# Optimization of theAedes aegypti Control Strategies for Integrated Vector Management

Authors: Marat Rafikov; Elvira Rafikova; Hyun Mo Yang
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

We formulate an infinite-time quadratic functional minimization problem ofAedes aegypti mosquito population. Three techniques of mosquito population management, chemical insecticide control, sterile insect technique control, and environmental carrying capacity reduction, are combined in order to obtain the most sustainable strategy to reduce mosquito population and consequently dengue disease. The solution of the optimization control problem is based on the ideas of the Dynamic Programming and Lyapunov Stability using State-Dependent Riccati Equation (SDRE) control method. Different scenarios are analyzed combining three mentioned population management efforts in order to assess the most sustainable policy to reduce the mosquito population.


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## Body

## 1. Introduction

According to World Health Organization, dengue is reported to be the most rapidly spreading mosquito-borne disease in the world [1]. Recent estimates are that 50 million dengue infections occur each year, with 2.5 billion people at risk of infection in dengue endemic countries.Aedes aegypti is a domesticated urban mosquito and is the vector responsible for the transmission of some infectious diseases. The most common of them is dengue disease-virus infection caused by four distinct but related single-strand RNA viruses of the family Flaviviridae. Each of them causes a different type of clinical manifestation of dengue disease, varying from classic form to severe dengue shock syndrome and the fatal hemorrhagic dengue form.Integrated vector management (IVM) is a strategy which aims to achieve a maximum impact on vector borne diseases like dengue. The emphasis of IVM is on examining and analyzing the local situation, making decisions at decentralized levels, and utilizing the appropriate mosquito control tools [1]. One of the features of IVM is the use of a range of interventions, often in combination and simultaneously, that work together to reduce dengue transmission. For dengue control, there are three main categories of intervention: biological control, the use of chemicals to kill the adult and immature mosquito stages, and the physical (mechanical) control, eliminating possible breeding sites.The biological control includes the well-known sterile insect technique (SIT). The SIT is a biological control, firstly presented by Knipling [2], and was used in 1958 to control Screwworm fly (Cochliomyia hominivorax [3, 4]). SIT control is a technique in which natural male insects are exposed to radiations that eliminate their ability to fertilize eggs. The sterile males are released in the environment to mate with natural female population. Once irradiated, the sperms of sterile male mosquitoes fertilize the eggs of female mosquitoes producing unviable eggs, which do not hatch and disrupt the natural reproductive process of the population.Ultra low volume(ULV) method consists of aerial sprays of insecticide for adult mosquitoes control. Chemical insecticides are sprayed using portable or truck-mounted machines in order to kill adult insects. Although studies have been shown that space spraying alone is relatively ineffective as a routine control strategy [5], it should be reserved for use only during epidemics.Aedes aegypti mosquitoes lay their eggs in containers such as bottles, tires, fountains, barrels, and pots. By removing these habitats, mosquitoes have fewer opportunities to lay eggs. This strategy is called mechanical control. The mechanical control must be done both by public health officials and by residents in affected areas [1].Mathematical modeling of mosquito population in order to assist SIT can be found in [6]. In [7] an optimal control of theA. aegypti population problem was formulated in terms of Pontryagin Maximum Principle, where a quadratic functional was minimized in finite time interval. It is known that in many cases the application of the Pontryagin Maximum Principle does not guarantee the long time stability of the controlled system.In this paper we formulate an infinite-time quadratic functional minimization problem ofA. aegyptimosquito population. The solution of this problem is based on the ideas of the Dynamic Programming and Lyapunov Stability using State-Dependent Riccati Equation (SDRE) control method [8, 9]. The reduction in the mosquito population is achieved by applying three control mechanisms: chemical insecticides control, biological control by release of the sterilized male insects, and mechanical control based on the reduction of the breeding sites.


## 2. Population Dynamics Model

The mosquito population dynamics model, proposed in [6], represents the interaction among four different stages of the natural mosquito population, and a sterile male mosquito group artificially was introduced into the environment as a control strategy.The population size of the immature phase of the insect (eggs, larvae and pupae) is considered as one compartment denoted byA. The natural adult or mature insects are divided into three compartments, which are denoted as I-unmated female population (before copulation), F-fertilized female adult population (after copulation with natural male mosquito), and M-natural male population.The remaining two compartments are S-sterile male population andU-females mated with sterile males resulting in unviable insects with dynamics uncoupled from the rest of the population. The dynamics of the population described above is represented by the following mathematical model:(1)dAdt=ϕ1-ACF-γ+μAA,dIdt=rγA-βMIM+S-βSSIM+S-μII,dFdt=βMIM+S-μFF,dMdt=1-rγA-μMM,dSdt=α-μSS,plus one equation uncoupled from the rest(2)dUdt=βSSIM+S-μUU.In (1) the mortality rates of aquatic phase, immature female adults, fertilized female adults, male adults, sterile male adults, and unmated female adults are represented by μA, μI, μF, μM, μS, and μu, respectively.An adult female mosquito mates only once during its lifespan and lays eggs in different places every three days (gonadotrophic cycle) during entire life. Therefore, the aquatic population growth is regulated by the parameterϕ, which represents the oviposition rate per female mosquito and depends on the environmental carrying capacity C. The term ϕ1-A/C is the per capita oviposition rate. The aquatic population becomes winged adult mosquitoes at rate γ. The female portion of these winged adults is represented by coefficient r, while the male portion is represented by coefficient (1-r). Unmated female mosquitoes I transform into fertilized female F or fertilized but unviable female U mosquitoes only after mating a natural male or a sterile male, respectively. It is assumed that the probability of the female I and natural male M encounter is given by M/(M+S). Therefore the per capita mating rate is given by βM/(M+S), where β represents the intrinsic mating rate of natural mosquitoes. For sterile male, this intrinsic rate could be diminished by physiological modification of the sterilization technique. So another βs is considered and the per capita mating rate of female I and sterile male is given by βSSI/(M+S). The rate α represents the artificial release of the sterile male population S in the environment.The dynamics of system (1) was considered in [6]. According to Esteva and Yang [6] the trivial equilibrium point P0=0,0,0,0,0 of system (1) without SIT control is stable if R=ϕrγβ/μA+γμI+βμF<1; that is, in the absence of sterile insects (α=0), the condition for existence of natural insects is R>1. In affected areas the last inequality is satisfied, and an application of IVM is necessary.In next section, the integrated vector management of the mosquito population is formulated as an optimal control problem.


## 3. Formulation of the Optimal Control Problem ofAedes aegypti Mosquitoes

Now, it is possible to formulate a control problem where the main goal is to minimize the fertile female mosquito population, and, consequently, all other mosquito populations are reduced by the action of three different control techniques: mechanical, chemical (insecticide spraying), and biological (sterile insect introduction).The mechanical control is related to educational campaigns, and it is essential to remove water from domestic recipients, eliminating possible breeding sites (such as bottles, tires, fountains, barrels, and pots). This control decreases the environmental carrying capacityC in the initial time of the educational campaign, and it can be considered constant for some periods of time.Let the insecticide control effort be denoted byu1, and it affects only adult phase of mosquito population. The sterile male insects release is represented by u2. Then the control model is given by (3)dAdt=ϕ1-ACF-γ+μAA,dIdt=rγA-βMIM+S-βSSIM+S-μI+u1I,dFdt=βMIM+S-μF+u1F,dMdt=1-rγA-μM+u1M,dSdt=u2-μS+u1S.For this system, the functional to be minimized can be represented as(4)J=12∫0∞q1x12+q2x22+q3x32+q4x42+q5x52+r1u12+r2u22dt,where q1,q2,q3,q4, and q5 represent the cost of control effort to minimize specific population compartment. The parameters r1 and r2 are the cost of insecticide application and cost of production and release of sterile mosquitoes, respectively. We assume a quadratic functional cost [7, 8] since we believe that the performance index is a nonlinear function. The quadratic terms act as a penalization [9, 10], amplifying the effects of great variations of the variables. Each quadratic term is multiplied by a coefficient, which establishes the relative importance of the term on dengue control cost.The optimization problem of the control of theAedes aegypti mosquito population by the sterile insect technique and insecticide can be formulated as determination of the strategy u which leads nonlinear system (3) from a given initial to a final state:(5)x∞=0,minimizing cost functional (4) and satisfying constraints:(6)0≤u1≤umas,0≤umas≤1.The formulated control problem can be solved by State-Dependent Riccati Equation (SDRE) method [11, 12]. SDRE approach is explained in more detail in the Appendix.Defining the vectorsx and u as (7)x=AIFMST,u=u1u2T,this results in the following system: (8)x˙=Axx+Bxu,where(9)Ax=-γ-μA0ϕ1-x1C00rγ-μI0-βx2x4+x5-βsx2x4+x50βsx4x4+x5-μF001-rγ00-μM00000-μS,Bx=00-x20-x30-x40-x51.According to SDRE method the control u was determined by (10)u=-R-1BTPxx,where a matrix P is a solution of the following State-Dependent Riccati Equation:(11)PxAx+ATxPx-PxBxR-1xBTxPx+Qx=0.


## 4. Numerical Simulation Results

For the solution of the control problem and attainment of controlu determined by (10), it is necessary to solve State-Dependent Riccati Equation (11). For this purpose the MATLAB software intrinsic lqr function was used. Once obtaining control u, system (5) is solved as initial value problem using numeric, fourth-order Runge-Kutta integrator in MATLAB.The parameter values of system (5) are shown in Table 1. The values for ϕ, μA, μI, μM, and γ are taken from [7].Table 1
Parameter values (units are days−1, except for r and C).
μ
A
μ
I
μ
F
μ
U
μ
M
μ
S
γ
r
β
β
S
C
φ
0.0583
0.0337
0.0337
0.0337
0.06
0.07
0.121
0.5
0.7
0.5
3
6.353The numerical simulations showed that SIT control alone cannot be a sustainable control strategy due to the lack of effectiveness in reducing mosquito population and high costs. In the same manner, the application of insecticide spraying alone cannot be seen as such an alternative because of its high ecological toxicity and mainly because it demands every day application at, almost, the maximum level of insecticide, which is obviously difficult for many reasons (weather conditions, restricted access of houses, lack of personal, and machinery).Figures1 and 2 present a scenario where both chemical (insecticide) and biological (sterile male insect release) controls are considered. The environmental carrying capacity has the original value, C=3, as in scenario 1.Figure 1
Insecticide control (a) and SIT control (b).
(a)
(b)Figure 2
Aquatic population (a), adult immature female population (b), adult female fertilized population (c), and natural male population (d).
(a)
(b)
(c)
(d)As it is seen from Figure1, the control u1 grows to its limit at the beginning of the period, acting at its maximum for approximately 10 days, and then asymptotically decreases. The total effort of u1 control is 14.91. The u2 control increases when u1 starts to decrease, forming the bell shaped pattern. The total cost of this control is 46.24.All four compartments of the mosquito population asymptotically tend to zero (Figure2). The aquatic population and the unmated female population decrease more slowly than fertilized female and natural male population. It means that a practically similar investment in SIT control (compared to previous scenario) plus a little more investment in insecticide reduced significantly the mosquito population.Next scenario illustrates the combination of insecticide control, SIT control, and reduction of the environmental carrying capacity toC=1.5 (Figures 3 and 4). This mainly affects aquatic phase and unmated female population, reducing them more efficiently than in previous scenario (Figures 4(a) and 4(b)). The sum of efforts of the insecticide control and SIT control are 13.51 and 36.9, respectively (Figure 3). These costs are lower than in previous scenario pointing out that the combination of three types of mosquito population management increases the efficacy of mosquito population control and reduces its costs.Figure 3
Insecticide control (a) and SIT control (b).
(a)
(b)Figure 4
Aquatic phase population (a), immature female population (b), fertilized female population (c), and normal male population (d).
(a)
(b)
(c)
(d)


## 5. Discussion and Conclusion

This paper considers optimization control problem regardingAedes aegypti that combines three techniques of mosquito population management, chemical insecticide control, sterile insect technique control, and environmental carrying capacity reduction, in order to obtain the most sustainable strategy to reduce mosquito population and consequent dengue disease reduction.When one seeks a control mechanism or a strategy that combines different but sometimes antagonistic mechanisms aiming at the mosquito population control, some criteria should be considered and some concerns arise. First of them is to find a control strategy that significantly and immediately reduces the adult mosquito population in order to reduce the disease propagation. Second concern is the use of the minimum chemical insecticide spraying due to its toxicity. And the last concern is to find a strategy that can be feasible, or easily employable by the government, which implies minimization of costs.In general, optimal control problems are addressed in specific situations, for instance, when public policy authorities have limited budget or when one type of controls is dangerous to public health (as insecticide, which presents another inconvenience of generating resistant strains). Discussions are presented bearing these features in mind.The numerical simulations in this paper showed that SIT control alone and the application of insecticide spraying alone cannot be sustainable control strategies due to many reasons noted in Section4.The combination of SIT control and insecticide spraying is an alternative to a sustainable mosquito population control. The numerical simulation of this scenario showed that all four compartments of natural population are minimized (Figure2). With respect to the optimal control, the insecticide application is active during only few days (which is attractive in terms of implementation) and the SIT control is also optimized.Scenario, presented in Figures3 and 4, reflects the combination of three kinds of effort in controlling the mosquito population: the insecticide control, the SIT control, and reduction of environmental carrying capacity. All the population compartments are minimized in the first week and the cost of insecticide and SIT controls is lower than in all previous scenarios. This strategy can be seen as the ideal control strategy of mosquito population in this optimization problem.In the case of presence of antagonistic control mechanism such as the combined release of sterile male insects and the application of the insecticide, optimal control problem provides management strategies for both interventions. The considered scenarios show that insecticide application must take place in early times with the maximum amount, but should be soon decreased. Meanwhile, sterile insect must be released in small amounts at the early times when insecticide is applied (as the insecticide also kills sterile insects) and must increase while insecticide application decreases. Then the release of sterile insects should decrease with the reduction in total mosquito population.



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*Source: 918194-2015-06-08.xml*