Volume 2011, Article ID 828509,19pages doi: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 Yang**

^{2}*1**College of Computer Science, Chongqing University, Chongqing 400044, China*

*2**College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China*

Correspondence should be addressed to Xiaofan Yang,xf yang1964@yahoo.com Received 20 April 2011; Revised 19 August 2011; Accepted 19 August 2011 Academic Editor: Antonia Vecchio

Copyrightq2011 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.

By incorporating the chemotherapy into a previous model describing the interaction of the im- mune system with the human immunodeficiency virusHIV, 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 diseasesespecially, the acquired immunodeficiency syndromeAIDS 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
virus13. Some of these models investigate how to avoid an excessive use of drugs because
it might be toxic to human body and, hence, cause damages1,4–6,8–11,14,15,17,18.

Recently, Sedaghat et al.13proposed a model, which describes the law governing
the transition of two populations of target cells, the T cellsthe abbreviation of the CD4^{}T
lymphoblasts and the M cellssay, macrophages, T cells in a lower state of activation, or
another cell type, in the eﬀect of free virusseeFigure 1. The T cells produce most of the
plasma virus and are responsible for the first-phase decay, while the M cells are responsible

*T**U* *T*1 *T*2

*k**T*

*k**M*

*N**M*

*N**T*

*β**T*

*β**M*

*M*2

*M*1

*M**U*

*V*
*δ**T**U* *δ**T*1 *δ**T*2

*δ**M**U* *δ**M*1 *δ**M*2

**Figure 1: The HIV model.**

for the second-phase decay. T cells are classified into three categories:*T**U*cellsuninfected T
cells,*T*_{1}cellsearly-stage infected T cells, and*T*_{2}cellslate-stage infected T cells. Let*T** _{U}*,

*T*

_{1}and

*T*2denote the numbers of

*T*

*U*cells,

*T*1cells, and

*T*2cells, respectively. Likewise, M cells are classified into three categories:

*M*

*u*cellsuninfected M cells,

*M*1 cellsearly-stage infected M cells, and

*M*

_{2}cellslate-stage infected M cells. Let

*M*

*,*

_{u}*M*

_{1}, and

*M*

_{2}denote the numbers of

*M*

*cells,*

_{U}*M*

_{1}cells and

*M*

_{2}cells, respectively. Besides, let

*V*denote the number of free viruses. Sedaghat et al.13made the following reasonable assumptions.

A1*T**U* cells are produced with constant rate*θ**T*.*M**U* cells are produced with constant
rate*θ**M*.

A2*T**U* cells become *T*1 cells with constant rate *β**T*. *M**U* cells become*M*1 cells with
constant rate*β**M*.

A3*T*1 cells become *T*2 cells with constant rate *k**T*. *M*1 cells become *M*2 cells with
constant rate*k**M*.

A4These cells die with constant rates*δ**T**U*,*δ**T*1,*δ**T*2,*δ**M**U*,*δ**M*1, and*δ**M*2respectively.

A5Free virusesVare cleared at a rate*c, produced byT*_{2}cells with a burst size of*N** _{T}*,
and produced by

*M*2cells with a burst size of

*N*

*M*, respectively.

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

*dT*_{U}

*dt* *θ** _{T}*−

*δ*

_{T}

_{U}*T*

*−*

_{U}*β*

_{T}*T*

_{U}*V,*

*dT*

_{1}

*dt* *β*_{T}*T*_{U}*V*−δ*T*1*k** _{T}*T1

*,*

*dT*

_{2}

*dt* *k*_{T}*T*_{1}−*δ*_{T}_{2}*T*_{2}*,*

*dM**U*

*dt* *θ**M*−*δ**M**U**M**U*−*β**M**M**U**V,*
*dM*1

*dt* *β**M**M**U**V*−δ*M*1*k**M*M1*,*
*dM*2

*dt* *k**M**M*1−*δ**M*2*M*2*,*
*dV*

*dt* *N*_{T}*T*_{2}*N*_{M}*M*_{2}−*cV.*

1.1 For a highly simplified version of this system, Sedaghat et al. 13 derived its analytic solution.

It is well known 5,6,8–11,13,15,17that there are mainly two categories of anti- HIV drugs: the reverse transcriptase inhibitorsRTIs, 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 notationsseeFigure 2:

*u*1t: the dosage of RTI at time*t, which is assumed to take values in the interval*0,1;

*u*2t: the dosage of PI at time*t, which is assumed to take values in*0,1;

*γ*1: the capability of preventing*T**U*cells from becoming*T*1cells with per unit dosage of
RTI;

*γ*_{2}: the capability of preventing*M** _{U}*cells from becoming

*M*

_{1}cells with per unit dosage of RTI;

*α*1: the capability of preventing*T*2cells from producing viruses with per unit dosage of
PI;

*α*_{2}: the capability of preventing*M*_{2}cells from producing viruses with per unit dosage
of PI.

Next, let us consider the following assumptions.

A6Due to the eﬀect of RTIs,*T**U*cells become*T*1cells with rate*β**T*1−*u*1tγ1, and*M**U*

cells become*M*1cells with rate*β**M*1−*u*1tγ2, where*γ*1and*γ*2are constants.

A7Due to the eﬀect of PIs, Free virusesVare produced by*T*_{2} and*M*_{2}cells with a
burst size of*α*11−*u*2tN*T*and*α*21−*u*2tN*M*, respectively, where*α*1and*α*2are
constants.

*T**U* *T*1 *T*2

*M*2

*M*1

*M**U*

*β**T*

*α*1

*β**M*

*α*2

*V*
*δ**T**U* *δ**T*1 *δ**T*2

*k*_{M}

*δ**M**U* *δ**M*1 *δ**M*2

*k**T*

(1−*u*1)*γ*1 (1−*u*2)*N**T*

(1−*u*1)*γ*2

(1−*u*2)N*M*

**Figure 2: The HIV model with therapy strategy.**

Under assumptionsA1–A7, we can derive the following system of ordinary diﬀerential equations:

*dT**U*

*dt* *θ**T*−*δ**T**U**T**U*−*β**T**V T**U*1−*u*1γ1*,*
*dT*1

*dt* *β**T**V T**U*1−*u*1γ1−δ*T*1*k**T*T1*,*
*dT*2

*dt* *k**T**T*1−*δ**T*2*T*2*,*
*dM**U*

*dt* *θ**M*−*δ**M**U**M**U*−*β**M**V M**U*1−*u*1γ2*,*
*dM*_{1}

*dt* *β**M**V M**U*1−*u*1γ2−δ*M*1*k**M*M1*,*
*dM*_{2}

*dt* *k*_{M}*M*_{1}−*δ*_{M}_{2}*M*_{2}*,*
*dV*

*dt* *α*_{1}*N*_{T}*T*_{2}1−*u*_{2} *α*_{2}*N*_{M}*M*_{2}1−*u*_{2}−*cV.*

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*

Ju1t, u2t
_{t}_{1}

*t*0

*B*_{1}*T*_{U}*B*_{2}*M** _{U}*−

*A*_{1}*u*^{2}_{1}*A*_{2}*u*^{2}_{2}

*dt,* 2.2

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 pairu^{∗}_{1}*, u*^{∗}_{2}such that

J
*u*^{∗}_{1}*, u*^{∗}_{2}

max{Ju1*, u*_{2}:u1*, u*_{2}∈ U}*,* 2.3

whereUis the admissible control set defined by

U*U*_{1}×*U*_{2}*,*

*U*_{1}*U*_{2}{ut:*u*measurable,0≤*ut*≤1, t∈t0*, t*_{1}}. 2.4

**3. Existence of an Optimal Control Pair**

For our purpose, let us introduce the following four assumptions.

A8The set of control and corresponding state variables is nonempty.

A9The admissible control setUis closed and convex.

A10All the right hand sides of equations of system2.1are 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.

A11There exist positive constants *c*1*, c*2 and *β* *>* 1 such that the integranddenoted
by*Ly, u, t*of the objective functional2.2is concave and satisfies the condition
*Ly, u, t*≤*c*_{1}−*c*_{2}u^{2}_{1}*u*^{2}_{2}* ^{β/2}*.

In what follows, it is always assumed that assumptionsA1–A7hold.

* Theorem 3.1. Consider system*2.1

*with initial conditions, and the objective functional*2.2. There

*existsu*

^{∗}u

^{∗}

_{1}

*, u*

^{∗}

_{2}

*such that*

J
*u*^{∗}_{1}*, u*^{∗}_{2}

max

*u∈U*Ju1*, u*2. 3.1

*Proof. It suﬃces to verify the assumptions* A8–A11 with respect to the seven ODEs of
system2.1.

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 resultseeAppendix Afrom 19we can obtain the existence to the solution of the system2.1.

The control setU*U*1×*U*2is obviously closed and convex, because both*U*1and*U*2

are closed and convex sets.

By definition, each right hand side of the ODEs of system2.1is 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*, andUare bounded ont0

*, t*

_{1}, implies the rest of assumptionA10.

It is easy to see that*Ly, u, t*is concave inU. By setting *c*1 max{B1*T**U**B*2*M**U*},
*c*_{2}infA1*, A*_{2}and*β*2, we can derive

*L*
*y, u, t*

*B*1*T**U**B*2*M**U*−

*A*1*u*^{2}_{1}*A*2*u*^{2}_{2}

≤*c*1−*c*2 *u*^{2}_{1}*u*^{2}_{2}
*.*

3.2

The proof is complete.

**4. Optimally Controlling Chemotherapy**

In this section, we discuss the theorem related to the characterization of the optimal con-
*trol. This result depends on the Pontryagin’s Maximum Principle, which gives necessary con-*
ditions for the optimal control. First, we rewrite the system 2.1 in the following vector
notation:

*dyt*

*dt* *A*

*y, u, t*

; ∀t > t0*,*∀u∈*U,*
*yt*0 *y*_{0}*,*

4.1

where*yt*and*Ay, u, t*are given by

*yt T**U*t,*T*1t,*T*2t,*M**U*t,*M*1t,*M*2t,*Vt*^{T}*,*
*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*
*H*

*y, u, p, t*
*L*

*y, u, t*

*λt*^{T}*A*
*y, u, t*

*,* 4.3

where the adjoint vector*λt*is defined by the adjoint equation
*dλt*

*dt* −A*y**λt*−*L**y**,*
*λt*1 0.

4.4

Here

*A*_{y}

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

*∂g*_{1}

*∂T**U*

*∂g*_{2}

*∂T**U*

*∂g*_{3}

*∂T**U*

*∂g*_{4}

*∂T**U*

*∂g*_{5}

*∂T**U*

*∂g*_{6}

*∂T**U*

*∂g*_{7}

*∂T**U*

*∂g*1

*∂T*1

*∂g*2

*∂T*1

*∂g*3

*∂T*1

*∂g*4

*∂T*1

*∂g*5

*∂T*1

*∂g*6

*∂T*1

*∂g*7

*∂T*1

*∂g*_{1}

*∂T*_{2}

*∂g*_{2}

*∂T*_{2}

*∂g*_{3}

*∂T*_{2}

*∂g*_{4}

*∂T*_{2}

*∂g*_{5}

*∂T*_{2}

*∂g*_{6}

*∂T*_{2}

*∂g*_{7}

*∂T*_{2}

*∂g*_{1}

*∂M*_{U}

*∂g*_{2}

*∂M*_{U}

*∂g*_{3}

*∂M*_{U}

*∂g*_{4}

*∂M*_{U}

*∂g*_{5}

*∂M*_{U}

*∂g*_{6}

*∂M*_{U}

*∂g*_{7}

*∂M*_{U}

*∂g*1

*∂M*1

*∂g*2

*∂M*1

*∂g*3

*∂M*1

*∂g*4

*∂M*1

*∂g*5

*∂M*1

*∂g*6

*∂M*1

*∂g*7

*∂M*1

*∂g*1

*∂M*_{2}

*∂g*2

*∂M*_{2}

*∂g*3

*∂M*_{2}

*∂g*4

*∂M*_{2}

*∂g*5

*∂M*_{2}

*∂g*6

*∂M*_{2}

*∂g*7

*∂M*_{2}

*∂g*_{1}

*∂V*

*∂g*_{2}

*∂V*

*∂g*_{3}

*∂V*

*∂g*_{4}

*∂V*

*∂g*_{5}

*∂V*

*∂g*_{6}

*∂V*

*∂g*_{7}

*∂V*

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

*E, F,* 4.5

where

*E*

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

−δ*T**U*−*β**T**V*1−*u*1γ1 *β**T**V*1−*u*1γ1 0

0 −δ*T*1*k*_{T}*k*_{T}

0 0 −δ*T*2

0 0 0

0 0 0

0 0 0

0 0 0

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠
*,*

*F*

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

0 0 0 0

0 0 0 0

0 0 0 *α*_{1}*N** _{T}*1−

*u*

_{2}

−δ*M**U*−*β**M**V*1−*u*1γ2 *β**M**V*1−*u*1γ2 0 0

0 −δ*M*1*k**M* *k**M* 0

0 0 −δ*M*2 *α*2*N**M*1−*u*2

0 0 0 −c

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠
*.*

4.6

In addition, the*L** _{y}*in system4.3is

*L*_{y}*∂L*

*∂T*_{U}*,* *∂L*

*∂T*_{1}*,* *∂L*

*∂T*_{2}*,* *∂L*

*∂M*_{U}*,* *∂L*

*∂M*_{1}*,* *∂L*

*∂M*_{2}*,∂L*

*∂V*
*T*

*,*
*B*1*,*0,0, B2*,*0,0,0^{T}*.*

4.7

Next, adding the penalty term will give us the optimality condition
*ξ*

*y, u, λ, t*
*H*

*y, u, λ, t*

Γutωt, 4.8

whereΓis an operator from^{Ê}^{2} to^{Ê}^{4} defined by

Γut 1−*u*1t, u1t,1−*u*2t, u2t,

*ωt *

⎛

⎜⎜

⎜⎜

⎜⎝
*ω*11t
*ω*12t
*ω*21t
*ω*_{22}t

⎞

⎟⎟

⎟⎟

⎟⎠*,* 4.9

where all*ω**ij**, i, j* 1,2 are nonnegative penalty multipliers satisfying the following condi-
tions:

1−*u*^{∗}_{1}t

*ω*_{11}t *u*^{∗}_{1}tω12t

1−*u*^{∗}_{2}t

*ω*_{21}t *u*^{∗}_{2}tω22t 0. 4.10
*According to the Pontryagin’s Maximum Principle, if the control* *u*^{∗}t and the corre-
sponding state *y*^{∗}tconstitute an optimal pair, there exists an adjoint vector *λt*defined
system4.4such that the function*ξy, u, λ, t*defined by4.8reaches its maximum on the
setUat the point*u*^{∗}. This gives the following result.

**Theorem 4.1. Given an optimal control pair**u^{∗}t u^{∗}_{1}t, u^{∗}_{2}t*and a solutiony*^{∗}t T_{U}^{∗}t,
*T*_{1}^{∗}t,*T*_{2}^{∗}t,*M*^{∗}* _{U}*t,

*M*

^{∗}

_{1}t,

*M*

^{∗}

_{2}t,

*V*

^{∗}t

*of the corresponding system, then there exist seven adjoint*

*variablesλ*

_{1}t, λ2t, . . . , λ7t

*satisfying*

*dλ*1

*dt*

*δ**T**U**β**T**V*1−*u*1γ1

*λ*1−*β**T**V*1−*u*1γ1*λ*2−*B*1*,*

*dλ*2

*dt* δ*T*1*k**T*λ2−*k**T**λ*3*,*
*dλ*3

*dt* *δ**T*2*λ*3−*α*1*N**T*1−*u*2λ7*,*
*dλ*_{4}

*dt*

*δ**M**U**β**M**V*1−*u*1γ2

*λ*4−*β**M**V*1−*u*1γ2*λ*5−*B*2*,*
*dλ*_{5}

*dt* *δ*_{M}_{1}*k** _{M}*λ5−

*k*

_{M}*λ*

_{6}

*,*

*dλ*

_{6}

*dt* *δ*_{M}_{2}*λ*_{6}−*α*_{2}*N** _{M}*1−

*u*

_{2}λ7

*,*

*dλ*7

*dt* *cλ*7*β**T**T**U*1−*u*1γ1*λ*1−*β**T**T**U*1−*u*1γ1*λ*2

*β*_{M}*M** _{U}*1−

*u*

_{1}γ2

*λ*

_{4}−

*β*

_{M}*M*

*1−*

_{U}*u*

_{1}γ2

*λ*

_{5}

*,*

4.11

*with the final conditions*

*λ*_{1}t1 *λ*_{2}t1 · · ·*λ*_{7}t1 0. 4.12
*Furthermore,u*^{∗}_{1}t min{max{0, R1t},1}, u^{∗}_{2}t min{max{0, R2t},1}, where

*R*1t *V*^{∗}
2A1

*β**T**T*_{U}^{∗}*γ*1λ1−*λ*2 *β**M**M*^{∗}_{U}*γ*2λ4−*λ5*
*,*
*R*_{2}t − *λ*_{7}

2A2α1*N*_{T}*T*_{2}*α*_{2}*N*_{M}*M*_{2}.

4.13

*Proof. According to the previous section, an optimal couple* y^{∗}t, u^{∗}t exists for maxi-
mizing the objective functional2.2subject to the system2.1. Therefore, by Pontryagin’s
*Maximum Principle, there exists a vectorλt λ*1t, . . . , λ7t* ^{T}*satisfying

*λt*

*dt* −*∂H*

*∂y* −L*y*−*A**y**λt.* 4.14

That yields

*λ*_{1}t

*dt* −

*∂g*_{1}

*y*^{∗}*, u*^{∗}*, t*

*∂T**U* *, . . . ,∂g*_{7}

*y*^{∗}*, u*^{∗}*, t*

*∂T**U*

*λt*−*∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂T**U* *,*
*λ*_{2}t

*dt* −

*∂g*_{1}

*y*^{∗}*, u*^{∗}*, t*

*∂T*1 *, . . . ,∂g*_{7}

*y*^{∗}*, u*^{∗}*, t*

*∂T*1

*λt*−*∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂T*1 *,*
*λ*3t

*dt* −

*∂g*_{1}

*y*^{∗}*, u*^{∗}*, t*

*∂T*2 *, . . . ,∂g*_{7}

*y*^{∗}*, u*^{∗}*, t*

*∂T*2

*λt*−*∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂T*2 *,*
*λ*4t

*dt* −

*∂g*_{1}

*y*^{∗}*, u*^{∗}*, t*

*∂M**U* *, . . . ,∂g*_{7}

*y*^{∗}*, u*^{∗}*, t*

*∂M**U*

*λt*−*∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂M**U* *,*
*λ*5t

*dt* −

*∂g*_{1}

*y*^{∗}*, u*^{∗}*, t*

*∂M*1 *, . . . ,∂g*_{7}

*y*^{∗}*, u*^{∗}*, t*

*∂M*1

*λt*−*∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂M*2 *,*
*λ*6t

*dt* −

*∂g*_{1}

*y*^{∗}*, u*^{∗}*, t*

*∂M*2 *, . . . ,∂g*_{7}

*y*^{∗}*, u*^{∗}*, t*

*∂M*2

*λt*−*∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂M*2 *,*
*λ*7t

*dt* −

*∂g*_{1}

*y*^{∗}*, u*^{∗}*, t*

*∂V* *, . . . ,∂g*_{7}

*y*^{∗}*, u*^{∗}*, t*

*∂V*

*λt*−*∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂V* *.*

4.15

Through simple calculations, we derive system4.11. The Pontryagin’s Maximum Prin-
*ciple gives the following necessary conditions to obtain the optimal pair*y^{∗}*, u*^{∗}:

*∂ξ*

*y*^{∗}*, u*^{∗}*, λ, t*

*∂u*_{1} 0, *∂ξ*

*y*^{∗}*, u*^{∗}*, λ, t*

*∂u*_{2} 0, 4.16

where*ξy*^{∗}*, u*^{∗}*, λ, t Hy*^{∗}*, u*^{∗}*, λ, t Γu*^{∗}tωt. From4.10and4.16, we have

*∂ξ*

*y*^{∗}*, u*^{∗}*, λ, t*

*∂u*1 0

⇒ *∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂u*1 *∂* *λt*^{T}*A*

*y*^{∗}*, u*^{∗}*, t*

*∂u*1 *∂Γu*^{∗}tωt

*∂u*1 0,

4.17

which implies

*u*^{∗}_{1}t *V*^{∗}
2A_{1}

*β**T**T*_{U}^{∗}*γ*1λ1−*λ*2 *β**M**M*^{∗}_{U}*γ*2λ4−*λ5*

*.* 4.18

On the other hand,

*∂ξ*

*y, u*^{∗}*, λ, t*

*∂u*_{2} 0

⇒ *∂L*

*y*^{∗}*, u*^{∗}*, t*

*∂u*2 *∂* *λt*^{T}*A*

*y, u*^{∗}*, t*

*∂u*2 *∂Γu*^{∗}tωt

*∂u*2 0,

4.19

which indicates

*u*^{∗}_{2}t − *λ*7

2A2α1*N**T**T*2*α*2*N**M**M*2. 4.20

Now from the constraint condition, the following three cases arise.

*Case 1.* *t*∈ {t: 0*< u*^{∗}_{1}t*<*1}and*ω*11t *ω*12t 0. Then*u*^{∗}_{1}t *R*1t.

*Case 2.* *t* ∈ {t : *u*^{∗}_{1}t 0} and*ω*11t 0. Then 0 *u*^{∗}_{1}t *R*1t *ω*12t, which implies
*u*^{∗}_{1}t≥*R*_{1}tbecause*ω*_{12}t≥0.

*Case 3.* *t*∈ {t:*u*^{∗}_{1}t 1}and*ω*12t 0. Then*u*^{∗}_{1}t *R*1t−*ω*11t, which leads to 1*u*^{∗}_{1}t≤
*R*1t, owing to*ω*11≥0.

Hence, we have *u*^{∗}_{1}t min{max{0, R1t},1}. Similarly, we can get that *u*^{∗}_{2}t
min{max{0, R2t},1}.

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

*dy*^{∗}t

*dt* *A*

*y*^{∗}*, u*^{∗}*, t*

; ∀t > t0*,*
*dλt*

*dt* −A*y*^{∗}*λt*−*L**y*^{∗}*,*
*y*^{∗}t0 *y*_{0}^{∗}*,*

*λt*1 0.

4.21

We substitute the expressions*u*^{∗} u^{∗}_{1}*, u*^{∗}_{2}in the above system. The uniqueness of the solu-
tion of the optimality system can be derived by a standard methodrefer to6for more de-
tails 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 by7and denoted by GSS1 method. It consists on dis-
cretizing the intervalt0*, t*1at the points*t**k**klt*0k0,1, . . . , n, where*l*is the time step.

In the following, we define the state and adjoint variables*T** _{U}*t,

*T*

_{1}t,

*T*

_{2}t,

*M*

*t,*

_{U}*M*1t,

*M*2t,

*Vt,λ*1t∼

*λ*7tand the controls

*u*1tand

*u*2tin terms of nodal points

*T*

_{U}*,*

^{k}*T*

_{1}

*,*

^{k}*T*

_{2}

*,*

^{k}*M*

_{U}*,*

^{k}*M*

^{k}_{1},

*M*

^{k}_{2},

*V*

*,*

^{k}*λ*

^{k}_{1}∼

*λ*

^{k}_{7},

*u*

^{k}_{1},

*u*

^{k}_{2}as the state and adjoint variables and the controls at initial time

*t*

_{0}, while

*T*

_{U}*,*

^{n}*T*

_{1}

*,*

^{n}*T*

_{2}

*,*

^{n}*M*

_{U}*,*

^{n}*M*

^{n}_{1},

*M*

^{n}_{2},

*V*

*,*

^{n}*λ*

^{n}_{1}∼

*λ*

^{n}_{7},

*u*

^{n}_{1},

*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

*dT**U*t

*dt* lim

*l*→0

*T**U*t*l*−*T**U*t

*l* *.* 5.1

We use the scheme developed by Gumel et al.7in the following way:

*T*_{U}* ^{k1}*−

*T*

_{U}

^{k}*l* *θ** _{T}*−

*δ*

_{T}

_{U}*T*

_{U}*−*

^{k1}*β*

_{T}*γ*

_{1}

*V*

*1−*

^{k}*u*

^{k}_{1}

*T*_{U}^{k1}*.* 5.2

Analogously, we have
*T*_{1}* ^{k1}*−

*T*

_{1}

^{k}*l* *β**T**γ*1*V** ^{k}* 1−

*u*

^{k}_{1}

*T*_{U}* ^{k1}*−δ

*T*1

*k*

*T*T

_{1}

^{k1}*,*

*T*

_{2}

*−*

^{k1}*T*

_{2}

^{k}*l* *k**T**T*_{1}* ^{k1}*−

*δ*

*T*2

*T*

_{2}

^{k1}*,*

*M*

^{k1}*−*

_{U}*M*

^{k}

_{U}*l* *θ** _{M}*−

*δ*

_{M}

_{U}*M*

^{k1}*−*

_{U}*β*

_{M}*γ*

_{2}

*V*

*1−*

^{k}*u*

^{k}_{1}

*M*

_{U}

^{k1}*,*

*M*^{k1}_{1} −*M*^{k}_{1}

*l* *β*_{M}*γ*_{2}*V** ^{k}* 1−

*u*

^{k}_{1}

*M*^{k1}* _{U}* −δ

*M*1

*k*

*M*

_{M}

^{k1}_{1}

*,*

*M*

^{k1}_{2}−

*M*

^{k}_{2}

*l* *k**M**M*^{k1}_{1} −*δ**M*2*M*^{k1}_{2} *,*
*V** ^{k1}*−

*V*

^{k}*l* *α*_{1}*N*_{T}*T*_{2}* ^{k1}* 1−

*u*

^{k}_{2}

*α*_{2}*N*_{M}*M*^{k1}_{2} 1−*u*^{k}_{2}

−*cV*^{k1}*.*

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

*λ*^{n−k}_{1} −*λ*^{n−k−1}_{1}

*l*

*δ*_{T}_{U}*β*_{T}*V** ^{k1}* 1−

*u*

^{k}_{1}

*γ*

_{1}

*λ*^{n−k−1}_{1} − *β*_{T}*V** ^{k1}* 1−

*u*

^{k}_{1}

*γ*_{1}*λ*^{n−k}_{2} −*B*_{1}*,*
*λ*^{n−k}_{2} −*λ*^{n−k−1}_{2}

*l* δ*T*1*k**T*λ^{n−k−1}_{2} −*k**T**λ*^{n−k}_{3} *,*
*λ*^{n−k}_{3} −*λ*^{n−k−1}_{3}

*l* *δ*_{T}_{2}*λ*^{n−k−1}_{3} −*α*_{1}*N** _{T}* 1−

*u*

^{k}_{2}

*λ*

^{n−k}_{7}

*,*

*λ*

^{n−k}_{4}−

*λ*

^{n−k−1}_{4}

*l*

*δ**M**U**β**M**V** ^{k1}* 1−

*u*

^{k}_{1}

*γ*2

*λ*^{n−k−1}_{4} − *β**M**V** ^{k1}* 1−

*u*

^{k}_{1}

*γ*2*λ*^{n−k}_{5} −*B*2*,*
*λ*^{n−k}_{5} −*λ*^{n−k−1}_{5}

*l* δ*M*1*k**M*λ^{n−k−1}_{5} −*k**M**λ*^{n−k}_{6} *,*
*λ*^{n−k}_{6} −*λ*^{n−k−1}_{6}

*l* *δ*_{M}_{2}*λ*^{n−k−1}_{6} −*α*_{2}*N** _{M}* 1−

*u*

^{k}_{2}

*λ*

^{n−k}_{7}

*,*

*λ*

^{n−k}_{7}−

*λ*

^{n−k−1}_{7}

*l* *cλ*^{n−k−1}_{7} *β**T**T*_{U}* ^{k1}* 1−

*u*

^{k}_{1}

*γ*1 *λ*^{n−k−1}_{1} −*λ*^{n−k−1}_{2}
*β**M**M*_{U}* ^{k1}* 1−

*u*

^{k}_{1}

*γ*2 *λ*^{n−k−1}_{4} −*λ*^{n−k−1}_{5}

*.*

5.4

Hence, we can establish an algorithm to solve the optimality system and then to com- pute the optimal control pair by employing the GSS1 method5.2–5.4that we denote by IGSS1 method hereseeAppendix 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

**Table 1**

Timedays *T**U*BT *T**U*AT *M**U*BT *M**U*AT *V* BT *V* 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

5 10 15 20 25 30 35 40 45 50

0 0 100 200 300 400 500 600 700 800 900 1000

Before treatment After treatment

**Figure 3: Uninfected T cells.**

after introducing the two optimal control variables*u*^{∗}_{1}t,*u*^{∗}_{2}t. For the following parameters
and initial values:

*θ**T* 10,*θ**M* 10,*δ**T**U* 0.02,*δ**T*1 0.5,*δ**T*2 1,*δ**M**U* 0.0495,*δ**M*1 0.0495,*δ**M*2 0.0495,
*β** _{T}* 0.00008,

*β*

*0.00008,*

_{M}*k*

*0.1,*

_{T}*k*

*0.1,*

_{M}*N*

*100,*

_{T}*N*

*100,*

_{M}*T*

_{U}^{0}1000,

*T*

_{1}

^{0}0,

*T*

_{2}

^{0}0,

*M*

^{0}

*1000,*

_{U}*M*

^{0}

_{1}0,

*M*

^{0}

_{2}0,

*V*

^{0}1,

*c*0.03.

The experimental results obtained are listed inTable 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 decreasesee Figures3and4.

The viruses do not stop to proliferate and so its abundance dramatically increasessee 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 virusesseeFigure 6.

Finally, the optimal controls*u*^{∗}_{1}t,*u*^{∗}_{2}tfor drug administration are presented through
Figures7and8.

5 10 15 20 25 30 35 40 45 50 00

100 200 300 400 500 600 700 800 900 1000

Before treatment After treatment

**Figure 4: Uninfected M cells.**

5 10 15 20 25 30 35 40 45 50

0 0 0.5

1 1.5 2 2.5 3 3.5 4

×10^{5}

**Figure 5: Virus population before optimal controls.**

**6. Conclusions**

By incorporating the chemotherapy into a previous model describing the interaction of the immune system with the human immunodeficiency virusHIV, 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.

5 10 15 20 25 30 35 40 45 50 0

0 50 100 150 200 250 300 350

**Figure 6: Virus population after optimal controls.**

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5 10 15 20 25 30 35 40 45 50

**Figure 7: Optimal control variable***u*^{∗}_{1}t.

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5 10 15 20 25 30 35 40 45 50

**Figure 8: Optimal control variable***u*^{∗}_{2}t.

**Appendices**

**A. The Theorem Used in** **Theorem 3.1**

The equations

*xt *˙ *ft, xt,*

*x|**tτ* *ξ,* A.1

whereτ, ξ ∈ *D, withD*a nonempty open subset ofR × R* ^{n}*and

*f*:

*D*→ R

*, are called a*

^{n}*Cauchy problem or initial-value problem.*

*A solution to the Cauchy Problem is defined to be any pair*I, φin which*I* is an open
subinterval ofRcontaining*τ, φ*:*I* → R* ^{n}*is absolutely continuous,t, φt∈

*D*for all

*t*∈

*I,*and

*φ*satisfies the above two equations at

*a.e.t*∈

*I.*

For*x*∈ R* ^{n}*with coordinates

*x*

*i*, define a norm onR

*by*

^{n}|x|max

1≤i≤n|x*i*|. A.2

The following theorem applies the Lebesgue integral and the hypothesis is stated in
terms of the rectangular subset ofR × R* ^{n}*centered aboutτ, ξ,

*R**a,b*{t, x:|t−*τ| ≤a,*|x−*ξ| ≤b},* *a >*0, b >0. A.3
**Theorem A.1**see19, p.182. The Cauchy problem has a solution if for some*R**a,b* ⊂ *Dcentered*
*about*τ, ξ*the restriction off* *toR**a,b**is continuous inxfor fixedt, measurable intfor fixedx, and*
*satisfies*

*f*t, x≤*mt,* t, x∈*R*_{a,b}*,* A.4

*for somemintegrable over the interval*τ−*a, τa.*

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

*Algorithm B.1.*

*Step 1.*

*T**U*t0←−*T*_{U}^{0}*,* *T*1t0←−*T*_{1}^{0}*,* *T*2t0←−*T*_{2}^{0}*,*

*M** _{U}*t0←−

*M*

^{0}

_{U}*,*

*M*

_{1}t0←−

*M*

^{0}

_{1}

*,*

*M*

_{2}t0←−

*M*

^{0}

_{2}

*,*

*V*t0←−

*V*

^{0}

*,*

*λ*1t

*n*←−0,

*λ*2t

*n*←−0,

*λ*3t

*n*←−0,

*λ*4t

*n*←−0,

*λ*_{5}t*n*←−0, *λ*_{6}t*n*←−0, *λ*_{7}t*n*←−0,
*u*_{1}t0←−0, *u*_{2}t0←−0.

B.1

*Step 2. fork*1, . . . , ndo

*T*_{U}* ^{k}* ←−

*lθ*

_{T}*T*

_{U}

^{k−1}1*lδ*_{T}_{U}*lβ*_{T}*V** ^{k−1}* 1−

*u*

^{k−1}_{1}

*γ*

_{1}

*,*

*T*_{1}* ^{k}*←−

*lβ*

*T*

*γ*1

*V*

*1−*

^{k−1}*u*

^{k−1}_{1}

*T*_{U}^{k}*T*_{1}* ^{k−1}*
1 δ

*T*1

*k*

*T*l

*,*

*T*

_{2}

*←−*

^{k}*lk*

*T*

*T*

_{1}

^{k}*T*

_{2}

^{k−1}1*lδ**T*2

*,*

*M*^{k}* _{U}*←−

*lθ*

_{M}*M*

_{U}

^{k−1}1*lδ**M**U**lβ**M**γ*2*V** ^{k−1}* 1−

*u*

^{k−1}_{1},

*M*^{k}_{1} ←−*lβ**M**γ*2*V** ^{k−1}* 1−

*u*

^{k−1}_{1}

*M*^{k}_{U}*M*^{k−1}_{1}
1 δ*M*1*k**M*l *,*
*M*^{k}_{2} ←−*lk**M**M*_{1}^{k}*M*^{k−1}_{2}

1*lδ**M*2

*,*

*V** ^{k}*←−

*lα*

_{1}

*N*

_{T}*T*

_{2}

*1−*

^{k}*u*

^{k−1}_{2}

*lα*_{2}*N*_{M}*M*_{2}* ^{k}* 1−

*u*

^{k−1}_{2}

*V*

^{k−1}1*cl* *,*

*λ*^{n−k}_{1} ←−*lB*1*lβ**T**V** ^{k}* 1−

*u*

^{k−1}_{1}

*γ*1*λ*^{n−k1}_{2} *λ*^{n−k1}_{1}
1*lδ*_{T}_{U}*lβ*_{T}*V** ^{k}* 1−

*u*

^{k−1}_{1}

*γ*_{1} *,*

*λ*^{n−k}_{2} ←−*λ*^{n−k1}_{2} *lk**T**λ*^{n−k1}_{3}
1 *δ**T*1*k**T*l *,*

*λ*^{n−k}_{3} ←−*λ*^{n−k1}_{3} *lα*_{1}*N** _{T}* 1−

*u*

^{k−1}_{2}

*λ*

^{n−k1}_{7}

1*lδ*_{T}_{2} *,*

*λ*^{n−k}_{4} ←−*lB*2*lβ**M**V** ^{k}* 1−

*u*

^{k−1}_{1}

*γ*2*λ*^{n−k1}_{5} *λ*^{n−k1}_{4}
1*lδ*_{M}_{U}*lβ*_{M}*V** ^{k}* 1−

*u*

^{k−1}_{1}

*γ*_{2}
*,*

*λ*^{n−k}_{5} ←−*λ*^{n−k1}_{5} *lk**M**λ*^{n−k1}_{6}
1 δ*M*1*k**M*l *,*

*λ*^{n−k}_{6} ←−*λ*^{n−k1}_{6} *lα*2*N**M* 1−*u*^{k−1}_{2}
*λ*^{n−k1}_{7}
1*lδ**M*2

*,*

*λ*^{n−k}_{7} ←−*λ*^{n−k1}_{7} *l* 1−*u*^{k−1}_{1}

*β*_{T}*T*_{U}^{k}*γ*_{1} *λ*^{n−k}_{2} −*λ*^{n−k}_{1}

*β*_{M}*M*^{k}_{U}*γ*_{2} *λ*^{n−k}_{5} −*λ*^{n−k}_{4}

1*lc* *,*