A control strategy for the Sterile Insect Technique using exponentially decreasing releases to avoid the hair-trigger effect

In this paper, we introduce a control strategy for applying the Sterile Insect Technique (SIT) to eliminate the population of Aedes mosquitoes which are vectors of various deadly diseases like dengue, zika, chikungunya... in a wide area. We use a system of reaction-diffusion equations to model the mosquito population and study the effect of releasing sterile males. Without any human intervention, and due to the so-called hair-trigger effect, the introduction of only a few individuals (eggs or fertilized females) can lead to the invasion of mosquitoes in the whole region after some time. To avoid this phenomenon, our strategy is to keep releasing a small number of sterile males in the treated zone and move this release forward with a negative forcing speed c to push back the invasive front of wild mosquitoes. By using traveling wave analysis, we show in the present paper that the strategy succeeds in repulsing the population while consuming a finite amount of mosquitoes in any finite time interval even though we treat a moving half-space. Moreover, we succeed in constructing a 'forced' traveling wave for our system moving at the same speed as the releases. We also provide some numerical illustrations for our results.