An Efficient Therapy Strategy under a Novel HIV Model

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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

1College of Computer Science, Chongqing University, Chongqing 400044, China

2College 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 immune 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 CD4T 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 CD4T 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

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TU T1 T2

kT

kM

NM

NT

βT

βM

M2

M1

MU

V δTU δT1 δT2

δMU δM1 δM2

Figure 1: The HIV model.

for the second-phase decay. T cells are classified into three categories:TUcellsuninfected T cells,T₁cellsearly-stage infected T cells, andT₂cellslate-stage infected T cells. LetT_U,T₁ andT2denote the numbers ofTUcells,T1cells, andT2cells, respectively. Likewise, M cells are classified into three categories:Mu cellsuninfected M cells,M1 cellsearly-stage infected M cells, andM₂cellslate-stage infected M cells. LetM_u,M₁, andM₂denote the numbers ofM_U cells,M₁ cells andM₂ cells, respectively. Besides, let V denote the number of free viruses. Sedaghat et al.13made the following reasonable assumptions.

A1TU cells are produced with constant rateθT.MU cells are produced with constant rateθM.

A2TU cells become T1 cells with constant rate βT. MU cells becomeM1 cells with constant rateβM.

A3T1 cells become T2 cells with constant rate kT. M1 cells become M2 cells with constant ratekM.

A4These cells die with constant ratesδTU,δT1,δT2,δMU,δM1, andδM2respectively.

A5Free virusesVare cleared at a ratec, produced byT₂cells with a burst size ofN_T, and produced byM2cells with a burst size ofNM, respectively.

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

dT_U

dt θ_T−δ_T_UT_U−β_TT_UV, dT₁

dt β_TT_UV−δT1k_TT1, dT₂

dt k_TT₁−δ_T₂T₂,

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dMU

dt θM−δMUMU−βMMUV, dM1

dt βMMUV−δM1kMM1, dM2

dt kMM1−δM2M2, dV

dt N_TT₂N_MM₂−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:

u1t: the dosage of RTI at timet, which is assumed to take values in the interval0,1;

u2t: the dosage of PI at timet, which is assumed to take values in0,1;

γ1: the capability of preventingTUcells from becomingT1cells with per unit dosage of RTI;

γ₂: the capability of preventingM_Ucells from becomingM₁cells with per unit dosage of RTI;

α1: the capability of preventingT2cells from producing viruses with per unit dosage of PI;

α₂: the capability of preventingM₂cells from producing viruses with per unit dosage of PI.

Next, let us consider the following assumptions.

A6Due to the eﬀect of RTIs,TUcells becomeT1cells with rateβT1−u1tγ1, andMU

cells becomeM1cells with rateβM1−u1tγ2, whereγ1andγ2are constants.

A7Due to the eﬀect of PIs, Free virusesVare produced byT₂ andM₂cells with a burst size ofα11−u2tNTandα21−u2tNM, respectively, whereα1andα2are constants.

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TU T1 T2

M2

M1

MU

βT

α1

βM

α2

V δTU δT1 δT2

k_M

δMU δM1 δM2

kT

(1−u1)γ1 (1−u2)NT

(1−u1)γ2

(1−u2)NM

Figure 2: The HIV model with therapy strategy.

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

dTU

dt θT−δTUTU−βTV TU1−u1γ1, dT1

dt βTV TU1−u1γ1−δT1kTT1, dT2

dt kTT1−δT2T2, dMU

dt θM−δMUMU−βMV MU1−u1γ2, dM₁

dt βMV MU1−u1γ2−δM1kMM1, dM₂

dt k_MM₁−δ_M₂M₂, dV

dt α₁N_TT₂1−u₂ α₂N_MM₂1−u₂−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 _t₁

t0

B₁T_UB₂M_U−

A₁u²₁A₂u²₂

dt, 2.2

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whereB1, B2 represent the benefit of perTU cell and per MU cell, respectively, andA1, A2

represent the cost of per unit RTI and per unit PI, respectively. Our goal is to obtain an optimal control pairu^∗₁, u^∗₂such that

J u^∗₁, u^∗₂

max{Ju1, u₂:u1, u₂∈ U}, 2.3

whereUis the admissible control set defined by

UU₁×U₂,

U₁U₂{ut:umeasurable,0≤ut≤1, t∈t0, t₁}. 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 ofu with coeﬃcients depending on time and state.

A11There exist positive constants c1, c2 and β > 1 such that the integranddenoted byLy, u, tof the objective functional2.2is concave and satisfies the condition Ly, u, t≤c₁−c₂u²₁u²₂^β/2.

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

Theorem 3.1. Consider system2.1with initial conditions, and the objective functional2.2. There existsu^∗ u^∗₁, u^∗₂such that

J u^∗₁, u^∗₂

max

u∈UJu1, u2. 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 setUU1×U2is obviously closed and convex, because bothU1andU2

are closed and convex sets.

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By definition, each right hand side of the ODEs of system2.1is continuous and can be written as a linear function ofuwith coeﬃcients depending on time and states. The fact that all state variablesT_U,T₁,T₂,M_U,M₁,M₂,V, andUare bounded ont0, t₁, implies the rest of assumptionA10.

It is easy to see thatLy, u, tis concave inU. By setting c1 max{B1TUB2MU}, c₂infA1, A₂andβ2, we can derive

L y, u, t

B1TUB2MU−

A1u²₁A2u²₂

≤c1−c2 u²₁u²₂ .

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, yt0 y₀,

4.1

whereytandAy, u, tare given by

yt TUt,T1t,T2t,MUt,M1t,M2t,Vt^T, A

y, u, t

g1

y, u, t , g2

y, u, t , . . . , g6

y, u, t , g7

y, u, t_T .

4.2

The Hamiltonian associated with our problem is H

y, u, p, t L

y, u, t

λt^TA y, u, t

, 4.3

where the adjoint vectorλtis defined by the adjoint equation dλt

dt −Ayλt−Ly, λt1 0.

4.4

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Here

A_y

⎛

⎜⎜

⎜⎝

∂g₁

∂TU

∂g₂

∂TU

∂g₃

∂TU

∂g₄

∂TU

∂g₅

∂TU

∂g₆

∂TU

∂g₇

∂TU

∂g1

∂T1

∂g2

∂T1

∂g3

∂T1

∂g4

∂T1

∂g5

∂T1

∂g6

∂T1

∂g7

∂T1

∂g₁

∂T₂

∂g₂

∂T₂

∂g₃

∂T₂

∂g₄

∂T₂

∂g₅

∂T₂

∂g₆

∂T₂

∂g₇

∂T₂

∂g₁

∂M_U

∂g₂

∂M_U

∂g₃

∂M_U

∂g₄

∂M_U

∂g₅

∂M_U

∂g₆

∂M_U

∂g₇

∂M_U

∂g1

∂M1

∂g2

∂M1

∂g3

∂M1

∂g4

∂M1

∂g5

∂M1

∂g6

∂M1

∂g7

∂M1

∂g1

∂M₂

∂g2

∂M₂

∂g3

∂M₂

∂g4

∂M₂

∂g5

∂M₂

∂g6

∂M₂

∂g7

∂M₂

∂g₁

∂V

∂g₂

∂V

∂g₃

∂V

∂g₄

∂V

∂g₅

∂V

∂g₆

∂V

∂g₇

∂V

⎞

⎟⎟

⎟⎠

E, F, 4.5

where

E

⎛

⎜⎜

⎝

−δTU−βTV1−u1γ1 βTV1−u1γ1 0

0 −δT1k_T k_T

0 0 −δT2

0 0 0

⎞

⎟⎟

⎠ ,

F

⎛

⎜⎜

⎝

0 0 0 0

0 0 0 α₁N_T1−u₂

−δMU−βMV1−u1γ2 βMV1−u1γ2 0 0

0 −δM1kM kM 0

0 0 −δM2 α2NM1−u2

0 0 0 −c

⎞

⎟⎟

⎠ .

4.6

In addition, theL_yin system4.3is

L_y ∂L

∂T_U, ∂L

∂T₁, ∂L

∂T₂, ∂L

∂M_U, ∂L

∂M₁, ∂L

∂M₂,∂L

∂V T

, B1,0,0, B2,0,0,0^T.

4.7

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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^Ê² to^Ê⁴ defined by

Γut 1−u1t, u1t,1−u2t, u2t,

ωt

⎛

⎜⎜

⎜⎝ ω11t ω12t ω21t ω₂₂t

⎞

⎟⎟

⎟⎠, 4.9

where allωij, i, j 1,2 are nonnegative penalty multipliers satisfying the following conditions:

1−u^∗₁t

ω₁₁t u^∗₁tω12t

1−u^∗₂t

ω₂₁t u^∗₂tω22t 0. 4.10 According to the Pontryagin’s Maximum Principle, if the control u^∗t and the corresponding state y^∗tconstitute an optimal pair, there exists an adjoint vector λtdefined system4.4such that the functionξy, u, λ, tdefined by4.8reaches its maximum on the setUat the pointu^∗. This gives the following result.

Theorem 4.1. Given an optimal control pairu^∗t u^∗₁t, u^∗₂tand a solutiony^∗t T_U^∗t, T₁^∗t,T₂^∗t,M^∗_Ut,M^∗₁t,M^∗₂t,V^∗tof the corresponding system, then there exist seven adjoint variablesλ₁t, λ2t, . . . , λ7tsatisfying

dλ1

dt

δTUβTV1−u1γ1

λ1−βTV1−u1γ1λ2−B1,

dλ2

dt δT1kTλ2−kTλ3, dλ3

dt δT2λ3−α1NT1−u2λ7, dλ₄

dt

δMUβMV1−u1γ2

λ4−βMV1−u1γ2λ5−B2, dλ₅

dt δ_M₁k_Mλ5−k_Mλ₆, dλ₆

dt δ_M₂λ₆−α₂N_M1−u₂λ7, dλ7

dt cλ7βTTU1−u1γ1λ1−βTTU1−u1γ1λ2

β_MM_U1−u₁γ2λ₄−β_MM_U1−u₁γ2λ₅,

4.11

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with the final conditions

λ₁t1 λ₂t1 · · ·λ₇t1 0. 4.12 Furthermore,u^∗₁t min{max{0, R1t},1}, u^∗₂t min{max{0, R2t},1}, where

R1t V^∗ 2A1

βTT_U^∗γ1λ1−λ2 βMM^∗_Uγ2λ4−λ5 , R₂t − λ₇

2A2α1N_TT₂α₂N_MM₂.

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 λ1t, . . . , λ7t^Tsatisfying

λt

dt −∂H

∂y −Ly−Ayλt. 4.14

That yields

λ₁t

dt −

∂g₁

y^∗, u^∗, t

∂TU , . . . ,∂g₇

y^∗, u^∗, t

∂TU

λt−∂L

y^∗, u^∗, t

∂TU , λ₂t

dt −

∂g₁

y^∗, u^∗, t

∂T1 , . . . ,∂g₇

y^∗, u^∗, t

∂T1

λt−∂L

y^∗, u^∗, t

∂T1 , λ3t

dt −

∂g₁

y^∗, u^∗, t

∂T2 , . . . ,∂g₇

y^∗, u^∗, t

∂T2

λt−∂L

y^∗, u^∗, t

∂T2 , λ4t

dt −

∂g₁

y^∗, u^∗, t

∂MU , . . . ,∂g₇

y^∗, u^∗, t

∂MU

λt−∂L

y^∗, u^∗, t

∂MU , λ5t

dt −

∂g₁

y^∗, u^∗, t

∂M1 , . . . ,∂g₇

y^∗, u^∗, t

∂M1

λt−∂L

y^∗, u^∗, t

∂M2 , λ6t

dt −

∂g₁

y^∗, u^∗, t

∂M2 , . . . ,∂g₇

y^∗, u^∗, t

∂M2

λt−∂L

y^∗, u^∗, t

∂M2 , λ7t

dt −

∂g₁

y^∗, u^∗, t

∂V , . . . ,∂g₇

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 pairy^∗, u^∗:

∂ξ

y^∗, u^∗, λ, t

∂u₁ 0, ∂ξ

y^∗, u^∗, λ, t

∂u₂ 0, 4.16

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whereξy^∗, u^∗, λ, t Hy^∗, u^∗, λ, t Γu^∗tωt. From4.10and4.16, we have

∂ξ

y^∗, u^∗, λ, t

∂u1 0

⇒ ∂L

y^∗, u^∗, t

∂u1 ∂ λt^TA

y^∗, u^∗, t

∂u1 ∂Γu^∗tωt

∂u1 0,

4.17

which implies

u^∗₁t V^∗ 2A₁

βTT_U^∗γ1λ1−λ2 βMM^∗_Uγ2λ4−λ5

. 4.18

On the other hand,

∂ξ

y, u^∗, λ, t

∂u₂ 0

⇒ ∂L

y^∗, u^∗, t

∂u2 ∂ λt^TA

y, u^∗, t

∂u2 ∂Γu^∗tωt

∂u2 0,

4.19

which indicates

u^∗₂t − λ7

2A2α1NTT2α2NMM2. 4.20

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

Case 1. t∈ {t: 0< u^∗₁t<1}andω11t ω12t 0. Thenu^∗₁t R1t.

Case 2. t ∈ {t : u^∗₁t 0} andω11t 0. Then 0 u^∗₁t R1t ω12t, which implies u^∗₁t≥R₁tbecauseω₁₂t≥0.

Case 3. t∈ {t:u^∗₁t 1}andω12t 0. Thenu^∗₁t R1t−ω11t, which leads to 1u^∗₁t≤ R1t, owing toω11≥0.

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

The proof is complete.

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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 −Ay^∗λt−Ly^∗, y^∗t0 y₀^∗,

λt1 0.

4.21

We substitute the expressionsu^∗ u^∗₁, u^∗₂in the above system. The uniqueness of the solution 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, t1at the pointstkklt0k0,1, . . . , n, wherelis the time step.

In the following, we define the state and adjoint variablesT_Ut,T₁t,T₂t,M_Ut, M1t,M2t,Vt,λ1t∼λ7tand the controlsu1tandu2tin terms of nodal pointsT_U^k, T₁^k,T₂^k,M_U^k,M^k₁,M^k₂,V^k,λ^k₁ ∼ λ^k₇,u^k₁,u^k₂ as the state and adjoint variables and the controls at initial timet₀, whileT_Uⁿ,T₁ⁿ,T₂ⁿ,M_Uⁿ,Mⁿ₁,Mⁿ₂,Vⁿ,λⁿ₁ ∼ λⁿ₇,uⁿ₁,uⁿ₂ as the state and adjoint variables and the controls at final timet1. As it is well known that the approximation of the time derivative by its first-order forward-diﬀerence is given, for the first state variableTU, by

dTUt

dt lim

l→0

TUtl−TUt

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_UT_U^k1−β_Tγ₁V^k 1−u^k₁

T_U^k1. 5.2

Analogously, we have T₁^k1−T₁^k

l βTγ1V^k 1−u^k₁

T_U^k1−δT1kTT₁^k1, T₂^k1−T₂^k

l kTT₁^k1−δT2T₂^k1, M^k1_U −M^k_U

l θ_M−δ_M_UM^k1_U −β_Mγ₂V^k 1−u^k₁ M_U^k1,

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M^k1₁ −M^k₁

l β_Mγ₂V^k 1−u^k₁

M^k1_U −δM1k_MM^k1₁ , M^k1₂ −M^k₂

l kMM^k1₁ −δM2M^k1₂ , V^k1−V^k

l α₁N_TT₂^k1 1−u^k₂

α₂N_MM^k1₂ 1−u^k₂

−cV^k1.

5.3 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:

λ^n−k₁ −λ^n−k−1₁

l

δ_T_Uβ_TV^k1 1−u^k₁ γ₁

λ^n−k−1₁ − β_TV^k1 1−u^k₁

γ₁λ^n−k₂ −B₁, λ^n−k₂ −λ^n−k−1₂

l δT1kTλ^n−k−1₂ −kTλ^n−k₃ , λ^n−k₃ −λ^n−k−1₃

l δ_T₂λ^n−k−1₃ −α₁N_T 1−u^k₂ λ^n−k₇ , λ^n−k₄ −λ^n−k−1₄

l

δMUβMV^k1 1−u^k₁ γ2

λ^n−k−1₄ − βMV^k1 1−u^k₁

γ2λ^n−k₅ −B2, λ^n−k₅ −λ^n−k−1₅

l δM1kMλ^n−k−1₅ −kMλ^n−k₆ , λ^n−k₆ −λ^n−k−1₆

l δ_M₂λ^n−k−1₆ −α₂N_M 1−u^k₂ λ^n−k₇ , λ^n−k₇ −λ^n−k−1₇

l cλ^n−k−1₇ βTT_U^k1 1−u^k₁

γ1 λ^n−k−1₁ −λ^n−k−1₂ βMM_U^k1 1−u^k₁

γ2 λ^n−k−1₄ −λ^n−k−1₅

.

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

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Table 1

Timedays TUBT TUAT MUBT MUAT 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 variablesu^∗₁t,u^∗₂t. For the following parameters and initial values:

θT 10,θM 10,δTU 0.02,δT1 0.5,δT2 1,δMU 0.0495,δM1 0.0495,δM2 0.0495, β_T 0.00008,β_M 0.00008,k_T 0.1,k_M 0.1,N_T 100,N_M 100,T_U⁰ 1000,T₁⁰ 0, T₂⁰ 0,M⁰_U1000,M⁰₁0,M⁰₂ 0,V⁰1,c0.03.

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 controlsu^∗₁t,u^∗₂tfor drug administration are presented through Figures7and8.

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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

×10⁵

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.

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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 variableu^∗₁t.

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 variableu^∗₂t.

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Appendices

A. The Theorem Used in Theorem 3.1

The equations

xt ˙ ft, xt,

x|tτ ξ, A.1

whereτ, ξ ∈ D, withDa nonempty open subset ofR × Rⁿandf : D → Rⁿ, are called a Cauchy problem or initial-value problem.

A solution to the Cauchy Problem is defined to be any pairI, φin whichI is an open subinterval ofRcontainingτ, φ:I → Rⁿis absolutely continuous,t, φt∈Dfor allt∈I, andφsatisfies the above two equations ata.e.t∈I.

Forx∈ Rⁿwith coordinatesxi, define a norm onRⁿby

|x|max

1≤i≤n|xi|. A.2

The following theorem applies the Lebesgue integral and the hypothesis is stated in terms of the rectangular subset ofR × Rⁿcentered aboutτ, ξ,

Ra,b{t, x:|t−τ| ≤a,|x−ξ| ≤b}, a >0, b >0. A.3 Theorem A.1see19, p.182. The Cauchy problem has a solution if for someRa,b ⊂ Dcentered aboutτ, ξthe restriction off toRa,bis continuous inxfor fixedt, measurable intfor fixedx, and satisfies