aedes aegypti (diptera: culicidae) with control · 312 juli an alejandro olarte garc a et al....

Applied Mathematical Sciences, Vol. 12, 2018, no. 7, 311 - 325HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2018.8120

A Model for the Population Dynamics of the Vector

Aedes aegypti (Diptera: Culicidae) with Control

Julian Alejandro Olarte Garcıa, Anıbal Munoz Loaizaand Carlos Alberto Abello Munoz

Grupo de Modelacion Matematica en Epidemiologıa (GMME)Facultad de Educacion

Universidad del QuindıoArmenia-Quindıo, Colombia

Copyright c© 2018 Julian Alejandro Olarte Garcıa et al. This article is distributed under

the Creative Commons Attribution License, which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

Abstract

The epidemics of dengue, zika, chikunguya and yellow fever can beprevented by fighting against their main vector Aedes aegypti. This pa-per presents a mathematical model described by a set of ordinary non-linear differential equations, which depend on the dynamics of the vectorand uses the theory of optimal control considering methods and associ-ated costs, application of insecticides, environmental sanitation and useof vector traps, to deduce the most sustainable strategy that eliminatesthe population of female mosquitoes. Optimal control is achieved byapplying the Pontryagin Maximum Principle.

Keywords: Aedes aegypti, optimal control, Pontryagin Maximum Princi-ple, simulation

1 Introduction

Historically, World Health Organization efforts to control vectors in the Ameri-cas region resulted in the elimination of Aedes aegypti populations in many ofthe tropical and subtropical countries by the 1970s, however, as is often thecase in the field of public health, when a health threat disappears, the controlprogram ceases to exist. Resources dwindled, control programmes collapsed,

312 Julian Alejandro Olarte Garcıa et al.

infrastructures dismantled, and fewer specialists were trained and deployed.The mosquitoes and the diseases that they transmit returned with force to anenvironment in which they remained few intact defenses [6].

The consequences of this dramatic reappearance are best illustrated by therecent history of dengue, zika, chikungunya, and yellow fever and recognizingthe limited existence of effective drugs and vaccines for the treatment of al-most all these diseases, prevention of infections and control of vectors becomea component essential for reducing the burden of vector-borne diseases [17].Therefore, the main purpose of most programs contemporary is to reduce thedensities of vector populations as much as possible and keep them at low le-vels, and where feasible, efforts should also be made to reduce the longevity ofadult female mosquitoes [11].

There are several control procedures used to inhibit the proliferation ofthe vector, all of which have their pros and cons and they can be classified asmechanical, chemical and biological control [16]. In mechanical control, publichealth officials are responsible to visit households and destroy, alter, removeor recycle non-essential containers that can harbor eggs, larvae or pupae of thevector. The chemical control uses pesticide on surface treatments with residualeffect or as spatial treatments and aims to eliminate the adult and inmaturemosquitoes. Although very effective, it can kill different mosquitoes and causehealth problems in population; however, choosing those products that are safer,highly efficient, with a very low degree of toxicity and with minimal or nopossibility of contamination of the environment usually generate acceptanceof communities [9]. Biological control consists of the introduction of livingorganisms that feed, compete, eliminate and parasitize the immature vectorstates in the water deposits. Another method of control is vectorial traps,which although originated for entomological surveillance, in several countriesthey have been modified to eliminate immature or adult populations of A.aegypti [6, 16].

In all these vector control efforts, minimizing labor and costs while maxi-mizing reduction of mosquitoes is a priority and control mathematical modelsare a very valuable tool for identifying the optimum solution. Optimal con-trol theory has been used successfully to make decisions involving biologicalor medical models, whose desired outcome, goal and performance of controlactions depend on the particular situation [12]. For example, Rodrigues, Mon-teiro and Torres (2010) present an application of the optimal control theoryto dengue epidemics, where the cost functional depends not only on the costsof medical treatment of the infected people but also on the costs related toeducational and sanitation campaigns [4]. Thome, Yang and Esteva (2010) an-alyze the minimal effort to reduce the fertile female mosquitoes, searching forthe optimal control by means of the Pontryagin’s Maximum Principle, whensterile male mosquitoes (produced by irradiation) are introduced as biological

A model for the population dynamics of the vector Aedes aegypti ... 313

control, besides the application of insecticide [10].Dias, Wanner and Cardoso (2015) analyze the dengue vector control pro-

blem in a multiobjective optimization approach, in which the intention is tominimize both social and economic costs, using a dynamic mathematical modelrepresenting the mosquitoes population [18]. Rafikov, Rafikova and Yang(2015) formulate an infinite-time quadratic functional minimization problem ofA. aegypti mosquito population and they analyze different scenarios in whichthe three efforts of management of the population (chemical insecticide control,sterile insect technique control, and environmental carrying capacity reduc-tion) are combined in order to assess the most sustainable policy to reduce themosquito population [7]. Masud, Kim, B. and Kim, Y. (2017) analyzed optimalimplementations of control with specific prevention measures for dengue trans-mission in the standard host(humans)-vector(mosquito) SIR-SI model [8].

In this article, we consider the model proposed in [5] and extend it to amathematical control model that describes the dynamics of the population oftwo states –aquatic and aerial– of A. aegypti, covering only female mosquitoes(since they represent the threat in the transmission of diseases), whose formsof intervention to reduce the infestation consist of environmental sanitationand the application of insecticides, combined with the use of vector traps.

2 The model

The mathematical model describes the dynamics of an Aedes aegypti po-pulation (female only) when a certain control program implements integratedeconomically sustainable methods that have an impact on the reproductionand ecological plasticity of mosquitoes. Here, we consider the life cycle of thefemale mosquito population divided into two compartments: the aquatic phaseor immature stages (egg, larva and pupa) and the aerial phase or adult stages.

We denote as x1(t): average number of adult mosquitoes at time t, x2(t):average number of immature mosquitoes at time t, and Ci(t) for i = 1, 2, 3:average number of epidemiologically important breeding places at time t. Thelife cycle of the mosquito and their breeding sites involve the following constantparameters: ω, rate of development of the immature stage to the adult stage; ε,mortality of the mature mosquito; φ, rate of oviposition of female mosquitoes;f , fraction of eggs that give birth to female mosquitoes; π, mortality rate ofimmature stages; ri, i = 1, 2, 3, intrinsic growth rate of the i-th breeding site;Ki, carrying capacity of the breeding site in category i in the environment.

The two management efforts of the mosquito population are insecticidesand lethal traps, which we will call control perifocal ; or insecticides and envi-ronmental sanitation, which we will call control focal. The application of thefocal control, u2(t), implies to proportionally decrease the variable x2(t) perunit time; and the variable x1(t) decreases proportionally per unit time, due


to the application of the perifocal control, u1(t). The system of differentialequations that describes the growth dynamics of the A. aegypti populationusing these controls is:

x1(t)

dt= ωx2(t)− εx1(t)− u1(t)x1(t) ≡ h1(·)

x2(t)

dt= fφx1(t)

(1− x2(t)

K(t)

)− (π + ω)x2(t)− u2(t)x2 ≡ h2(·)

(1a)

{Ci(t)

dt= ri

(1− Ci(t)

Ki

)Ci(t) ≡ h3(·) (i = 1, 2, 3) (1b)

with {ω, ε,φ, π, θi, ri,Ki} ⊂ R+, f ∈ (0, 1), u1(t) ∈ [0, 1], u2(t) ∈ [0, 1] andinitial conditions:

x1(0) = x10, x2(0) = x20, Ci(0) = Ci0 (i = 1, 2, 3)

The trajectories of the solutions of the system exist in the compact and posi-tively invariant set:

{xxx>=

x1

x2

C1

C2

C3

∈ R5+ : 0 ≤ xj ≤

(ωε

)2−jK, j = 1, 2; 0 < Ci ≤ Ki, i = 1, 2, 3

},

in which K =3∑i=1

θiKi > 0 denotes the fixed carrying capacity for the popu-

lation of immature mosquitoes. This expression represents the maximumamount of immature forms that can lodge the three types of more produc-tive breeding places in a locality, taking into account for each category ofcontainer its carrying capacity (Ki) and its average pupal productivity (θi).

The model (1a)-(1b) can be simplified in the following way. First, thecompact equation for the three types of breeding sites can be decoupled fromthe system. Secondly, the size of K(t) is controlled by human intervention andindependent of the mosquito population. In fact, dependent on the analyticalsolution of the nonlinear equation (1b), K(t) is given by:

K(t) =3∑i=1

θiCi0Ki

Ci0 + (Ki − Ci0)e−rit(i = 1, 2, 3)

Thus, assuming that ri > 0, K(t) will converge (rapidly or not, dependingon ri) to the maximum capacity of eggs-larvae-pupae in the middle, K, then


system (1a)-(1b) become the following autonomous system:x1

dt= ωx2(t)− εx1(t)− u1(t)x1(t) ≡ h1(·), x1(0) = x10

x2

dt= fφx1(t)

(1− x2(t)

K

)− (π + ω + u2(t))x2(t) ≡ h2(·), x2(0) = x20

(2)

The simplest model will be analyzed in the compact subset:{zzz>=

[x1

x2

]∈ R2

+: 0 ≤ x1 ≤ω

εK, 0 ≤ x2 ≤ K

}(3)

2.1 Controllability

Some systems can be brought into a position of equilibrium, by applying aninput and after a finite period of time. In order to know if the solution to aproblem of design of a control system exists, one must arrive at specific con-clusions regarding its controllability. Since there is no theory of controllabilityapplicable to non-linear systems, the system (2) is linearized in the neighbor-hood of the stable stationary solution zzz =

[x1, x2, x3

], considering deviations

of said stationary solution and the deviation of the control uuu =[u1, u2

]:

∆z∆z∆z = zzz − zzz, ∆u∆u∆u = uuu− uuu

to have the system in deviations of the form:

(∆z∆z∆z)> = A(∆z∆z∆z)> +B(∆u∆u∆u)> (4)

equivalently,

d

dt

[∆x∆y

]=

[a11 a12

a21 a22

] [∆x∆y

]+

[b11 00 b21

] [∆u1

∆u2

]where

a11 =∂h1(zzz, uuu)

∂x1

= −(ε+ u1), a21 =∂h2(zzz, uuu)

∂x1

= fφ

(1− x2

K

),

a12 =∂h1(zzz, uuu)

∂x2

= ω, a22 =∂h2(zzz, uuu)

∂x2

= −fφx1

K− (π + ω + u2),

b11 =∂h1(zzz, uuu)

∂u1

= −x1, b22 =∂h3(zzz, uuu)

∂u2

= −x2.

The system (4) is said to be completely controllable if the state of the systemcan be transfered from the zero state zzz0 at any initial time t0 to any terminalstate zzzf within a finite time tf− t0. Here, when we say that the system can he


transferred from one state to another, we mean that there exists a piecewisecontinuous input uuu ∈ L2([0, tf];R2), 0 ≤ t ≤ tf, which brings the system fromone state to the other [2].

A necessary and sufficient condition, due to R. E. Kalman (1956), for com-plete controllability of the state of a finite-dimensional linear system withinvariant matrices A and B, similar to (4), is that

rg [B AB . . . An−1B] = n (5)

where [B AB . . . An−1B] is the Kalman controllability matrix [2]. As analternative to this condition, one may determine the number of columns of C(or rows) that are linearly independent [13]. For the system (2), the contro-llability matrix is

C =

[b11 0 a11b11 a12a21

0 b22 b11a21 b21a22

]This matrix has a non-zero diagonal because bii = xi 6= 0. Then the tworows of C are non-zero and form a linearly independent set, or equivalently,the first two columns of C contain pivot and are then linearly independent.Therefore, rgC ′ = rgC = 3. As the dimension of the linear system of state(4) is 3 = rgC, then said system is completely controllable.

3 The optimal control problem

In this section, we use the optimal control theory to analyze the populationgrowth of Aedes aegypti. Our objective is to solve the following problem: giventhe initial population size in the two compartments, x1, x2 and x3, find the beststrategy in terms of combined efforts, insecticides, lethal traps and suppressionof mosquito breeding foci that would minimize the number of mosquitoes in aninfested area, while also keeping the execution of the control program at lowcost.

Mathematically, the problem is to minimize the following functional objec-tive:

J(uuu(t)) =

∫ tf

0

(x1(t) + x2(t) +

κ0

2u2

1(t) +κ1

2u2

2(t))dt (6)

where the coefficients κ0, κ1 > 0 are constant weights on the controls, andui ∈ L2

([0, tf]; [0, 1]

), subject to the initial value problem (2). In other words,

it is a question of determining a vectorial function uuu∗ = uuu∗(t) in the set ofpermissible functions Γ such that J(uuu∗) ≤ J(uuu), ∀uuu ∈ Γ, or equivalently:

J(u∗1, u∗2) = minuuu∈Γ

J(u1, u2) (7)


donde Γ ={uuu ∈ L2

([0, tf];R2

): uuu = [u1,u2], 0 ≤ ui ≤ 1

}.

This functional objective includes the costs that health providers have bud-geted for the application of control(s) during a specified period of time; κ1u

22 is

the necessary cost to carry out focal control activities and κ0u21 is the necessary

cost to carry out perifocal control activities.The basic framework of an optimal control problem is to prove the existence

of the optimal control and then characterize the optimal control through theoptimality system.

Proposition 3.1 Given the control problem with system (2), there existsuuu∗ = [u∗1, u∗2] ∈ Γ such that min

uuu∈ΓJ(uuu) = J(uuu∗).

Proof. The existence of an optimal control pair can be tested using theresults of Fleming and Rishel ([15], Theorem 4.1.), which asks to check thefollowing hypothesis. i) The set of controls and corresponding state variablesis not empty: a result of existence by Lukes ([1], Theorem 9.2.1) gives theexistence of the system solution (2) with bounded coefficients. ii) The controlset is convex and closed by definition. iii) The right side of the system (2)is bounded by a linear function in the state and the control: since the statesolutions are a priori bounded. iv) The integrand in the functional target(6), L, is clearly convex in Γ and, moreover, there are constants δ0, δ1 > 0and σ > 1 such that L satisfies L(t, zzz(t),uuu(t)) ≥ δ0|uuu(t)|

σ2 − δ1, just choose

δ0 =1

2min{κ0,κ1}, σ = 2 and δ1 > 0 is arbitrary.

3.1 Maximum Principle

In 1956 S. L. Pontryagin established conditions necessary to find optimal con-trols, what is known as the Maximum Principle [14]. This principle converts(2),(6) and (7) into the problem of minimizing pointwise a Hamiltonian, H,with respect to u1 and u2:

H(zzz(t), uuu(t), λλλ(t)

)= L

(zzz(t), uuu(t)

)+

2∑i=1

λihi

where zzz(t) =[x1(t), x2(t)

]is the vector of state variables, uuu(t) =

[u1(t),

u2(t)]

is the vector of controls, λλλ(t) =[λ1(t), λ2(t)

]is the vector of adjoint or

conjugated variables, hi is the right side of the ith differential equation of thesystem (2) and L

(zzz(t), uuu(t)

)is the integrand defined in (6). Namely,

H(zzz, uuu, λλλ

)= x1 + x2 +

1

2

(κ0u

21 + κ1u

22

)+ λ1

(ωx2 − (ε+ u1)x1

)= + λ2

(fφx1

(1− x2

K

)− (π + ω + u2)x2

)


We obtain the optimality condition with the help of the Lagrangian, whichis formed by adding a penalized term to the criterion. So, the LagrangianL(zzz,uuu,λλλ

)is

L(zzz,uuu,λλλ

)= H

(zzz,uuu,λλλ

)+

2∑i=1

ρiui +2∑i=1

ρi+2(1− ui),

where ρj ≥ 0 (j = 1, 2, 3, 4) are multipliers of penalty and satisfy

ρiui = 0, ρi+2(1− ui) = 0 (i = 1, 2) (8)

By applying the Pontryagin’s maximum principle [14] and the existence of anoptimal control, we obtain the following proposition:

Proposition 3.2 If u∗1 and u∗2 is an optimal control couple that minimizesJ(u1,u2) over Γ with corresponding states x∗1 and x∗2, then there exists adjointfunctions λ1 and λ2 verifying

dλ1

dt= (ε+ u1)λ1 − fφ

(1− x2

K

)λ2 − 1

dλ2

dt= −ωλ1 +

(fφx1

K+ π + ω + u2

)λ2 − 1

with the transversality conditions: λ1(tf) = λ2(tf) = 0. Futhermore, the opti-mal control uuu∗ = [u∗1,u∗2] is given by

uuu∗ =

[min

{max

{0,λ1

κ0

x∗1

}, 1

}, min

{max

{0,λ2

κ1

x∗2

}, 1

}](9)

Proof. The adjoint equations and the transversality conditions can beobtained using

dλλλ

dt= −Hzzz

(zzz,uuu,λλλ

), λλλ(tf) = 0

From the first order condition, Luuu = 0, the optimal control is solved. This is,

κi−1ui − λixi + ρi − ρi+2 = 0 (i = 1, 2)

or equivalently,

u∗i =λix∗i − ρi + ρi+2

κi−1

(i = 1, 2) (10)

According to (8), we distinguish three cases:


(i) On the set {t : 0 < u∗i (t) < 1}; ρi = 0, ρi+2 = 0. Replacing in (10),

u∗i =λiκi−1

x∗i .

(ii) On the set {t : u∗i (t) = 0}; ρi ≥ 0, ρi+2 = 0. Replacing in (10),ρiκi−1

=λiκi−1

x∗i ≥ 0.

(iii) On the set {t : u∗i (t) = 1}; ρi = 0, ρi+2 ≥ 0. Replacing in (10),ρi+2

κi−1

= 1− λiκi−1

x∗i ≥ 0.

Clearly by case (i), the inequalities of cases (ii) and (iii) must satisfy 0 <λiκi−1

x∗i

< 1, for which it is necessary, with respect to case (ii) that u∗i = min

{λiκi−1

x∗i , 1

},

and with respect to case (iii) that u∗i = max

{0,

λiκi−1

x∗i

}. The joint analysis of

the three cases allows us to characterize the control u∗i = u∗i (t) for i = 1, 2 ofthe form (9).

3.2 Optimality System

The optimality system arises by incorporating the state system with giveninitial conditions and the adjoint system with the transversality conditions,coupled with the characterization of the control vector. Thus, we have

x1 = ωx2 − (ε+ u1)x1

x2 = fφx1

(1− x2

K

)− (π + ω + u2)x2

λ1 = (ε+ u1)λ1 − fφ(

1− x2

K

)λ2 − 1

λ2 = −ωλ1 +

(fφx1

K+ π + ω + u2

)λ2 − 1

u∗1 = min

{max

{0,λ1

κ0

x∗1

}, 1

}, u∗2 = min

{max

{0,λ2

κ1

x∗2

}, 1

}Initial conditions: x10 = x1(0), x20 = x2(0).Terminal conditions: λ1(tf) = 0, λ2(tf) = 0.

(11)

4 Numerical illustration

Numerical solutions to the optimality system (11) are executed using MA-TLAB with the parameter values and initial conditions reported in la Table 1.It is important note that the parameters values were chosen such that the total


population never goes into extinction and it yields fφωε(π+ω)

> 1 in the absence

of control strategies (i.e. when u1 = u2 = 0). The entomological parametersof the vector were determined by evaluating the polynomial functions of Yanget al. (2009) at 22◦C derived from empirical data [3], while the remainingparameters are hypothetical.

An iterative method is used for solving the optimality system. We start tosolve the state equations with a initial guess for the controls over the simu-lated time using a forward fourth order Runge-Kutta scheme. Because ofthe transversality conditions, the adjoint equations are solved by a backwardfourth order Runge-Kutta scheme using the current iteration solution of thestate equations. Then, the controls are updated by using a convex combinationof the previous controls and the value from the characterization (9). This isdone iteratively until we obtain convergence. For more detail see [12].

Table 1: Parameters for the life cycle model. modelmmodlmldl. ofParameters and initial conditions for the life cycle model and their values.o.f

Parameter φ ε π ω

Value 3.985816 0.036082704 0.042800864 0.1017712

Parameter f x10 x20 K

Value 0.5 4× 106 2× 106 3.2× 106

Figure 1-(I) represents the population of immature mosquitoes in the systemwithout control (dashed line) and (2) (continuous line) using perifocal controlas the only elimination method. We appreciate in the scenario with control,that the aquatic phase increases until about the 30th hour of development andsurvival of the offspring and reaches its maximum number of x2(t) = 2750000,then decreases severely, reaching x2(t) = 1055500 on the 30th day when thecontrol disappears. In omission of any of the controls, the aquatic phase growsprolifically and begins to stabilize after the 30th hour.

Figure 1-(II) represents the population of mature mosquitoes in the systemwithout control (dashed line) and (2) (continuous line) using perifocal controlas the only elimination method. We appreciate in the scenario with control,that the maximum density of the aerial phase is x2(0) = 4000000 during thewhole vector control campaign, then it drops sharply in the period 0-5 daysand, in the course of the next 25 days, flocks of mosquitoes are slowly disap-pearing. In omission of any of the controls, the aerial phase grows prolificallyfrom time zero.

In Figure 2-(III) we see that before the marginal application of focal control,the number of immature mosquitoes increases rapidly at the beginning, untilday 1.25, but its growth becomes quite slow later; after the application of


the focal control, the size of the aquatic phase increases during the entireimplementation period, more quickly until before the first day and at the endof the control campaign. The difference detected between the two scenarios isthat the focal control manages to reduce the density of the aquatic phase in anon-controlled model by up to 26%.

In Figure 2-(IV) we see that with the application of focal control as the onlycontrol measure against any control effort, the number of mature mosquitoesincreases monotonously within 30 days of the eradication campaign. The no-table difference between the two scenarios is that the focal control managesto reduce the density of the aerial phase in the non-controlled model by up to20%.

(I) (II)

0 5 10 15 20 25 30

Time(day)

1

1.5

2

2.5

3

3.5

Aq

uat

ic p

has

e

#106

Without control With control

0 5 10 15 20 25 30

Time(day)

0

1

2

3

4

5

6

7

8

Aer

ial p

has

e

#106

Without controlWith control

Figure 1: The graphs show densities of mature x1(t) and immature x2(t) mosquito popu-

lations with perifocal control and without control.

(III) (VI)

0 5 10 15 20 25 30

Time(day)

2

2.2

2.4

2.6

2.8

3

3.2

Aq

uat

ic p

has

e

#106

Without controlWith control

0 5 10 15 20 25 30

Time(day)

4

4.5

5

5.5

6

6.5

7

7.5

Aer

ial p

has

e

#106

With controlWithout control

Figure 2: The graphs show densities of mature mosquitoes x1(t) and immature x2(t) with

focal control and without control.


The controls u1(t) and u2(t) plotted in Figure 3 are functions of time withκ0 = 1 and κ1 = 1, respectively. The control u1(t) depends largely on theadjoint variable λ1(t) and the values of κ0; the control u2(t) depends to a largeextent on the adjoint variable λ2(t) and the values of κ1.

(V) (VI)

0 5 10 15 20 25 30

Time(day)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Co

ntr

ol

Control u1(t)

Control u2(t)

0 5 10 15 20 25 30

Time(day)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Co

ntr

ol

Control u2(t)

Control u1(t)

Figure 3: Graphs of control functions applying (V): focal control without perifocal control;

(VI): perifocal control without focal control.

Figure 4 illustrates the responses of the evolutionary phases to the samecontrol method; in subfigure (VII), x2(t) (dashed line) and x1(t) (solid line)were mapped under the action of the perifocal control, and in subfigure (VIII),x2(t) (dashed line) and x1(t) (solid line) were mapped under the action of thefocal control. We observe that in situation (VII) both curves decline from day1.25, but x1(t) becomes slowly smaller and reduces its distance of x2(t), whichexceeds it by 750000 at the end of the campaign. Very different is the situation(VIII) where both curves rise from the first day, showing a future tendency torecover its steady state beyond the thirtieth day.

Figure 5 shows how the subpopulations decrease due to integrated control(focal-perifocal) or perifocal control only. The continuous curve reveals thatboth subpopulations fall to zero level in an approximate time of 11 days.With respect to the segmented curve, it can be seen that the densities ofthe immature phase and the adult phase are greater by implementing a singlemeasurement than integrating both controls. The abysmal dissimilarity occursbetween the curves of subfigure (IX), while the curves of subfigure (X) remainalmost the same for the first 30 hours and separate almost parallel.

The controls u1(t) and u2(t) plotted in Figure 6 are functions of time withκ0 = κ1 = 1, respectively. The control u1(t) depends largely on the adjointvariable λ1(t) and the values of κ0; the control u2(t) depends to a large extenton the adjoint variable λ2(t) and the values of κ1.


(VII) (VIII)

0 5 10 15 20 25 30

Time(day)

0

0.5

1

1.5

2

2.5

3

3.5

4

Su

bp

op

ula

tio

n

#106

Aquatic phaseAerial phase

0 5 10 15 20 25 30

Time(day)

2

2.5

3

3.5

4

4.5

5

5.5

6

Su

bp

op

ula

tio

n

#106

Aquatic phaseAerial phase

Figure 4: The graphs represent the biological phases of A. aegypti in the same cartesian

plane using a single control in each situation: (VII) perifocal control, (VIII) focal control.

(IX) (X)

0 5 10 15 20 25 30

Time(day)

0

0.5

1

1.5

2

2.5

3

Aq

uat

ic p

has

e

#106

0 5 10 15 20 25 30

Time(day)

0

0.5

1

1.5

2

2.5

3

3.5

4

Aer

ial p

has

e

#106

Figure 5: Each graph represents a single biological phase of A. aegypti in the same cartesian

plane with combined controls (continuous line) and perifocal control (dotted line).

5 Conclusion

In this document, we study the optimal combination of control strategies tolead to the reduction of Aedes aegypti infestation within a specific period andarea. We consider a stratified model in the aquatic and aerial phases of thelife cycle and its densities, including focal and perifocal controls as measuresagainst infestation by public health agents. We apply the Pontryagin Maxi-mum Principle to characterize the controls and derive the optimization system.The numerical simulations of the resulting optimality system showed that theimplementation of combined controls (focal-perifocal) is the best strategy tochoose, since it reduces the number of female mosquitoes in both phases fastenough and in greater abundance, in addition, the application must remain


(XI) (XII)

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Figure 6: Graphs of the control functions by applying the focal control with the perifocal

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highly effective for 25 days for certain appropriate parameter values such thatequilibrium with infestation is asymptotically stable.

Acknowledgements. The authors are very grateful with Grupo de Mode-lacion Matematica en Epidemiologıa (GMME) and its institution, Universidaddel Quindıo.

References

[1] D.L. Lukes, Differential Equations: Classical to Controlled, Mathematicsin Science and Engineering, Academic Press, New York, 1982.

[2] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Vol. 1,Wiley-Interscience, New York, 1972.

[3] H.M. Yang, M.L.G. Macoris, K.C. Galvani, M.T.M. Andrighetti andD.M.V. Wanderley, Assessing the effects of temperature on the popu-lation of Aedes aegypti, the vector of dengue, Epidemiology & Infection,137 (2009), 1188-1202. https://doi.org/10.1017/s0950268809002040

[4] H.S. Rodrigues, M.T.T. Monteiro and D.F. Torres, Dynamics of dengueepidemics when using optimal control, Mathematical and Computer Mod-elling, 52 (2010), 1667-1673. https://doi.org/10.1016/j.mcm.2010.06.034

[5] J.A.O Garcıa and A.M. Loaiza, Un modelo de crecimiento poblacionalde Aedes aegypti con capacidad de carga logıstica, Revista de Matemtica:Teorıa y Aplicaciones, 1 (2018), 79-113.


[6] M.J. Nelson, Aedes Aegypti: Biologıa y Ecologıa, Washington, D.C: OPS,1986.

[7] M. Rafikov, E. Rafikova and H.M. Yang, Optimization of the Aedes aegypticontrol strategies for integrated vector management, Journal of AppliedMathematics, 2015 (2015), 1-8. https://doi.org/10.1155/2015/918194

[8] M.A. Masud, B.N. Kim and Y. Kim, Optimal control problems ofmosquito-borne disease subject to changes in feeding behavior of Aedesmosquitoes, Biosystems, 156 (2017), 23-39.https://doi.org/10.1016/j.biosystems.2017.03.005

[9] R. Rodrıguez, Estrategias para el control del dengue y del Aedes aegypti enlas Americas, Revista Cubana de Medicina Tropical, 54 (2002), 189-201.

[10] R.C. Thome, H.M. Yang and L. Esteva, Optimal control of Aedes aegyptimosquitoes by the sterile insect technique and insecticide, MathematicalBiosciences, 223 (2010), 12-23.https://doi.org/10.1016/j.mbs.2009.08.009

[11] R. Barrera, Recomendaciones para la vigilancia de Aedes aegypti,Biomedica, 36 (2016), 454-462.https://doi.org/10.7705/biomedica.v36i3.2892

[12] S. Lenhart and J.T. Workman, Optimal Control Applied to Biological mo-dels, Crc Press, 2007.

[13] S.I. Grossman and J.J. Flores, Algebra Lineal, McGraw-Hill, 2012.

[14] T.L. Vincent and W.J. Grantham, Nonlinear and Optimal Control Sys-tems, John Wiley & Sons, 1997.

[15] W.H. Fleming and R.W. Rishel, Deterministic and stochastic OptimalControl, Springer-Verlag, New York, 1975.https://doi.org/10.1007/978-1-4612-6380-7

[16] World Health Organization, Dengue: Guidelines for Diagnosis TreatmentPrevention and Control, New Edition 2009, World Health Organization.

[17] World Health Organization, A Global Brief on Vector-Borne Diseases,Geneva: WHO, 2014.

[18] W.O. Dias, E.F. Wanner and R.T. Cardoso, A multiobjective optimiza-tion approach for combating Aedes aegypti using chemical and biologicalalternated step-size control, Mathematical Biosciences, 269 (2015), 37-47.https://doi.org/10.1016/j.mbs.2015.08.019

Received: February 11, 2018; Published: March 2, 2018

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