979. View at Google Scholar - H. Chang and A. Astolfi, “Control of HIV infection dynamics: approximating high-order dynamics by adapting reduced-order model parameters,”
*IEEE Control Systems Magazine*, vol. 28, no. 2, pp. 28–39, 2008. View at Publisher · View at Google Scholar - R. V. Culshaw, S. G. Ruan, and R. J. Spiteri, “Optimal HIV treatment by maximising immune response,”
*Journal of Mathematical Biology*, vol. 48, no. 5, pp. 545–562, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. R. Fister, S. Lenhart, and J. S. McNally, “Optimizing chemotherapy in an HIV model,”
*Electronic Journal of Differential Equations*, vol. 1998, no. 32, pp. 1–12, 1998. View at Google Scholar · View at Zentralblatt MATH - A. B. Gumel, P. N. Shivakumar, and B. M. Sahai, “A mathematical model for the dynamics of HIV-1 during the typical course of infection,” in
*Proceedings of the 3rd World Congress of Nonlinear Analysts*, vol. 47, pp. 2073–2083, 2001. - H.-D. Kwon, “Optimal treatment strategies derived from a HIV model with drug-resistant mutants,”
*Applied Mathematics and Computation*, vol. 188, no. 2, pp. 1193–1204, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Kirschner, S. Lenhart, and S. Serbin, “Optimal control of the chemotherapy of HIV,”
*Journal of Mathematical Biology*, vol. 35, no. 7, pp. 775–792, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Karrakchou, M. Rachik, and S. Gourari, “Optimal control and infectiology: application to an HIV/AIDS model,”
*Applied Mathematics and Computation*, vol. 177, no. 2, pp. 807–818, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - U. Ledzewicz and H. Schättler, “Optimal bang-bang controls for a two-compartment model in cancer chemotherapy,”
*Journal of Optimization Theory and Applications*, vol. 114, no. 3, pp. 609–637, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. M. Rouzine, R. A. Sergeev, and A. I. Glushtsov, “Two types of cytotoxic lymphocyte regulation explain kinetics of immune response to human immunodeficiency virus,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 103, no. 3, pp. 666–671, 2006. View at Publisher · View at Google Scholar - A. R. Sedaghat, J. B. Dinoso, L. Shen, C. O. Wilke, and R. F. Siliciano, “Decay dynamics of HIV-1 depend on the inhibited stages of the viral life cycle,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 105, no. 12, pp. 4832–4837, 2008. View at Publisher · View at Google Scholar - H. Shim, S.-J. Han, C. C. Chung, S. W. Nam, and J. H. Seo, “Optimal scheduling of drug treatment for HIV infection: continuous dose control and receding horizon control,”
*International Journal of Control, Automation and Systems*, vol. 1, no. 3, pp. 282–288, 2003. View at Google Scholar - V. V. Velichenko and D. A. Pritykin, “Numerical methods of optimal control of the HIV-infection dynamics,”
*Journal of Computer and Systems Sciences International*, vol. 45, no. 6, pp. 894–905, 2006. View at Publisher · View at Google Scholar - X. Xia and C. H. Moog, “Identifiability of nonlinear systems with application to HIV/AIDS models,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 2, pp. 330–336, 2003. View at Publisher · View at Google Scholar - X. F. Yan and Y. Zou, “Optimal and sub-optimal quarantine and isolation control in SARS epidemics,”
*Mathematical and Computer Modelling*, vol. 47, no. 1-2, pp. 235–245, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Zurakowski and A. R. Teel, “A model predictive control based scheduling method for HIV therapy,”
*Journal of Theoretical Biology*, vol. 238, no. 2, pp. 368–382, 2006. View at Publisher · View at Google Scholar - D. L. Lukes,
*Differential Equations: Classical to Controlled, Mathematics in Science and Engineering*, vol. 162 of*Mathematics in Science and Engineering*, Academic Press, New York, NY, USA, 1982.