  This work is devoted to the mathematical study of an optimization problem
regarding control strategies of mosquito population in a heterogeneous
environment. Mosquitoes are well known to be vectors of diseases, but, in some
cases, they have a reduced vector capacity when carrying the endosymbiotic
bacterium Wolbachia. We consider a mathematical model of a replacement
strategy, consisting in rearing and releasing Wolbachia-infected mosquitoes to
replace the wild population. We investigate the question of optimizing the
release protocol to have the most effective replacement when the environment is
heterogeneous. In other words we focus on the question: where to release, given
an inhomogeneous environment, in order to maximize the replacement across the
domain. To do so, we consider a simple scalar model in which we assume that the
carrying capacity is space dependent. Then, we investigate the existence of an
optimal release profile and prove some interesting properties. In particular,
neglecting the mobility of mosquitoes and under some assumptions on the
biological parameters, we characterize the optimal releasing strategy for a
short time horizon, and provide a way to reduce to a one-dimensional
optimization problem the case of a long time horizon. Our theoretical results
are illustrated with several numerical simulations.
