Control strategies on mosquitos population for the fight against arboviruses

In the fight against vector-borne arboviruses, an important strategy of control of epidemic consists in controlling the population of vector, \textit{Aedes} mosquitoes in this case. Among possible actions, two techniques consist in releasing mosquitoes to reduce the size of the population (Sterile Insect Technique) or in replacing the wild population by a population carrying a bacteria, called \textit{Wolbachia}, blocking the transmission of viruses from mosquitoes to human. This paper is concerned with the question of optimizing the release protocol for these two strategies with the aim of getting as close as possible to the objectives. Starting from a mathematical model describing the dynamics of the population, we include the control function and introduce the cost functional for both \textit{population replacement} and \textit{Sterile Insect Technique} problems. Next, we establish some properties of the optimal control and illustrate them with some numerical simulations.