Mathematical Biosciences 242 (2013) 9–16
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Mathematical Biosciences
journal homepage: www.elsevier.com/locate/mbs
An optimal control problem arising from a dengue disease transmission model
Dipo Aldila a, Thomas Götz b, Edy Soewono a,⇑
a
b
Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia
Institute of Mathematics, University of Koblenz, 56070 Koblenz, Germany
a r t i c l e
i n f o
Article history:
Received 11 April 2011
Received in revised form 30 August 2012
Accepted 26 November 2012
Available online 27 December 2012
Keywords:
Dengue disease transmission
Basic reproductive ratio
Optimal control problem
a b s t r a c t
An optimal control problem for a host-vector Dengue transmission model is discussed here. In the model,
treatments with mosquito repellent are given to adults and children and those who undergo treatment
are classified in treated compartments. With this classification, the model consists of 11 dynamic equations. The basic reproductive ratio that represents the epidemic indicator is obtained from the largest
eigenvalue of the next generation matrix. The optimal control problem is designed with four control
parameters, namely the treatment rates for children and adult compartments, and the drop-out rates
from both compartments. The cost functional accounts for the total number of the infected persons,
the cost of the treatment, and the cost related to reducing the drop-out rates. Numerical results for
the optimal controls and the related dynamics are shown for the case of epidemic prevention and outbreak reduction strategies.
Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction
Dengue fever is one of the major world concerns in terms of
vector borne diseases and about 1.5 billion people at risk [22]. It
is estimated that there may be 50–100 million 10 dengue infections worldwide every year. It is also reported that vector control
is the only available method for control and prevention of Dengue
transmission [22].
Basic host-vector models for Dengue transmission were first
proposed in [8–10,18] with the focus on the construction of the basic reproductive ratio (see in [6]) and related stability analysis of
the disease free and endemic equilibria. Model development with
a variety of multistrain viruses is shown in [2,11,16]. Further developments involving vaccination with variation of compartments are
discussed in detail in [5,17,20]. Simple models to control the
spread of Dengue with the use of optimal control theory have been
done in [1,19]. These studies focus on modeling the spread of Dengue fever and its control using medication and insecticides related
to epidemic removal. Based on data from WHO [22], the worldwide
majority of patients suffering from Dengue fever are children. This
suggests that children are more exposed to contact with mosquitos
and more susceptible to contracting Dengue fever. Hence, it is necessary to analyze the effect of different treatment scenarios on
both the child and adult subpopulations.
⇑ Corresponding author.
E-mail address: [email protected] (E. Soewono).
0025-5564/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.mbs.2012.11.014
This paper describes a more realistic control model that can be
used to study how to reduce the spread and prevent the possible
occurrence of Dengue epidemics. The approach uses two age classes, i.e. children and adult classes. Unlike the two-age class model
in [20], here we classify the treated subpopulations in separate
compartments to give better relevance to the implementation of
mosquito repellent. The optimal control problem considers portions of both the child and adult populations who are treated with
mosquito repellent. The use of repellent is designed to reduce the
contact between humans and mosquitoes. By minimizing a suitable cost functional, we are able to compute the optimal treatment
and drop-out rates for both age compartments.
The paper is organized as follows. In Section 2 we present the
two age class Dengue transmission model and in Section 3 we carry out some analysis to determine the basic reproductive ratio. An
optimal control problem to design treatment rates is formulated in
Section 4. Section 5 shows some numerical results based on the
optimization algorithm presented in the previous section.
2. The mathematical model
A control problem for dengue disease transmission with a two
age class SIR-model for the human population and an SI-model
for the mosquito population is presented. The human population
is composed of the untreated susceptible children x1 , treated susceptible children x2 , untreated susceptible adults x3 , treated susceptible adults x4 , untreated infected and infectious children x5 ,
treated infected and infectious children x6 , untreated infected
10
D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16
and infectious adults x7 , treated infected and infectious adults x8 and
recovered human x9 . The vector population is divided into two compartments, susceptible vectors v 1 and infectious vectors v 2 .
Let x ¼ ðx1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 ; x8 ; x9 ; v 1 ; v 2 ÞT 2 R11 . Transmission
diagrams between compartments are shown in Fig. 1. We consider
the following dynamical system
dx
¼ Mðt; xÞ
dt
ð1Þ
for the sake of shorter notation that we use. The complete dynamical system is given by
dx1
bb x1 v 2
lx x1 þ ðd0 þ d1 ðtÞÞx2 ;
¼ A ax1 h1 ðtÞx1 x
dt
Nx
dx2
¼ h1 ðtÞx1 ax2 lx x2 ðd0 þ d1 ðtÞÞx2 ;
dt
dx3
bb x3 v 2
lx x3 þ ðd0 þ d2 ðtÞÞx4 ;
¼ ax1 h2 ðtÞx3 x
dt
Nx
dx4
¼ ax2 þ h2 ðtÞx3 lx x4 ðd0 þ d2 ðtÞÞx4 ;
dt
dx5 bbx x1 v 2
ax5 cx5 lx x5 h1 ðtÞx5 þ ðd0 þ d1 ðtÞÞx6 ;
¼
dt
Nx
dx6
¼ h1 ðtÞx5 ax6 cx6 lx x6 ðd0 þ d1 ðtÞÞx6 ;
dt
dx7 bbx x3 v 2
þ ax5 cx7 lx x7 h2 ðtÞx7 þ ðd0 þ d2 ðtÞÞx8 ;
¼
dt
Nx
dx8
¼ h2 ðtÞx7 þ ax6 cx8 lx x8 ðd0 þ d2 ðtÞÞx8 ;
dt
dx9
¼ cðx5 þ x6 þ x7 þ x8 Þ lx x9 ;
dt
dv 1
bb ðx5 þ x7 Þv 1
lv v 1 ;
¼B v
dt
Nx
dv 2 bbv ðx5 þ x7 Þv 1
lv v 2 ;
¼
dt
Nx
ð2aÞ
ð2bÞ
ð2cÞ
dNv
¼ B lv Nv :
dt
ð4Þ
In general, the natural death rate depends on ages (see [15] for
related PDE model with age dependent). In this model, we only use
the average natural death rate which is commonly used both for
children and adult compartments. Assuming that human and mosquito population to be in equilibria, we obtain N x ¼ lA and N v ¼ lB .
x
v
With this assumption, we could scale the human subpopulations
by N x and the vector subpopulations by N v . Let xi ¼ xi =N x for
i ¼ 1; . . . ; 9 and v j ¼ v j =N v for j ¼ 1; 2. We could obtain the normalized system of (2) by replacing xi with xi ; v j with v j , and the corresponding
parameters
A; B; bx
with
A ¼ NAx ; B ¼ NBv ; bx ¼ bxNNx v ,
respectively. The rest of parameters remain the same. See Table 1
for further detail about parameters description.
ð2dÞ
3. Model analysis
ð2eÞ
Throughout this section, we first assume that hi ðtÞ hi and
d0 þ di ðtÞ di0 are both constant in time. Then, the system (2)
has a disease-free equilibrium
ð2fÞ
ð2gÞ
xeq ¼
ð2hÞ
ð2iÞ
ð2jÞ
ð2kÞ
with initial conditions xð0Þ ¼ x0 . For the sake of notation, we use
x10 ¼ v 1 and x11 ¼ v 2 . Note that the removal rate d0 þ di ðtÞ is the
sum of the returning rate from treated to untreated compartment
d0 (inverse of the implementation period of repellent to each treated
individual) and the drop-out rate di ðtÞ of treated persons who do not
succeed to complete the treatment. The drop-out rate di ðtÞ depends
on the effort being spent for socialization and campaign of the proP
gram. For the overall human population Nx ¼ 9i¼1 xi it holds that
dNx
¼ A lx N x
dt
and for the total vector population N v ¼ v 1 þ v 2 we have
ð3Þ
a þ lx þ d10
h1
d20 h1 þ ðlx þ d20 Þða þ lx þ d10 Þ
A;
aA;
K1 K2
K1
h2 ða þ lx þ d10 Þ þ ðlx þ h2 Þh1
B
aA; 0; 0; 0; 0; 0; ; 0
ð5Þ
K1 K2
lv
K1
A;
where
K 1 ¼ ða þ lx þ h1 Þða þ lx þ d10 Þ d10
and
K 2 ¼ lx ðlx þ h2 þ d20 Þ.
In the case of no treatment and no age class division, i.e.
h1 ¼ h2 ¼ 0; d1 ¼ d2 ¼ 0; x2 ¼ x3 ¼ x4 ¼ x6 ¼ x7 ¼ x8 ¼ 0, the system
(2) is reduced to a standard dengue transmission model [4] with
the known basic reproductive ratio
R00
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
b bx bv Nv
¼
:
lv ðlx þ cÞNx
ð6Þ
The basic reproductive ratio R00 represents the expected numbers of secondary cases produced by a typical infected individual
during its entire period of infectiousness in a completely susceptible population (see the detail in [6,7]).The parameter ðR00 Þ2 is in
fact a measure of two-stage infection as follows. A single infected
mosquito successfully infects bbx humans per unit time during
their infection period l1 . Each infected human then infects
v
Fig. 1. Host transmission diagram.
bbV Nv
Nx
D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16
11
Table 1
Variables and parameters used in system (2) and their description. All variables and
parameters are assumed to be non-negative.
Variable/
parameter
Description
Value
x1 ðtÞ; x2 ðtÞ
x3 ðtÞ; x4 ðtÞ
x5 ðtÞ; x6 ðtÞ
x1 ðtÞ; x2 ðtÞ P 0
x3 ðtÞ; x4 ðtÞ P 0
x5 ðtÞ; x6 ðtÞ P 0
A; B
Untreated/treated susceptible children
Untreated/treated susceptible adults
Untreated/treated infected and
infectious children
Untreated/treated infected and
infectious adults
Recovered and immune humans
Susceptible/infectious vectors
Human/vector recruitment rate
a
Rate of transition from children to adults
1
12365
h1 ðtÞ; h2 ðtÞ
d0
d1 ðtÞ; d2 ðtÞ
lx ; lv
Rate of treated children/adults
Treatment frequency
Drop-out rate of treated children/adults
Natural death rate of humans/vectors
½0; 1
1
½0; 1
c
Human recovery rate
bx ; bv
b
Transmission rate in humans/vectors
Mosquito biting rate
x7 ðtÞ; x8 ðtÞ
x9 ðtÞ
v 1 ðtÞ; v 2 ðtÞ
x7 ðtÞ; x8 ðtÞ P 0
x9 ðtÞ P 0
v 1 ðtÞ; v 2 ðtÞ P 0
1000
500
65365 and 30
1
65365
1
14
and
1
30
0.1 and 0.1
1
Fig. 2. Sensitivity of R0 with respect to the treatment rates h1 and h2 .
vectors per unit time during the infection period ðl 1þcÞ [3]. This
x
parameter R00 can be used to estimate whether a new infection
may end up in an epidemic and to estimate the severity of the epidemic when no treatment takes place.
For general constant controls, the next generation matrix G (see
[6] for general description of next generation matrix), of the system (2) is given by
0
0
Bh
B 10
B N1
B
B a
G ¼ B N1
B
B0
B
@
bB
N1
d10
N2
0
0
0
0
0
0
0
d20
N4
a
N2
h20
N3
0
0
bB
N3
f
ð7Þ
where
N1 ¼ a þ c þ lx þ h10 ;
N3 ¼ c þ lx þ h20 ;
N2 ¼ a þ c þ lx þ d10 ;
N4 ¼ c þ lx þ d20 ;
and
b ¼ bBlx bv ;
B
Al v
blx bx
a þ lx þ d10
f¼
;
lv ða þ lx Þ ða þ lx þ h10 þ d10 Þ
babx
ðlx þ d20 Þ ða þ lx þ d10 Þ þ h10 d20
s¼
:
lv ða þ lx Þ ða þ lx þ h10 þ d10 Þ ðlx þ h20 þ d20 Þ
The element Gjk represents the number of new infections in
compartment j being generated by a single infective from compartment k. Note that for j and k from 1 to 5 represents x5 ; x6 ; x7 ; x8 and
v 2 , respectively. For an example, element G12 represent that every
single infective person from compartment x5 can infect aþcþdl10 þd10
x
person in compartment x6 . The basic reproductive ratio R0 of the
system (2) is given by the spectral radius of G, i.e. R0 is the largest
eigenvalue of G. From the Perron theorem [13], the largest eigenvalue of G is real and positive. Moreover, the characteristic polynomial of G is given by
5
3
a1 ¼
b ðfd20 h20 N 2 þ sd10 h10 N4 Þ þ d10 d20 h10 h20
B
;
N1 N2 N3 N4
a0 ¼
afh10 d20 Bb
:
N1 N2 N3 N4
The basic reproductive ratio R0 which satisfies vG ðR0 Þ ¼ 0 may
not be obtained analytically. We find that the disease free equilibrium is stable if and only if R0 < 1 [6], or equivalently vG ðR0 Þ > 0 .
Before the treatment is given to the population ðh1 ¼ h2 ¼ d1 ¼
d2 ¼ 0Þ and if there is no age class division (a ¼ 0), R0 reduces to
R00 .
The sensitivities of the parameters that affect R0 are shown in
Figs. 2 and 3 (see the parameters value in Table 2). It can be seen,
that a relatively huge effort is required to reduce R0 , which is not
feasible in practice due to the limited budget. For instance, in Fig. 2,
with a constant rate of h2 in 2:5 103 , larger h1 will reduce R0
much better. As a consequence, the budget is quite large because
the drop-out rate ðd1 ; d2 Þ is quite small only 103 . For daily application of control treatment, the rates of treatment and drop-out will
depend on time. In this case, the basic reproductive ratio does not
exist. Related to the optimal control analysis, it is shown in the
next section that the result with optimal control is far more efficient (in terms of cost as well as outbreak reduction) in comparison
with constant control.
In the next section we consider the case of the non-autonomous
system (2), in which the epidemic condition is satisfied before the
treatment, i.e. R0 > 1. Due to further limitation of budget and resources the epidemic may not be removed within the admissible
control parameters. In this case, the optimal treatment has to be
designed such that the total number of the infected individuals is
kept as small as possible within admissible control parameters.
4. Optimal control problem
2
vG ðkÞ ¼ k a3 k a2 k þ a1 k a0 ;
where the coefficients ai are positive and given by
a3 ¼
b
fa B
;
N1 N3
1
C
0C
C
C
sC
C;
C
0C
C
A
0
0
a2 ¼
b
b sB
d20 h20 d10 h10 f B
þ
þ
þ
;
N3 N4
N1 N2 N1 N3
Given the disease model (2) we want to design the treatment
rates h1 ðtÞ; h2 ðtÞ and the drop-out rates d1 ðtÞ; d2 ðtÞ such that we
minimize the number of the infected individuals. Obviously, this
task can always be achieved by imposing extremely high treatment
rates and minimal drop-out rates. However, high treatment rates
12
D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16
Fig. 3. Sensitivity of R0 with respect to a and h1 (left hand side) or h2 (right hand side).
Table 2
Parameters value for sensitivity analysis (8).
Parameters
Nx
Nv
b
a
c
lx
lv
bx
bv
d1
d2
Value
100,000
5000
1
1
12365
1
14
1
65365
1
30
0.1
0.1
103
103
and low drop-out rates increase the costs of the campaign. To find
a suitable compromise between minimal number of the infected
individuals and the costs of the campaign, we consider the following cost-functional
Jðx; uÞ ¼
1
2T
Z
0
11
T X
4
X
i¼1
j¼1
xx;i x2i ðtÞ þ
xu;j u2j ðtÞ dt:
ð8Þ
The first order optimality condition states that an optimum of the
problem (9) is a stationary point of L, see [21,12]. Hence
@ x L ¼ @ z L ¼ @ u L ¼ 0. Taking the derivative with respect to the adjoint variable z yields the state system x_ ¼ Mðx; uÞ.
Next, taking the derivative with respect to the state variables x
yields the adjoint system
z_ ¼ @ x Mðx; uÞz þ @ x Jðx; uÞ zðTÞ ¼ 0:
The weighting parameters xx;i and xu;j are used for the state
variables x and the control variables u ¼ fh1 ðtÞ; h2 ðtÞ; d1 ðtÞ; d2 ðtÞg,
respectively. Since we are mainly interested in minimizing the
number of the infected humans, we may set xx;i > 0 for
i ¼ 5; . . . ; 8. The number of susceptible humans, i.e. x1 ; . . . x4 as
well as the number of recovered individuals x9 and mosquito
population v 1 and v 2 is not taken into account, hence we choose
the weights xx;i ¼ 0 for those compartments. On the other hand
we want to minimize the costs for our treatment and campaign.
The costs of the treatment use are more-or-less proportional to
the treatment rates u1 and u2 , hence we choose the weights
xu;1 ; xu;2 > 0, as usual. To the second set u3 ; u4 of control variables, i.e. the drop-out-rates, the situation is just the other way
round: small drop-out rates require high costs and hence we
have to choose negative weights xu;3 ; xu;4 , if we aim to minimize
the costs.
Now, the optimal control task reads as
minJðx; uÞ s:t: Pðx; uÞ ¼ 0;
ð9Þ
u
ð11Þ
In the case of the disease model (2), we obtain the following set of
equations
z_ 1 ¼ lx z1 þ aðz1 z3 Þ þ bx x11 ðz1 z5 Þ þ h1 ðz1 z2 Þ;
z_ 2 ¼ lx z2 þ aðz2 z4 Þ þ ðd0 þ h1 Þðz2 z1 Þ;
z_ 3 ¼ lx z3 þ bx x11 ðz1 z5 Þ þ h2 ðz3 z4 Þ;
ð12aÞ
ð12bÞ
ð12cÞ
z_ 4 ¼ lx z4 þ ðd0 þ d2 Þðz4 z3 Þ;
z_ 5 ¼ lx z5 þ aðz5 z7 Þ þ cðz5 z9 Þ þ bv x10 ðz10 z11 Þ
ð12dÞ
þ h1 ðz5 z6 Þ þ xx;5 x5 ;
z_ 6 ¼ lx z6 þ aðz6 z8 Þ þ cðz6 z9 Þ þ ðd0 þ d1 Þðz6 z5 Þ
ð12eÞ
þ xx;6 x6 ;
z_ 7 ¼ lx z7 þ cðz7 z9 Þ þ bv x10 ðz10 z11 Þ þ h2 ðz7 z8 Þ
ð12fÞ
þ xx;7 x7 ;
_z8 ¼ lx z8 þ cðz8 z9 Þ þ ðd0 þ d2 Þðz8 z7 Þ þ xx;8 x8 ;
z_ 9 ¼ lx z9 ;
z_ 10 ¼ lv z10 þ bv ðx5 þ x7 Þðz10 z11 Þ;
ð12gÞ
z_ 11 ¼ lv z11 þ bx ðx1 ðz1 z5 Þ þ x3 ðz3 z7 ÞÞ:
ð12hÞ
ð12iÞ
ð12jÞ
ð12kÞ
where Pðx; uÞ ¼ 0 is a short hand notation for the state system (2),
i.e. x_ ¼ Mðt; x; uÞ; xð0Þ ¼ x0 . To derive the optimality conditions, we
introduce the Lagrangian
All of these equations are subject to the condition zi ðTÞ ¼ 0.
Finally, taking the derivative with respect to the control variable
u yields the gradient equation
Lðx; u; zÞ ¼ Jðx; uÞ þ hPðx; uÞ; zi:
@ u Jðx; uÞ @ u Mðx; uÞz ¼ 0:
ð10Þ
2
The duality product hPðx; uÞ; zi is induced by the L -inner product
hPðx; uÞ; zi ¼
1
T
Z
T
ðx_ Mðx; uÞÞz dt
0
¼ xðTÞ zðTÞ x0 zð0Þ þ
Z
0
T
x z_ Mðx; uÞz dt:
ð13Þ
In the setting of the disease model (2) this reads as
xu;1 u1 þ x1 ðz1 z2 Þ þ x5 ðz5 z6 Þ ¼ 0;
xu;2 u2 þ x3 ðz3 z4 Þ þ x7 ðz7 z8 Þ ¼ 0;
xu;3 u3 x2 ðz1 z2 Þ x6 ðz5 z6 Þ ¼ 0;
xu;4 u4 x4 ðz3 z4 Þ x8 ðz7 z8 Þ ¼ 0:
ð14aÞ
ð14bÞ
ð14cÞ
ð14dÞ
13
D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16
Choose some initial guess u for the control.
Solve the state Eq. (2) to compute x.
Compute the actual cost J.
Solve the adjoint Eq. (12) to compute z.
Solve the gradient Eq. (14) to compute the negative gradient
Du J.
6. Solve an approximate line search to find s . mins Jðxðus Þ; us Þ,
where us ¼ u þ s Du Jðx; uÞ.
7. Set u ¼ u þ s Du Jðx; uÞ.
8. Go to 2, unless termination criteria are met.
This algorithm in pseudo-code requires some additional
comments.
Steps 2 and 4 require the solution of a system of ordinary differential equations by any suitable numerical scheme. The state equation has to be solved forward in time, integrating from 0 to T while
the adjoint equation requires backward integration from T downto
0.
Step 6 is one of the crucial steps within the algorithm. Of course,
performing an exact line search, i.e. determining ^s such that
minJðxðus Þ; us Þ ¼ Jðxðu^s Þ; u^s Þ
s
would be the optimal choice regarding accuracy. However, determining ^s usually requires many evaluations of J, i.e. one has to solve
the state equation quite a number of times. To avoid the time consuming line search one has to resort to some heuristics. The simplest one is using a fixed stepsize s0 in the direction of the
negative gradient. Being very simple, this method faces convergence problems. Another, quite popular, choice is the Armijo-rule
[14], starting with a stepsize s0 , one tries the sequence bk s0 for some
0 < b < 1 of step sizes, until the cost functional is reduced for the
first time. The reasoning behind this heuristics is the following.
The overall aim is to reduce the cost functional in each step to avoid
oscillations. Knowing, that going a small step in the direction of
negative gradient will always lead to a decrease of the cost functional, we have to reduce step sizes that are too large and lead to
an increase of J. More advanced methods for determining a suitable
value s can be thought of, including local polynomial approximations
of the cost functional, see [14] for further details.
5. Numerical results
In this section we present some numerical simulations to the
optimal control problem. Let xx;i ; i ¼ 1; 2 . . . 9 be the weight for social cost in human (hospitalization), xx;i ; i ¼ 10; 11 for mosquito
and xu;j ; j ¼ 1; 2 be the weight cost in treatment rates for children
and adults. For j ¼ 3; 4, we define xu;j as the weight for the government campaign cost on the importance of repellent for children
and adults, respectively. The choice of these weights reflects the
different scales of the costs for counselling and treatment. Here
we used xu;j ¼ ð1; 2; 7; 10Þ for weight in control parameters
and the following weights in Table 3 for state variables in the cost
functional (8). The weight control treatment parameter for adults
(xu;2 ) is twice as large as that for children (xu;1 ). This is related
to the fact that children are much easier to target for treatment
than adults.
Numerical simulations of the optimal control problem are carried out for two scenarios, namely the case of prevention and the
case of epidemic reduction. The case of prevention occurs at the
early state of transmission, i.e. when the number of infected people
Weight
parameters
Compartment
x1
x2
x3
x4
x5
x6
x7
x8
x9
v1
v2
xx;i
0
0
0
0
200
200
400
400
0
0
0
Table 4
Initial conditions for both scenario.
Scenario
Compartment
Prevention
Reduction
x1
x2
x3
x4
x5
x6
x7
x8
x9
v1
v2
175
150
0
0
822
794
0
0
1
5
0
0
1
50
0
0
1
1
499
480
1
20
25
Total infected comp. before treatments
Total infected comp. with optimal control
20
Infected Compartment
1.
2.
3.
4.
5.
ð0Þ
Table 3
Weight parameters in Eq. (8).
15
10
5
0
0
50
100
150
Day
Fig. 4. Dynamics of the infected compartment ðx5 þ x6 þ x7 þ x8 Þ in the prevention
scenario.
1000
980
Susceptible Compartment
Solving these equations for the vector u, we obtain the direction of
the negative gradient Du Jðx; uÞ at the point ðx; uÞ.
With these ingredients at hand, we are now in position to setup
a gradient method for solving the minimization problem.
960
940
Total susceptible comp. before treatments
Total susceptible comp. with optimal control
920
900
880
0
50
100
150
Day
Fig. 5. Dynamics of the susceptible compartment ðx1 þ x2 þ x3 þ x4 Þ in the prevention scenario.
is still relatively small before the start of the treatment. In the case
of epidemic reduction, the treatment starts during the outbreak
period, i.e. when the number of infected people is significantly
14
D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16
0.14
0.1008
0.12
0.1007
δ1
2
0.1006
θ2
0.1
Drop−out rate
Treatment rate
θ1
δ
0.08
0.06
0.04
0.1005
0.1004
0.1003
0.1002
0.02
0.1001
0
0.1
0
50
100
150
0
50
100
Day
150
Day
Fig. 6. Treatment and drop out rates in the prevention scenario.
70
1000
Total susceptible comp. before treatments
Total susceptible comp. with optimal control
65
900
Susceptible Compartment
Infected Compartment
60
55
50
Total infected comp. before treatments
Total infected comp. with optimal control
45
40
35
800
700
600
500
400
30
25
0
50
100
150
Day
300
0
50
100
150
Day
Fig. 7. Dynamics of the infected compartment ðx5 þ x6 þ x7 þ x8 Þ in the reduction
scenario.
Fig. 8. Dynamics of the susceptible compartment ðx1 þ x2 þ x3 þ x4 Þ in the reduction scenario.
large. As an initial condition, we assume that in the field the ratio
between the mosquito and human population is 12. With this ratio,
the corresponding basic reproductive ratio is about 1.414 before
the start of the treatment. The initial conditions for both scenarios
are shown in Table 4.
In the case of prevention, the main goal of the treatment is to
prevent the occurrence of an epidemic in the population or to reduce the intensity of the epidemic if it already started. Hence we
want to suppress the outbreaks as much as possible within the given budget constraints. The dimensionless value of the cost functional (8) for one unit period of treatment is 4829. Simulation
results for total infected compartment and total susceptible compartment in the prevention case are shown in Figs. 4 and 5, respectively. It can be seen in Fig. 7, with the related optimal control
function shown in Fig. 6, the total infected population at the peak
of the outbreak could be suppressed to a level two times smaller
then without treatment along with delaying of the outbreak.
For the case of epidemic reduction, the optimal controls are
shown in Fig. 9. In the epidemic reduction scenario, the treatment
rate of the population is higher than in the prevention case to prevent the return of the outbreak. The achieved reduction of the total
number of infected persons and the susceptible population as a
consequence of the high rate of the treatment is shown in Figs. 7
and 8. In this scenario, the total cost for one unit period of treatment equals 90; 440 being almost twenty times greater than the
minimal cost in the prevention scenario. Hence, prevention of the
disease should be preferred to just later reduction of the epidemic.
Neglecting the age structure, i.e. a ¼ 0, the mathematical model
(2) is reduces to a 7-dimensional system. The corresponding
weight cost for control parameters h1 and d1 are 2 and 10 respectively, and weight parameters for state variables are given in Table 5. The state variables x3 ; x4 ; x7 ; x8 and the control variables
h2 ; d2 are eliminated since there is no age class in the model.
The comparison between the dynamics of the two models
(with and without age class) for the prevention and epidemic
reduction scenario are shown in Fig. 10, left and right, respectively. The corresponding actual control function for the prevention case is shown in Fig. 11. It can be seen that although the
dynamic of the infected compartment is only slightly different between the models with and without age class, the total cost for
each case is significantly different, i.e. 4829 and 5528, respectively. Similarly, in the epidemic reduction case (see Fig. 12),
15
D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16
0.5
0.112
θ1
0.45
δ1
θ2
0.35
Drop−out rate
Treatment rate
δ2
0.11
0.4
0.3
0.25
0.2
0.108
0.106
0.104
0.15
0.1
0.102
0.05
0
0.1
0
50
100
150
0
50
100
Day
150
Day
Fig. 9. Treatment and drop out rates in the epidemic reduction scenario.
25
70
65
20
Infected compartment
Infected compartment
Total infected with no age structured and before treatments
Total infected with no age structured, with treatment
Total infected with age structured and treatment
15
10
5
60
55
50
45
Total infected with no age structured and before treatments
Total infected with no age structured, with treatment
Total infected with age structured and treatment
40
35
30
25
0
20
0
50
100
150
0
50
100
Day
150
Day
Fig. 10. Dynamics of the infected compartment in prevention (left) and epidemic reduction case (right).
the corresponding costs are 90,440 and 98,906 respectively. The
significant difference in the cost value here is due to the absence
of a children’s compartment in the case with no age class, with
the consequence that everybody is treated using the adult rate
(which is more expensive).
Table 5
Weight parameters when a ¼ 0.
Weight parameters
x1
x2
x5
x6
x9
v1
v2
0
0
400
400
0
0
0
0.14
0.1008
0.12
0.1007
0.1006
0.1
Drop−out rate
Treatment rate
xx;i
Compartment
θ with age structured
1
θ2 with age structured
0.08
θ1 with no age structured
0.06
0.04
0.1005
δ1 with age structured
δ with age structured
0.1004
2
δ1 with no age structured
0.1003
0.1002
0.02
0.1001
0
0.1
0
50
100
150
0
Day
50
100
Day
Fig. 11. Treatment and drop out rates in prevention scenario.
150
16
D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16
0.5
0.114
θ with age structured
δ with age structured
1
0.45
1
θ with age structured
0.112
2
δ with age structured
2
θ with no age structured
0.4
δ with no age structured
1
Drop−out rate
Treatment rate
1
0.35
0.3
0.25
0.2
0.15
0.11
0.108
0.106
0.104
0.1
0.102
0.05
0
0.1
0
50
100
150
0
50
Day
100
150
Day
Fig. 12. Treatment and drop out rates in the epidemic reduction scenario.
6. Conclusion
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Acknowledgements
The authors would like to thanks the referees for their valuable
comments and suggestions. Parts of the research are funded by the
International Research Grant of the Indonesian Directorate General
for Higher Education and the German Academic Exchange Service
(DAAD).