Mathematical Biosciences 242 (2013) 9–16

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Mathematical Biosciences

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An optimal control problem arising from a dengue disease transmission model

Dipo Aldila a, Thomas Götz b, Edy Soewono a,⇑a Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesiab Institute of Mathematics, University of Koblenz, 56070 Koblenz, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 April 2011Received in revised form 30 August 2012Accepted 26 November 2012Available online 27 December 2012

Keywords:Dengue disease transmissionBasic reproductive ratioOptimal control problem

0025-5564/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.mbs.2012.11.014

⇑ Corresponding author.E-mail address: esoewono@math.itb.ac.id (E. Soew

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 treatmentare classified in treated compartments. With this classification, the model consists of 11 dynamic equa-tions. The basic reproductive ratio that represents the epidemic indicator is obtained from the largesteigenvalue of the next generation matrix. The optimal control problem is designed with four controlparameters, namely the treatment rates for children and adult compartments, and the drop-out ratesfrom 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 forthe optimal controls and the related dynamics are shown for the case of epidemic prevention and out-break reduction strategies.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Dengue fever is one of the major world concerns in terms ofvector borne diseases and about 1.5 billion people at risk [22]. Itis estimated that there may be 50–100 million 10 dengue infec-tions worldwide every year. It is also reported that vector controlis the only available method for control and prevention of Denguetransmission [22].

Basic host-vector models for Dengue transmission were firstproposed in [8–10,18] with the focus on the construction of the ba-sic reproductive ratio (see in [6]) and related stability analysis ofthe disease free and endemic equilibria. Model development witha variety of multistrain viruses is shown in [2,11,16]. Further devel-opments involving vaccination with variation of compartments arediscussed in detail in [5,17,20]. Simple models to control thespread of Dengue with the use of optimal control theory have beendone in [1,19]. These studies focus on modeling the spread of Den-gue fever and its control using medication and insecticides relatedto epidemic removal. Based on data from WHO [22], the worldwidemajority of patients suffering from Dengue fever are children. Thissuggests that children are more exposed to contact with mosquitosand more susceptible to contracting Dengue fever. Hence, it is nec-essary to analyze the effect of different treatment scenarios onboth the child and adult subpopulations.

ll rights reserved.

ono).

This paper describes a more realistic control model that can beused to study how to reduce the spread and prevent the possibleoccurrence of Dengue epidemics. The approach uses two age clas-ses, i.e. children and adult classes. Unlike the two-age class modelin [20], here we classify the treated subpopulations in separatecompartments to give better relevance to the implementation ofmosquito repellent. The optimal control problem considers por-tions of both the child and adult populations who are treated withmosquito repellent. The use of repellent is designed to reduce thecontact between humans and mosquitoes. By minimizing a suit-able cost functional, we are able to compute the optimal treatmentand drop-out rates for both age compartments.

The paper is organized as follows. In Section 2 we present thetwo age class Dengue transmission model and in Section 3 we car-ry out some analysis to determine the basic reproductive ratio. Anoptimal control problem to design treatment rates is formulated inSection 4. Section 5 shows some numerical results based on theoptimization algorithm presented in the previous section.

2. The mathematical model

A control problem for dengue disease transmission with a twoage class SIR-model for the human population and an SI-modelfor the mosquito population is presented. The human populationis composed of the untreated susceptible children x1, treated sus-ceptible children x2, untreated susceptible adults x3, treated sus-ceptible 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 andrecovered human x9. The vector population is divided into two com-partments, susceptible vectors v1 and infectious vectors v2.

Let x ¼ ðx1; x2; x3; x4; x5; x6; x7; x8; x9;v1;v2ÞT 2 R11. Transmissiondiagrams between compartments are shown in Fig. 1. We considerthe following dynamical system

dxdt¼ Mðt; xÞ ð1Þ

for the sake of shorter notation that we use. The complete dynam-ical system is given by

dx1

dt¼ A� ax1 � h1ðtÞx1 �

bbxx1v2

Nx� lxx1 þ d0 þ d1ðtÞð Þx2; ð2aÞ

dx2

dt¼ h1ðtÞx1 � ax2 � lxx2 � d0 þ d1ðtÞð Þx2; ð2bÞ

dx3

dt¼ ax1 � h2ðtÞx3 �

bbxx3v2

Nx� lxx3 þ d0 þ d2ðtÞð Þx4; ð2cÞ

dx4

dt¼ ax2 þ h2ðtÞx3 � lxx4 � d0 þ d2ðtÞð Þx4; ð2dÞ

dx5

dt¼ bbxx1v2

Nx� ax5 � cx5 � lxx5 � h1ðtÞx5 þ d0 þ d1ðtÞð Þx6; ð2eÞ

dx6

dt¼ h1ðtÞx5 � ax6 � cx6 � lxx6 � d0 þ d1ðtÞð Þx6; ð2fÞ

dx7

dt¼ bbxx3v2

Nxþ ax5 � cx7 � lxx7 � h2ðtÞx7 þ d0 þ d2ðtÞð Þx8; ð2gÞ

dx8

dt¼ h2ðtÞx7 þ ax6 � cx8 � lxx8 � d0 þ d2ðtÞð Þx8; ð2hÞ

dx9

dt¼ c x5 þ x6 þ x7 þ x8ð Þ � lxx9; ð2iÞ

dv1

dt¼ B� bbv x5 þ x7ð Þv1

Nx� lvv1; ð2jÞ

dv2

dt¼ bbv x5 þ x7ð Þv1

Nx� lvv2; ð2kÞ

with initial conditions xð0Þ ¼ x0. For the sake of notation, we usex10 ¼ v1 and x11 ¼ v2. Note that the removal rate d0 þ diðtÞ is thesum of the returning rate from treated to untreated compartmentd0 (inverse of the implementation period of repellent to each treatedindividual) and the drop-out rate diðtÞ of treated persons who do notsucceed to complete the treatment. The drop-out rate diðtÞ dependson the effort being spent for socialization and campaign of the pro-gram. For the overall human population Nx ¼

P9i¼1xi it holds that

dNx

dt¼ A� lxNx ð3Þ

Fig. 1. Host transm

and for the total vector population Nv ¼ v1 þ v2 we have

dNv

dt¼ B� lvNv : ð4Þ

In general, the natural death rate depends on ages (see [15] forrelated PDE model with age dependent). In this model, we only usethe average natural death rate which is commonly used both forchildren and adult compartments. Assuming that human and mos-quito population to be in equilibria, we obtain Nx ¼ A

lxand Nv ¼ B

lv.

With this assumption, we could scale the human subpopulationsby Nx and the vector subpopulations by Nv . Let xi ¼ xi=Nx fori ¼ 1; . . . ;9 and v j ¼ v j=Nv for j ¼ 1;2. We could obtain the normal-ized system of (2) by replacing xi with xi;v j with v j, and the corre-

sponding parameters A;B; bx with A ¼ ANx;B ¼ B

Nv; bx ¼ bxNv

Nx,

respectively. The rest of parameters remain the same. See Table 1for further detail about parameters description.

3. Model analysis

Throughout this section, we first assume that hiðtÞ � hi andd0 þ diðtÞ � di0 are both constant in time. Then, the system (2)has a disease-free equilibrium

xeq ¼ aþ lx þ d10

K1A;

h1

K1A;

d20h1 þ ðlx þ d20Þðaþ lx þ d10ÞK1 K2

aA;�h2ðaþ lx þ d10Þ þ ðlx þ h2Þh1

K1 K2aA; 0;0;0;0;0;

Blv

;0�

ð5Þ

where K1 ¼ ðaþ lx þ h1Þðaþ lx þ d10Þ � d10 andK2 ¼ lxðlx þ h2 þ d20Þ.

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] withthe known basic reproductive ratio

R00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2bxbvNv

lvðlx þ cÞNx

s: ð6Þ

The basic reproductive ratio R00 represents the expected num-bers of secondary cases produced by a typical infected individualduring its entire period of infectiousness in a completely suscepti-

ble population (see the detail in [6,7]).The parameter ðR00Þ2 is infact a measure of two-stage infection as follows. A single infectedmosquito successfully infects bbx humans per unit time duringtheir infection period 1

lv. Each infected human then infects bbV Nv

Nx

ission diagram.

Fig. 2. Sensitivity of R0 with respect to the treatment rates h1 and h2.

Table 1Variables and parameters used in system (2) and their description. All variables andparameters are assumed to be non-negative.

Variable/parameter

Description Value

x1ðtÞ; x2ðtÞ Untreated/treated susceptible children x1ðtÞ; x2ðtÞP 0x3ðtÞ; x4ðtÞ Untreated/treated susceptible adults x3ðtÞ; x4ðtÞP 0x5ðtÞ; x6ðtÞ Untreated/treated infected and

infectious childrenx5ðtÞ; x6ðtÞP 0

x7ðtÞ; x8ðtÞ Untreated/treated infected andinfectious adults

x7ðtÞ; x8ðtÞP 0

x9ðtÞ Recovered and immune humans x9ðtÞP 0v1ðtÞ; v2ðtÞ Susceptible/infectious vectors v1ðtÞ;v2ðtÞP 0A;B Human/vector recruitment rate 1000

65�365 and 50030

a Rate of transition from children to adults 112�365

h1ðtÞ; h2ðtÞ Rate of treated children/adults ½0; 1�d0 Treatment frequency 1d1ðtÞ; d2ðtÞ Drop-out rate of treated children/adults ½0; 1�lx;lv Natural death rate of humans/vectors 1

65�365 and 130

c Human recovery rate 114

bx; bv Transmission rate in humans/vectors 0.1 and 0.1b Mosquito biting rate 1

D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16 11

vectors per unit time during the infection period 1ðlxþcÞ [3]. This

parameter R00 can be used to estimate whether a new infectionmay end up in an epidemic and to estimate the severity of the epi-demic 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 sys-tem (2) is given by

G ¼

0 d10N2

0 0 fh10N1

0 0 0 0a

N10 0 d20

N4s

0 aN2

h20N3

0 0bBN1

0 bBN3

0 0

0BBBBBBBBB@

1CCCCCCCCCA; ð7Þ

where

N1 ¼ aþ cþ lx þ h10; N2 ¼ aþ cþ lx þ d10;

N3 ¼ cþ lx þ h20; N4 ¼ cþ lx þ d20;

and

bB ¼ bBlxbv

Alv;

f ¼ blxbx

lv� aþ lx þ d10

ðaþ lxÞ ðaþ lx þ h10 þ d10Þ;

s ¼ babx

lv� ðlx þ d20Þ ðaþ lx þ d10Þ þ h10d20

ðaþ lxÞ ðaþ lx þ h10 þ d10Þ ðlx þ h20 þ d20Þ:

The element Gjk represents the number of new infections incompartment j being generated by a single infective from compart-ment k. Note that for j and k from 1 to 5 represents x5; x6; x7; x8 andv2, respectively. For an example, element G12 represent that everysingle infective person from compartment x5 can infect d10

aþcþlxþd10

person in compartment x6. The basic reproductive ratio R0 of thesystem (2) is given by the spectral radius of G, i.e. R0 is the largesteigenvalue of G. From the Perron theorem [13], the largest eigen-value of G is real and positive. Moreover, the characteristic polyno-mial of G is given by

vGðkÞ ¼ k5 � a3k3 � a2k

2 þ a1k� a0;

where the coefficients ai are positive and given by

a3 ¼d20h20

N3N4þ d10h10

N1N2þ fbB

N1þ sbB

N3;

a2 ¼fabB

N1N3;

a1 ¼bB ðfd20h20N2 þ sd10h10N4Þ þ d10d20h10h20

N1N2N3N4;

a0 ¼afh10d20

bBN1N2N3N4

:

The basic reproductive ratio R0 which satisfies vGðR0Þ ¼ 0 maynot be obtained analytically. We find that the disease free equilib-rium is stable if and only if R0 < 1 [6], or equivalently vGðR0Þ > 0 .Before the treatment is given to the population ðh1 ¼ h2 ¼ d1 ¼d2 ¼ 0Þ and if there is no age class division (a ¼ 0), R0 reduces toR00.

The sensitivities of the parameters that affect R0 are shown inFigs. 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 notfeasible in practice due to the limited budget. For instance, in Fig. 2,with a constant rate of h2 in 2:5� 10�3, larger h1 will reduce R0

much better. As a consequence, the budget is quite large becausethe drop-out rate d1; d2ð Þ is quite small only 10�3. For daily applica-tion of control treatment, the rates of treatment and drop-out willdepend on time. In this case, the basic reproductive ratio does notexist. Related to the optimal control analysis, it is shown in thenext section that the result with optimal control is far more effi-cient (in terms of cost as well as outbreak reduction) in comparisonwith constant control.

In the next section we consider the case of the non-autonomoussystem (2), in which the epidemic condition is satisfied before thetreatment, i.e. R0 > 1. Due to further limitation of budget and re-sources the epidemic may not be removed within the admissiblecontrol parameters. In this case, the optimal treatment has to bedesigned such that the total number of the infected individuals iskept as small as possible within admissible control parameters.

4. Optimal control problem

Given the disease model (2) we want to design the treatmentrates h1ðtÞ; h2ðtÞ and the drop-out rates d1ðtÞ; d2ðtÞ such that weminimize the number of the infected individuals. Obviously, thistask can always be achieved by imposing extremely high treatmentrates and minimal drop-out rates. However, high treatment rates

Table 2Parameters value for sensitivity analysis (8).

Parameters Nx Nv b a c lx lv bx bv d1 d2

Value 100,000 5000 1 112�365

114

165�365

130

0.1 0.1 10�3 10�3

Fig. 3. Sensitivity of R0 with respect to a and h1 (left hand side) or h2 (right hand side).

12 D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16

and low drop-out rates increase the costs of the campaign. To finda suitable compromise between minimal number of the infectedindividuals and the costs of the campaign, we consider the follow-ing cost-functional

Jðx;uÞ ¼ 12T

Z T

0

X11

i¼1

xx;ix2i ðtÞ þ

X4

j¼1

xu;ju2j ðtÞdt: ð8Þ

The weighting parameters xx;i and xu;j are used for the statevariables x and the control variables u ¼ fh1ðtÞ; h2ðtÞ; d1ðtÞ; d2ðtÞg,respectively. Since we are mainly interested in minimizing thenumber of the infected humans, we may set xx;i > 0 fori ¼ 5; . . . ;8. The number of susceptible humans, i.e. x1; . . . x4 aswell as the number of recovered individuals x9 and mosquitopopulation v1 and v2 is not taken into account, hence we choosethe weights xx;i ¼ 0 for those compartments. On the other handwe want to minimize the costs for our treatment and campaign.The costs of the treatment use are more-or-less proportional tothe treatment rates u1 and u2, hence we choose the weightsxu;1;xu;2 > 0, as usual. To the second set u3;u4 of control vari-ables, i.e. the drop-out-rates, the situation is just the other wayround: small drop-out rates require high costs and hence wehave to choose negative weights xu;3;xu;4, if we aim to minimizethe costs.

Now, the optimal control task reads as

minu

Jðx;uÞ s:t: Pðx;uÞ ¼ 0; ð9Þ

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, weintroduce the Lagrangian

Lðx;u; zÞ ¼ Jðx;uÞ þ hPðx;uÞ; zi: ð10Þ

The duality product hPðx; uÞ; zi is induced by the L2-inner product

hPðx;uÞ; zi ¼ 1T

Z T

0ð _x�Mðx;uÞÞzdt

¼ xðTÞ � zðTÞ � x0 � zð0Þ þZ T

0�x � _z�Mðx; uÞzdt:

The first order optimality condition states that an optimum of theproblem (9) is a stationary point of L, see [21,12]. Hence@xL ¼ @zL ¼ @uL ¼ 0. Taking the derivative with respect to the ad-joint variable z yields the state system _x ¼ Mðx; uÞ.

Next, taking the derivative with respect to the state variables xyields the adjoint system

_z ¼ �@xMðx;uÞzþ @xJðx;uÞ zðTÞ ¼ 0: ð11Þ

In the case of the disease model (2), we obtain the following set ofequations_z1 ¼ lxz1 þ aðz1 � z3Þ þ bxx11ðz1 � z5Þ þ h1ðz1 � z2Þ; ð12aÞ_z2 ¼ lxz2 þ aðz2 � z4Þ þ ðd0 þ h1Þðz2 � z1Þ; ð12bÞ_z3 ¼ lxz3 þ bxx11ðz1 � z5Þ þ h2ðz3 � z4Þ; ð12cÞ_z4 ¼ lxz4 þ ðd0 þ d2Þðz4 � z3Þ; ð12dÞ_z5 ¼ lxz5 þ aðz5 � z7Þ þ cðz5 � z9Þ þ bvx10ðz10 � z11Þþ h1ðz5 � z6Þ þxx;5x5; ð12eÞ

_z6 ¼ lxz6 þ aðz6 � z8Þ þ cðz6 � z9Þ þ ðd0 þ d1Þðz6 � z5Þþxx;6x6; ð12fÞ

_z7 ¼ lxz7 þ cðz7 � z9Þ þ bvx10ðz10 � z11Þ þ h2ðz7 � z8Þþxx;7x7; ð12gÞ

_z8 ¼ lxz8 þ cðz8 � z9Þ þ ðd0 þ d2Þðz8 � z7Þ þxx;8x8; ð12hÞ_z9 ¼ lxz9; ð12iÞ_z10 ¼ lvz10 þ bvðx5 þ x7Þðz10 � z11Þ; ð12jÞ_z11 ¼ lvz11 þ bxðx1ðz1 � z5Þ þ x3ðz3 � z7ÞÞ: ð12kÞAll of these equations are subject to the condition ziðTÞ ¼ 0.

Finally, taking the derivative with respect to the control variableu yields the gradient equation

@uJðx;uÞ � @uMðx;uÞz ¼ 0: ð13ÞIn the setting of the disease model (2) this reads as

xu;1u1 þ x1ðz1 � z2Þ þ x5ðz5 � z6Þ ¼ 0; ð14aÞxu;2u2 þ x3ðz3 � z4Þ þ x7ðz7 � z8Þ ¼ 0; ð14bÞxu;3u3 � x2ðz1 � z2Þ � x6ðz5 � z6Þ ¼ 0; ð14cÞxu;4u4 � x4ðz3 � z4Þ � x8ðz7 � z8Þ ¼ 0: ð14dÞ

Table 3Weight parameters in Eq. (8).

Weightparameters

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 4Initial conditions for both scenario.

Scenario Compartment

x1 x2 x3 x4 x5 x6 x7 x8 x9 v1 v2

Prevention 175 0 822 0 1 0 1 0 1 499 1Reduction 150 0 794 0 5 0 50 0 1 480 20

0 50 100 1500

5

10

15

20

25

Day

Infe

cted

Com

partm

ent

Total infected comp. before treatmentsTotal infected comp. with optimal control

Fig. 4. Dynamics of the infected compartment ðx5 þ x6 þ x7 þ x8Þ in the preventionscenario.

980

1000

nt

D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16 13

Solving these equations for the vector �u, we obtain the direction ofthe negative gradient DuJðx;uÞ at the point ðx;uÞ.

With these ingredients at hand, we are now in position to setupa gradient method for solving the minimization problem.

1. Choose some initial guess uð0Þ for the control.2. Solve the state Eq. (2) to compute x.3. Compute the actual cost J.4. Solve the adjoint Eq. (12) to compute z.5. Solve the gradient Eq. (14) to compute the negative gradient

DuJ.6. Solve an approximate line search to find s�. minsJðxðusÞ;usÞ,

where us ¼ uþ s � DuJðx;uÞ.7. Set u ¼ uþ s� � DuJðx;uÞ.8. Go to 2, unless termination criteria are met.

This algorithm in pseudo-code requires some additionalcomments.

Steps 2 and 4 require the solution of a system of ordinary differ-ential equations by any suitable numerical scheme. The state equa-tion has to be solved forward in time, integrating from 0 to T whilethe adjoint equation requires backward integration from T downto0.

Step 6 is one of the crucial steps within the algorithm. Of course,performing an exact line search, i.e. determining s such that

mins

JðxðusÞ;usÞ ¼ JðxðusÞ;usÞ

would be the optimal choice regarding accuracy. However, deter-mining s usually requires many evaluations of J, i.e. one has to solvethe state equation quite a number of times. To avoid the time con-suming line search one has to resort to some heuristics. The sim-plest one is using a fixed stepsize s0 in the direction of thenegative gradient. Being very simple, this method faces conver-gence problems. Another, quite popular, choice is the Armijo-rule[14], starting with a stepsize s0, one tries the sequence bks0 for some0 < b < 1 of step sizes, until the cost functional is reduced for thefirst time. The reasoning behind this heuristics is the following.The overall aim is to reduce the cost functional in each step to avoidoscillations. Knowing, that going a small step in the direction ofnegative gradient will always lead to a decrease of the cost func-tional, we have to reduce step sizes that are too large and lead toan increase of J. More advanced methods for determining a suitablevalue s� can be thought of, including local polynomial approximationsof the cost functional, see [14] for further details.

0 50 100 150880

900

920

940

960

Day

Susc

eptib

le C

ompa

rtme

Total susceptible comp. before treatmentsTotal susceptible comp. with optimal control

Fig. 5. Dynamics of the susceptible compartment ðx1 þ x2 þ x3 þ x4Þ in the preven-tion scenario.

5. Numerical results

In this section we present some numerical simulations to theoptimal control problem. Let xx;i; i ¼ 1;2 . . . 9 be the weight for so-cial cost in human (hospitalization), xx;i; i ¼ 10;11 for mosquitoand xu;j; j ¼ 1;2 be the weight cost in treatment rates for childrenand adults. For j ¼ 3;4, we define xu;j as the weight for the govern-ment campaign cost on the importance of repellent for childrenand adults, respectively. The choice of these weights reflects thedifferent scales of the costs for counselling and treatment. Herewe used xu;j ¼ ð1; 2; �7; �10Þ for weight in control parametersand the following weights in Table 3 for state variables in the costfunctional (8). The weight control treatment parameter for adults(xu;2) is twice as large as that for children (xu;1). This is relatedto the fact that children are much easier to target for treatmentthan adults.

Numerical simulations of the optimal control problem are car-ried out for two scenarios, namely the case of prevention and thecase of epidemic reduction. The case of prevention occurs at theearly state of transmission, i.e. when the number of infected people

is still relatively small before the start of the treatment. In the caseof epidemic reduction, the treatment starts during the outbreakperiod, i.e. when the number of infected people is significantly

0 50 100 1500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Day

Trea

tmen

t rat

e

θ1

θ2

0 50 100 1500.1

0.1001

0.1002

0.1003

0.1004

0.1005

0.1006

0.1007

0.1008

Day

Dro

p−ou

t rat

e

δ1δ2

Fig. 6. Treatment and drop out rates in the prevention scenario.

0 50 100 15025

30

35

40

45

50

55

60

65

70

Day

Infe

cted

Com

partm

ent

Total infected comp. before treatmentsTotal infected comp. with optimal control

Fig. 7. Dynamics of the infected compartment ðx5 þ x6 þ x7 þ x8Þ in the reductionscenario.

0 50 100 150300

400

500

600

700

800

900

1000

Day

Susc

eptib

le C

ompa

rtmen

tTotal susceptible comp. before treatmentsTotal susceptible comp. with optimal control

Fig. 8. Dynamics of the susceptible compartment ðx1 þ x2 þ x3 þ x4Þ in the reduc-tion scenario.

14 D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16

large. As an initial condition, we assume that in the field the ratiobetween the mosquito and human population is 1

2. With this ratio,the corresponding basic reproductive ratio is about 1.414 beforethe start of the treatment. The initial conditions for both scenariosare shown in Table 4.

In the case of prevention, the main goal of the treatment is toprevent the occurrence of an epidemic in the population or to re-duce the intensity of the epidemic if it already started. Hence wewant to suppress the outbreaks as much as possible within the gi-ven budget constraints. The dimensionless value of the cost func-tional (8) for one unit period of treatment is 4829. Simulationresults for total infected compartment and total susceptible com-partment in the prevention case are shown in Figs. 4 and 5, respec-tively. It can be seen in Fig. 7, with the related optimal controlfunction shown in Fig. 6, the total infected population at the peakof the outbreak could be suppressed to a level two times smallerthen without treatment along with delaying of the outbreak.

For the case of epidemic reduction, the optimal controls areshown in Fig. 9. In the epidemic reduction scenario, the treatmentrate of the population is higher than in the prevention case to pre-vent the return of the outbreak. The achieved reduction of the total

number of infected persons and the susceptible population as aconsequence of the high rate of the treatment is shown in Figs. 7and 8. In this scenario, the total cost for one unit period of treat-ment equals 90;440 being almost twenty times greater than theminimal cost in the prevention scenario. Hence, prevention of thedisease 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 correspondingweight cost for control parameters h1 and d1 are 2 and �10 respec-tively, and weight parameters for state variables are given in Ta-ble 5. The state variables x3; x4; x7; x8 and the control variablesh2; 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 epidemicreduction scenario are shown in Fig. 10, left and right, respec-tively. The corresponding actual control function for the preven-tion case is shown in Fig. 11. It can be seen that although thedynamic of the infected compartment is only slightly different be-tween the models with and without age class, the total cost foreach case is significantly different, i.e. 4829 and 5528, respec-tively. Similarly, in the epidemic reduction case (see Fig. 12),

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Day

Trea

tmen

t rat

eθ1θ2

0 50 100 1500.1

0.102

0.104

0.106

0.108

0.11

0.112

Day

Dro

p−ou

t rat

e

δ1δ2

Fig. 9. Treatment and drop out rates in the epidemic reduction scenario.

Table 5Weight parameters when a ¼ 0.

Weight parameters Compartment

x1 x2 x5 x6 x9 v1 v2

xx;i 0 0 400 400 0 0 0

0 50 100 1500

5

10

15

20

25

Day

Infe

cted

com

partm

ent

Total infected with no age structured and before treatmentsTotal infected with no age structured, with treatmentTotal infected with age structured and treatment

0 50 100 15020

25

30

35

40

45

50

55

60

65

70

Day

Infe

cted

com

partm

ent

Total infected with no age structured and before treatmentsTotal infected with no age structured, with treatmentTotal infected with age structured and treatment

Fig. 10. Dynamics of the infected compartment in prevention (left) and epidemic reduction case (right).

0 50 100 1500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Day

Trea

tmen

t rat

e

θ1 with age structured

θ2 with age structured

θ1 with no age structured

Fig. 11. Treatment and drop out

D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16 15

the corresponding costs are 90,440 and 98,906 respectively. Thesignificant difference in the cost value here is due to the absenceof a children’s compartment in the case with no age class, withthe consequence that everybody is treated using the adult rate(which is more expensive).

0 50 100 1500.1

0.1001

0.1002

0.1003

0.1004

0.1005

0.1006

0.1007

0.1008

Day

Dro

p−ou

t rat

e

δ1 with age structured

δ2 with age structured

δ1 with no age structured

rates in prevention scenario.

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Day

Trea

tmen

t rat

e

θ1 with age structured

θ2 with age structured

θ1 with no age structured

0 50 100 1500.1

0.102

0.104

0.106

0.108

0.11

0.112

0.114

Day

Dro

p−ou

t rat

e

δ1 with age structured

δ2 with age structured

δ1 with no age structured

Fig. 12. Treatment and drop out rates in the epidemic reduction scenario.

16 D. Aldila et al. / Mathematical Biosciences 242 (2013) 9–16

6. Conclusion

We have presented a two-age class model for Dengue transmis-sion with mosquito repellent as a control mechanism. The progressof the outbreak of the disease is influenced by several controlparameters, namely the treatment and the drop-out rates for bothchildren and adults. The basic reproductive ratio before the treat-ment is obtained implicitly as the positive root of the fifth ordercharacteristic polynomial of the next generation matrix.

An optimal control problem has been formulated for the aboveDengue transmission model. We have derived a gradient algorithmto solve the optimization problem. For two typical scenarios, theprevention of an outbreak and the reduction of an existing epi-demic, we have carried out numerical simulations to computethe optimal treatment and drop-out rates. The significance of theage structure is indicated in the calculation of the optimal cost.The higher cost value in the case with no age structure is simplydue to the use of adults unit treatment cost for all persons.

With a limited budget, it is much better to apply the treatmentwell before the occurrence of the outbreak. This can be done if it isknown that an epidemic of Dengue will take place, assuming thatthe ratio between the mosquito and human population is knownand hence the basic reproductive ratio can be estimated. This workcould be considered as a first approach to finding a simple treat-ment for a complicated Dengue transmission problem. Further re-search could study integrated treatments for human as well as forvectors. In a more diverse population, the impact of spatial heter-ogeneity may also be taken into account.

Acknowledgements

The authors would like to thanks the referees for their valuablecomments and suggestions. Parts of the research are funded by theInternational Research Grant of the Indonesian Directorate Generalfor Higher Education and the German Academic Exchange Service(DAAD).

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