Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 828509, 19 pages
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 Yang2
1
2
College of Computer Science, Chongqing University, Chongqing 400044, China
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
Copyright q 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.
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
2
Discrete Dynamics in Nature and Society
NT
TU
T1
βT
T2
kT
δT2
δT1
δTU
V
MU
M2
M1
kM
βM
NM
δMU
δM1
δM2
Figure 1: The HIV model.
for the second-phase decay. T cells are classified into three categories: TU cells uninfected T
cells, T1 cells early-stage infected T cells, and T2 cells late-stage infected T cells. Let TU , T1
and T2 denote the numbers of TU cells, T1 cells, and T2 cells, respectively. Likewise, M cells are
classified into three categories: Mu cells uninfected M cells, M1 cells early-stage infected
M cells, and M2 cells late-stage infected M cells. Let Mu , M1 , and M2 denote the numbers
of MU cells, M1 cells and M2 cells, respectively. Besides, let V denote the number of free
viruses. Sedaghat et al. 13 made the following reasonable assumptions.
A1 TU cells are produced with constant rate θT . MU cells are produced with constant
rate θM .
A2 TU cells become T1 cells with constant rate βT . MU cells become M1 cells with
constant rate βM .
A3 T1 cells become T2 cells with constant rate kT . M1 cells become M2 cells with
constant rate kM .
A4 These cells die with constant rates δTU , δT1 , δT2 , δMU , δM1 , and δM2 respectively.
A5 Free viruses V are cleared at a rate c, produced by T2 cells with a burst size of NT ,
and produced by M2 cells with a burst size of NM , respectively.
Under these assumptions, Sedaghat et al. 13 deduced the following system of ordinary
differential equations:
dTU
θT − δTU TU − βT TU V,
dt
dT1
βT TU V − δT1 kT T1 ,
dt
dT2
kT T1 − δT2 T2 ,
dt
Discrete Dynamics in Nature and Society
3
dMU
θM − δMU MU − βM MU V,
dt
dM1
βM MU V − δM1 kM M1 ,
dt
dM2
kM M1 − δM2 M2 ,
dt
dV
NT T2 NM M2 − cV.
dt
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 antiHIV 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:
u1 t: the dosage of RTI at time t, which is assumed to take values in the interval 0, 1;
u2 t: the dosage of PI at time t, which is assumed to take values in 0, 1;
γ1 : the capability of preventing TU cells from becoming T1 cells with per unit dosage of
RTI;
γ2 : the capability of preventing MU cells from becoming M1 cells with per unit dosage
of RTI;
α1 : the capability of preventing T2 cells from producing viruses with per unit dosage of
PI;
α2 : the capability of preventing M2 cells from producing viruses with per unit dosage
of PI.
Next, let us consider the following assumptions.
A6 Due to the effect of RTIs, TU cells become T1 cells with rate βT 1 − u1 tγ1 , and MU
cells become M1 cells with rate βM 1 − u1 tγ2 , where γ1 and γ2 are constants.
A7 Due to the effect of PIs, Free viruses V are produced by T2 and M2 cells with a
burst size of α1 1 − u2 tNT and α2 1 − u2 tNM , respectively, where α1 and α2 are
constants.
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Discrete Dynamics in Nature and Society
β T (1 − u1 )γ1
α1 (1 − u2 )NT
TU
T1
T2
kT
δTU
δT2
δT1
β M (1 − u1 )γ2
MU
V
M1
M2
kM
δMU
δM1
α2 (1 − u2 )NM
δM2
Figure 2: The HIV model with therapy strategy.
Under assumptions A1 –A7 , we can derive the following system of ordinary differential
equations:
dTU
θT − δTU TU − βT V TU 1 − u1 γ1 ,
dt
dT1
βT V TU 1 − u1 γ1 − δT1 kT T1 ,
dt
dT2
kT T1 − δT2 T2 ,
dt
dMU
θM − δMU MU − βM V MU 1 − u1 γ2 ,
dt
2.1
dM1
βM V MU 1 − u1 γ2 − δM1 kM M1 ,
dt
dM2
kM M1 − δM2 M2 ,
dt
dV
α1 NT T2 1 − u2 α2 NM M2 1 − u2 − cV.
dt
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
Ju1 t, u2 t t1 t0
B1 TU B2 MU − A1 u21 A2 u22 dt,
2.2
Discrete Dynamics in Nature and Society
5
where B1 , B2 represent the benefit of per TU cell and per MU cell, respectively, and A1 , A2
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
J u∗1 , u∗2 max{Ju1 , u2 : u1 , u2 ∈ U},
2.3
where U is the admissible control set defined by
U U1 × U2 ,
U1 U2 {ut : u measurable, 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.
A8 The set of control and corresponding state variables is nonempty.
A9 The admissible control set U 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 u
with coefficients depending on time and state.
A11 There exist positive constants c1 , c2 and β > 1 such that the integrand denoted
by Ly, u, t of the objective functional 2.2 is concave and satisfies the condition
β/2
Ly, u, t ≤ c1 − c2 u21 u22 .
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 u∗ u∗1 , u∗2 such that
J u∗1 , u∗2 maxJu1 , u2 .
u∈U
3.1
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 U U1 × U2 is obviously closed and convex, because both U1 and U2
are closed and convex sets.
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Discrete Dynamics in Nature and Society
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 TU , T1 , T2 , MU , M1 , M2 , V , and U are bounded on t0 , t1 , implies the
rest of assumption A10 .
It is easy to see that Ly, u, t is concave in U. By setting c1 max{B1 TU B2 MU },
c2 infA1 , A2 and β 2, we can derive
L y, u, t B1 TU B2 MU − A1 u21 A2 u22
≤ c1 − c2 u21 u22 .
3.2
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:
dyt
A y, u, t ;
dt
∀t > t0 , ∀u ∈ U,
4.1
yt0 y0 ,
where yt and Ay, u, t are given by
yt TU t, T1 t, T2 t, MU t, M1 t, M2 t, V tT ,
T
A y, u, t g1 y, u, t , g2 y, u, t , . . . , g6 y, u, t , g7 y, u, t .
4.2
The Hamiltonian associated with our problem is
H y, u, p, t L y, u, t λtT A y, u, t ,
4.3
where the adjoint vector λt is defined by the adjoint equation
dλt
−Ay λt − Ly ,
dt
λt1 0.
4.4
Discrete Dynamics in Nature and Society
7
Here
⎛
∂g2
∂g3
∂g4
∂g5
∂g6
∂g7
∂g1
⎜ ∂TU ∂TU ∂TU ∂TU ∂TU ∂TU ∂TU
⎜
⎜ ∂g
∂g2
∂g3
∂g4
∂g5
∂g6
∂g7
1
⎜
⎜
⎜ ∂T1
∂T
∂T
∂T
∂T
∂T
∂T1
1
1
1
1
1
⎜
⎜ ∂g1
∂g
∂g
∂g
∂g
∂g
∂g7
2
3
4
5
6
⎜
⎜ ∂T
∂T2
∂T2
∂T2
∂T2
∂T2
∂T2
2
⎜
⎜
∂g2
∂g3
∂g4
∂g5
∂g6
∂g7
⎜ ∂g1
Ay ⎜
⎜ ∂MU ∂MU ∂MU ∂MU ∂MU ∂MU ∂MU
⎜
⎜ ∂g1
∂g2
∂g3
∂g4
∂g5
∂g6
∂g7
⎜
⎜
⎜ ∂M1 ∂M1 ∂M1 ∂M1 ∂M1 ∂M1 ∂M1
⎜
⎜ ∂g1
∂g2
∂g3
∂g4
∂g5
∂g6
∂g7
⎜
⎜ ∂M2 ∂M2 ∂M2 ∂M2 ∂M2 ∂M2 ∂M2
⎜
⎝ ∂g
∂g2
∂g3
∂g4
∂g5
∂g6
∂g7
1
∂V
∂V
∂V
∂V
∂V
∂V
∂V
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟ E, F,
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
4.5
where
⎛
−δTU − βT V 1 − u1 γ1 βT V 1 − u1 γ1
⎜
⎜
⎜
⎜
⎜
⎜
⎜
E⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
0
0
−δT1 kT kT
0
0
−δT2
0
0
0
0
0
0
0
0
0
0
0
0
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
0
0
0
0
⎜
⎜
0
0
0
0
⎜
⎜
⎜
0
0
0
α1 NT 1 − u2 ⎜
⎜
F⎜
0
0
⎜−δMU − βM V 1 − u1 γ2 βM V 1 − u1 γ2
⎜
⎜
0
−δM1 kM kM
0
⎜
⎜
⎜
0
0
−δM2 α2 NM 1 − u2 ⎝
0
0
0
−c
⎞
4.6
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
⎠
In addition, the Ly in system 4.3 is
Ly ∂L ∂L ∂L ∂L
∂L
∂L ∂L
,
,
,
,
,
,
∂TU ∂T1 ∂T2 ∂MU ∂M1 ∂M2 ∂V
T
B1 , 0, 0, B2, 0, 0, 0 .
T
,
4.7
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Discrete Dynamics in Nature and Society
Next, adding the penalty term will give us the optimality condition
ξ y, u, λ, t H y, u, λ, t Γutωt,
4.8
where Γ is an operator from Ê2 to Ê4 defined by
Γut 1 − u1 t, u1 t, 1 − u2 t, u2 t,
⎛
⎞
ω11 t
⎜
⎟
⎜ω12 t⎟
⎜
⎟
ωt ⎜
⎟,
⎜ω21 t⎟
⎝
⎠
4.9
ω22 t
where all ωij , i, j 1, 2 are nonnegative penalty multipliers satisfying the following conditions:
1 − u∗1 t ω11 t u∗1 tω12 t 1 − u∗2 t ω21 t u∗2 tω22 t 0.
4.10
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 λt defined
system 4.4 such that the function ξy, u, λ, t defined by 4.8 reaches its maximum on the
set U at 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 solution y∗ t TU∗ t,
∗
t, M1∗ t, M2∗ t, V ∗ t of the corresponding system, then there exist seven adjoint
T1∗ t, T2∗ t, MU
variables λ1 t, λ2 t, . . . , λ7 t satisfying
dλ1 δTU βT V 1 − u1 γ1 λ1 − βT V 1 − u1 γ1 λ2 − B1 ,
dt
dλ2
δT1 kT λ2 − kT λ3 ,
dt
dλ3
δT2 λ3 − α1 NT 1 − u2 λ7 ,
dt
dλ4 δMU βM V 1 − u1 γ2 λ4 − βM V 1 − u1 γ2 λ5 − B2 ,
dt
dλ5
δM1 kM λ5 − kM λ6 ,
dt
dλ6
δM2 λ6 − α2 NM 1 − u2 λ7 ,
dt
dλ7
cλ7 βT TU 1 − u1 γ1 λ1 − βT TU 1 − u1 γ1 λ2
dt
βM MU 1 − u1 γ2 λ4 − βM MU 1 − u1 γ2 λ5 ,
4.11
Discrete Dynamics in Nature and Society
9
with the final conditions
λ1 t1 λ2 t1 · · · λ7 t1 0.
4.12
Furthermore, u∗1 t min{max{0, R1 t}, 1}, u∗2 t min{max{0, R2 t}, 1}, where
R1 t V∗ ∗
γ2 λ4 − λ5 ,
βT TU∗ γ1 λ1 − λ2 βM MU
2A1
λ7
R2 t −
α1 NT T2 α2 NM M2 .
2A2
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 2.1. Therefore, by Pontryagin’s
Maximum Principle, there exists a vector λt λ1 t, . . . , λ7 tT satisfying
∂H
λt
−
−Ly − Ay λt.
dt
∂y
4.14
That yields
∂g1 y∗ , u∗ , t
∂g7 y∗ , u∗ , t
λ1 t
λt −
−
,...,
dt
∂TU
∂TU
∂g1 y∗ , u∗ , t
∂g7 y∗ , u∗ , t
λ2 t
−
,...,
λt −
dt
∂T1
∂T1
∂g7 y∗ , u∗ , t
∂g1 y∗ , u∗ , t
λ3 t
λt −
−
,...,
dt
∂T2
∂T2
∂g1 y∗ , u∗ , t
∂g7 y∗ , u∗ , t
λ4 t
λt −
−
,...,
dt
∂MU
∂MU
∂g1 y∗ , u∗ , t
∂g7 y∗ , u∗ , t
λ5 t
−
,...,
λt −
dt
∂M1
∂M1
∂g1 y∗ , u∗ , t
∂g7 y∗ , u∗ , t
λ6 t
λt −
−
,...,
dt
∂M2
∂M2
∂g1 y∗ , u∗ , t
∂g7 y∗ , u∗ , t
λ7 t
−
,...,
λt −
dt
∂V
∂V
∂L y∗ , u∗ , t
,
∂TU
∂L y∗ , u∗ , t
,
∂T1
∂L y∗ , u∗ , t
,
∂T2
∂L y∗ , u∗ , t
,
∂MU
4.15
∂L y∗ , u∗ , t
,
∂M2
∂L y∗ , u∗ , t
,
∂M2
∂L y∗ , u∗ , t
.
∂V
Through simple calculations, we derive system 4.11. The Pontryagin’s Maximum Principle gives the following necessary conditions to obtain the optimal pair y∗ , u∗ :
∂ξ y∗ , u∗ , λ, t
0,
∂u1
∂ξ y∗ , u∗ , λ, t
0,
∂u2
4.16
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Discrete Dynamics in Nature and Society
where ξy∗ , u∗ , λ, t Hy∗ , u∗ , λ, t Γu∗ tωt. From 4.10 and 4.16, we have
∂ξ y∗ , u∗ , λ, t
0
∂u1
∂ λtT Ay∗ , u∗ , t
∂L y∗ , u∗ , t
∂Γu∗ tωt
⇒
0,
∂u1
∂u1
∂u1
4.17
which implies
u∗1 t V∗ ∗
βT TU∗ γ1 λ1 − λ2 βM MU
γ2 λ4 − λ5 .
2A1
4.18
On the other hand,
∂ξ y, u∗ , λ, t
0
∂u2
∂ λtT Ay, u∗ , t
∂L y∗ , u∗ , t
∂Γu∗ tωt
0,
⇒
∂u2
∂u2
∂u2
4.19
which indicates
u∗2 t −
λ7
α1 NT T2 α2 NM M2 .
2A2
4.20
Now from the constraint condition, the following three cases arise.
Case 1. t ∈ {t : 0 < u∗1 t < 1} and ω11 t ω12 t 0. Then u∗1 t R1 t.
Case 2. t ∈ {t : u∗1 t 0} and ω11 t 0. Then 0 u∗1 t R1 t ω12 t, which implies
u∗1 t ≥ R1 t because ω12 t ≥ 0.
Case 3. t ∈ {t : u∗1 t 1} and ω12 t 0. Then u∗1 t R1 t − ω11 t, which leads to 1 u∗1 t ≤
R1 t, owing to ω11 ≥ 0.
Hence, we have u∗1 t min{max{0, R1 t}, 1}. Similarly, we can get that u∗2 t
min{max{0, R2 t}, 1}.
The proof is complete.
Discrete Dynamics in Nature and Society
11
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
A y ∗ , u∗ , t ;
dt
∀t > t0 ,
dλt
−Ay∗ λt − Ly∗ ,
dt
4.21
y∗ t0 y0∗ ,
λt1 0.
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.
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 t0 , t1 at the points tk kl t0 k 0, 1, . . . , n, where l is the time step.
In the following, we define the state and adjoint variables TU t, T1 t, T2 t, MU t,
M1 t, M2 t, V t, λ1 t ∼ λ7 t and the controls u1 t and u2 t in terms of nodal points TUk ,
k
T1k , T2k , MU
, M1k , M2k , V k , λk1 ∼ λk7 , uk1 , uk2 as the state and adjoint variables and the controls
n
, M1n , M2n , V n , λn1 ∼ λn7 , un1 , un2 as the state and adjoint
at initial time t0 , while TUn , T1n , T2n , MU
variables and the controls at final time t1 . 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 TU , by
TU t l − TU t
dTU t
lim
.
l→0
dt
l
5.1
We use the scheme developed by Gumel et al. 7 in the following way:
TUk1 − TUk
θT − δTU TUk1 − βT γ1 V k 1 − uk1 TUk1 .
l
Analogously, we have
T1k1 − T1k
βT γ1 V k 1 − uk1 TUk1 − δT1 kT T1k1 ,
l
T2k1 − T2k
kT T1k1 − δT2 T2k1 ,
l
k1
k
MU
− MU
k1
k1
θM − δMU MU
− βM γ2 V k 1 − uk1 MU
,
l
5.2
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Discrete Dynamics in Nature and Society
M1k1 − M1k
k1
βM γ2 V k 1 − uk1 MU
− δM1 kM M1k1 ,
l
M2k1 − M2k
kM M1k1 − δM2 M2k1 ,
l
V k1 − V k
α1 NT T2k1 1 − uk2 α2 NM M2k1 1 − uk2 − cV k1 .
l
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:
− λn−k−1
λn−k
1
1
δTU βT V k1 1 − uk1 γ1 λn−k−1
− βT V k1 1 − uk1 γ1 λn−k
− B1 ,
2
1
l
λn−k
− λn−k−1
2
2
δT1 kT λn−k−1
− kT λn−k
2
3 ,
l
λn−k
− λn−k−1
3
3
k
n−k
−
α
N
δT2 λn−k−1
1
−
u
1
T
3
2 λ7 ,
l
λn−k
− λn−k−1
4
4
k1
k
n−k
δMU βM V k1 1 − uk1 γ2 λn−k−1
1
−
u
−
β
V
− B2 ,
M
4
1 γ2 λ5
l
− λn−k−1
λn−k
5
5
δM1 kM λn−k−1
− kM λn−k
5
6 ,
l
− λn−k−1
λn−k
6
6
k
n−k
δM2 λn−k−1
1
−
u
−
α
N
2
M
6
2 λ7 ,
l
λn−k
− λn−k−1
7
7
n−k−1
cλn−k−1
βT TUk1 1 − uk1 γ1 λn−k−1
−
λ
7
2
1
l
k1
βM MU
− λn−k−1
1 − uk1 γ2 λn−k−1
.
5
4
5.4
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
Discrete Dynamics in Nature and Society
13
Table 1
Time days
0
2
4
6
8
10
20
30
40
50
TU BT
1000
980.2369
955.4799
839.4257
270.1388
11.92899
0.511601
0.379269
0.334284
0.322352
TU AT
1000
980.2012
953.1210
889.1900
833.1982
784.0453
628.5318
597.7372
574.2037
555.2309
MU BT
1000
926.3417
856.9665
754.0975
372.5725
41.26735
1.021524
0.756860
0.666976
0.643089
MU AT
1000
926.1494
852.1262
754.1006
671.2911
601.1371
388.0657
313.0676
268.0361
240.5969
V BT
1
15.33444
283.2647
4710.942
38782.17
99155.03
255003.3
335762.8
376288.7
387774.5
V AT
1
19.568136
334.65888
315.44809
297.34008
280.27154
208.54835
155.17956
115.46816
85.919158
1000
900
800
700
600
500
400
300
200
100
0
0
5
10
15
20
25
30
35
40
45
50
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:
θT 10, θM 10, δTU 0.02, δT1 0.5, δT2 1, δMU 0.0495, δM1 0.0495, δM2 0.0495,
βT 0.00008, βM 0.00008, kT 0.1, kM 0.1, NT 100, NM 100, TU0 1000, T10 0,
0
1000, M10 0, M20 0, V 0 1, c 0.03.
T20 0, MU
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.
14
Discrete Dynamics in Nature and Society
1000
900
800
700
600
500
400
300
200
100
0
0
5
10
15
20
25
30
35
40
45
50
Before treatment
After treatment
Figure 4: Uninfected M cells.
×105
4
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
40
45
50
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 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.
Discrete Dynamics in Nature and Society
15
350
300
250
200
150
100
50
0
0
5
10
15
20
25
30
35
40
45
50
Figure 6: Virus population after optimal controls.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
Figure 7: Optimal control variable
40
45
50
45
50
u∗1 t.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
40
Figure 8: Optimal control variable u∗2 t.
16
Discrete Dynamics in Nature and Society
Appendices
A. The Theorem Used in Theorem 3.1
The equations
ẋt ft, xt,
A.1
x|tτ ξ,
where τ, ξ ∈ D, with D a nonempty open subset of R × Rn and f : D → Rn , are called a
Cauchy problem or initial-value problem.
A solution to the Cauchy Problem is defined to be any pair I, φ in which I is an open
subinterval of R containing τ, φ : I → Rn is absolutely continuous, t, φt ∈ D for all t ∈ I,
and φ satisfies the above two equations at a.e.t ∈ I.
For x ∈ Rn with coordinates xi , define a norm on Rn by
A.2
|x| max|xi |.
1≤i≤n
The following theorem applies the Lebesgue integral and the hypothesis is stated in
terms of the rectangular subset of R × Rn centered about τ, ξ,
Ra,b {t, x : |t − τ| ≤ a, |x − ξ| ≤ b},
a > 0, b > 0.
A.3
Theorem A.1 see 19, p.182. The Cauchy problem has a solution if for some Ra,b ⊂ D centered
about τ, ξ the restriction of f to Ra,b is continuous in x for fixed t, measurable in t for fixed x, and
satisfies
ft, x ≤ mt,
t, x ∈ Ra,b ,
A.4
for some m integrable over the interval τ − a, τ a.
B. An Algorithm Using the GSS1 Method
Algorithm B.1.
Step 1.
TU t0 ←− TU0 ,
0
MU t0 ←− MU
,
λ1 tn ←− 0,
T1 t0 ←− T10 ,
M1 t0 ←− M10 ,
λ2 tn ←− 0,
λ5 tn ←− 0,
M2 t0 ←− M20 ,
λ3 tn ←− 0,
λ6 tn ←− 0,
u1 t0 ←− 0,
T2 t0 ←− T20 ,
V t0 ←− V 0 ,
λ4 tn ←− 0,
λ7 tn ←− 0,
u2 t0 ←− 0.
B.1
Discrete Dynamics in Nature and Society
17
Step 2. for k 1, . . . , n do
TUk ←−
T1k ←−
T2k ←−
k
MU
←−
M1k ←−
M2k ←−
V k ←−
λn−k
1
lβT γ1 V k−1 1 − uk−1
TUk T1k−1
1
1 δT1 kT l
,
lkT T1k T2k−1
,
1 lδT2
1 lδMU
k−1
lθM MU
,
lβM γ2 V k−1 1 − uk−1
1
k
M1k−1
lβM γ2 V k−1 1 − uk−1
MU
1
1 δM1 kM l
,
lkM M1k M2k−1
,
1 lδM2
lα1 NT T2k 1 − uk−1
lα2 NM M2k 1 − uk−1
V k−1
2
2
1 cl
,
γ1 λn−k1
λn−k1
lB1 lβT V k 1 − uk−1
2
1
1
←−
,
k−1
k
1 lδTU lβT V 1 − u1 γ1
λn−k
←−
2
λn−k
←−
3
λn−k
4
1 lδTU
lθT TUk−1
,
γ1
lβT V k−1 1 − uk−1
1
λn−k1
lkT λn−k1
2
3
,
1 δT1 kT l
k−1
1
−
u
λn−k1
lα
N
λn−k1
1
T
3
2
7
,
1 lδT2
λn−k1
lB2 lβM V k 1 − uk−1
γ2 λn−k1
5
1
4
←−
,
γ2
1 lδMU lβM V k 1 − uk−1
1
λn−k
←−
5
λn−k
←−
6
λn−k
←−
7
lkM λn−k1
λn−k1
5
6
,
1 δM1 kM l
lα2 NM 1 − uk−1
λn−k1
λn−k1
6
2
7
,
1 lδM2
k
l 1 − uk−1
− λn−k
γ2 λn−k
− λn−k
λn−k1
βT TUk γ1 λn−k
βM MU
7
2
5
1
1
4
1 lc
,
18
Discrete Dynamics in Nature and Society
Rk1 ←−
Vk k
βT TUk γ1 λn−k
βM MU
,
− λn−k
γ2 λn−k
− λn−k
2
5
1
4
2A1
λn−k
7
α1 NT T2k α2 NM M2k ,
2A2
uk1 ←− min max 0, Rk1 , 1 , uk2 ←− min max 0, Rk2 , 1 .
Rk2 ←− −
B.2
Step 3. for k 1, . . . , n, denote
∗
k
, u∗1 tk uk1 , u∗2 tk uk2 .
TU∗ tk TUk , MU
tk MU
B.3
It is easy to conclude that this algorithm takes On 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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