Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide

Abstract

We present a mathematical model to describe the dynamics of mosquito population when sterile male mosquitoes (produced by irradiation) are introduced as a biological control, besides the application of insecticide. In order to analyze the minimal effort to reduce the fertile female mosquitoes, we search for the optimal control considering the cost of insecticide application, the cost of the production of irradiated mosquitoes and their delivery as well as the social cost (proportional to the number of fertilized females mosquitoes). The optimal control is obtained by applying the Pontryagin’s Maximum Principle.

Introduction

Dengue disease is actually an important problem of public health in the tropical regions of the word. The infective agent is the Dengue virus of the family of Flaviviridae. Four serotypes have been recognized, denoted by DEN-I, DEN-II, DEN-III, and DEN-IV. Infection for any Dengue serotype produces permanent immunity to it, but apparently only temporary cross immunity to other serotypes [1]. Therefore, individuals that live in dengue endemic areas can have the disease more than one time. The virus is transmitted to humans by the bite of Aedes female mosquitoes, being Aedes aegypti its principal transmissor [2].

Dengue infection by any of the four serotypes causes a spectrum of illness in humans, ranging from clinically inapparent, to severe and fatal hemorrhagic disease [1]. Due to the geographical expansion of the vector and virus [3], the incidence of dengue infection in all of its manifestations has been increasing in the last decades. In 2005, dengue was considered the more important viral vector borne disease. Its world distribution is compared to malaria, and it is estimated that more than 2.5 billion of persons live in transmission risk areas.

Since there is not vaccine to control dengue disease, all efforts are directed to avoid the proliferation of the mosquito population. The control mechanisms include

• Chemical control of adult population by dichloro-diphenyl-trichloroethane (DDT) spraying, and ultra low volume (ULV) spraying.

• Chemical control of larvae by larvicides.

• Reduction of mosquito breeding sites by elimination of discarded tires, and litter, draining of unnecessary containers, etc.

• Biological control by using parasites or/and predators of mosquitoes.

• Genetic manipulation of mosquitoes to produce mosquitoes refractory to infection of transmission, or sterile insects.

The sterile insect technique (SIT) is a biological control in which the natural reproductive process of insects is disrupted by the use of mutagens such as gamma radiation thus rendering the insects sterile. These sterile insects are then released into the environment in very large numbers in order to mate with the native insects that are present in the environment. A native female that mates with a sterile male will produce eggs, but the eggs will not hatch (the same effect will occur for the reciprocal cross). If there is a sufficiently high number of sterile insects, most of the crosses are sterile, and as time goes on, the number of native insects decreases and the ratio of sterile to normal insects increases, thus driving the native population to extinction.

The SIT was first conceived by Knipling [4], and used successfully in 1958 in Florida to control Screwworm fly (Cochliomya omnivorax) [5], [6]. Since then, the release of sterile insects have been used with varying success. Some examples are screwworm fly in USA, Mexico and Libya; Mediterranean Fruit Fly (Ceratitis capitata Wiedemann) in USA and Mexico; Melon Fly (Dacus cucurbitae Coquillett) in Japan and Taiwan; Pink Bollworm (Pectinophora gossypiella Saunders) in USA; Tsetse Fly (Glossina species) in Tanzania, Zimbabwe and Upper Volta; Boll Weevil (Anthonomus grandis Boheman) in Southeastern USA; Mexican Fruit Fly (Anastrepha ludens Loew) in USA and Mexico; Gypsy Moth (Lymantria dispar Linnaeus) in USA and Canada [7].

Mathematical models have been done to assist the effectiveness of the SIT (see, e.g., [4], [8], [9], [10], [11], [12], [13]). Some of them contemplate combination of SIT with other control measures as pesticides [14], or release of parasitoids [15].

The goal of this paper is to use optimal control theory to evaluate the effectiveness of the application of both SIT and insecticide to mosquito population. We want to find the minimal effort necessary to reduce the fertile female mosquitoes considering the cost of insecticide application, the cost of the production of irradiated mosquitoes, and the social cost. By social cost we mean all the expenses related to the disease like infectives treatment, hospital care, and even eventual death. This work is a continuation of [16] where the authors formulated a model to analyze the application of the SIT for the control of Aedes aegypti mosquitoes.