Journal of Applied Mathematics

Volume 2015, Article ID 918194, 8 pages

http://dx.doi.org/10.1155/2015/918194

## Optimization of the *Aedes aegypti* Control Strategies for Integrated Vector Management

^{1}UFABC, Centro de Engenharia, Modelagem e Ciências Sociais Aplicadas, 09210-580 Santo André, SP, Brazil^{2}UNICAMP, IMECC, Departamento de Matemática Aplicada, 13081-970 Campinas, SP, Brazil

Received 2 February 2015; Accepted 25 May 2015

Academic Editor: Han H. Choi

Copyright © 2015 Marat Rafikov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We formulate an infinite-time quadratic functional minimization problem of *Aedes 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.

#### 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 the* A. 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 of* A. aegypti *mosquito 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 by . The natural adult or mature insects are divided into three compartments, which are denoted as -unmated female population (before copulation), -fertilized female adult population (after copulation with natural male mosquito), and -natural male population.

The remaining two compartments are S-sterile male population and -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:plus one equation uncoupled from the rest

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 , , , , , and , 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 parameter , which represents the oviposition rate per female mosquito and depends on the environmental carrying capacity . The term is the per capita oviposition rate. The aquatic population becomes winged adult mosquitoes at rate . The female portion of these winged adults is represented by coefficient , while the male portion is represented by coefficient (). Unmated female mosquitoes transform into fertilized female or fertilized but unviable female mosquitoes only after mating a natural male or a sterile male, respectively. It is assumed that the probability of the female and natural male encounter is given by . Therefore the per capita mating rate is given by , where 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 is considered and the per capita mating rate of female and sterile male is given by . The rate represents the artificial release of the sterile male population in the environment.

The dynamics of system (1) was considered in [6]. According to Esteva and Yang [6] the trivial equilibrium point of system (1) without SIT control is stable if ; that is, in the absence of sterile insects (), the condition for existence of natural insects is . 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 of* Aedes 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 capacity 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 by , and it affects only adult phase of mosquito population. The sterile male insects release is represented by . Then the control model is given by

For this system, the functional to be minimized can be represented aswhere *, **, **, *, and represent the cost of control effort to minimize specific population compartment. The parameters and 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 the* Aedes aegypti* mosquito population by the sterile insect technique and insecticide can be formulated as determination of the strategy which leads nonlinear system (3) from a given initial to a final state:minimizing cost functional (4) and satisfying constraints:

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 vectors and as this results in the following system: whereAccording to SDRE method the control was determined by where a matrix is a solution of the following State-Dependent Riccati Equation:

#### 4. Numerical Simulation Results

For the solution of the control problem and attainment of control determined by (10), it is necessary to solve State-Dependent Riccati Equation (11). For this purpose the MATLAB software intrinsic function was used. Once obtaining control , 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 , , , , and are taken from [7].