Optimal releases for population replacement strategies: Application to Wolbachia

Abstract

In this article, we consider a simplified model of time dynamics for a mosquito population subject to the artificial introduction of Wolbachia-infected mosquitoes in order to fight arboviruses transmission. Indeed, it has been observed that when some mosquito populations are infected by some Wolbachia bacteria, various reproductive alterations are induced in mosquitoes, including cytoplasmic incompatibility. Some of these Wolbachia bacteria greatly reduce the ability of insects to become infected with viruses such as the dengue ones, cutting down their vector competence and thus effectively stopping local dengue transmission. The behavior of infected and uninfected mosquitoes is assumed to be driven by a compartmental system enriched with the presence of an internal control source term standing for releases of infected mosquitoes, distributed in time. We model and design an optimal releasing control strategy with the help of a least squares problem. In a nutshell, one wants to minimize the number of uninfected mosquitoes at a given horizon of time, under some relevant biological constraints. We derive properties of optimal controls and highlight a limit problem providing useful asymptotic properties of optimal controls. We numerically illustrate the relevance of our approach.