Periodic patterns and Pareto efficiency of state dependent impulsive controls regulating interactions between wild and transgenic mosquito populations

Abstract

It is conceivable that genetically modified mosquitoes could stop the spread of malaria, by outcompeting the wild mosquitoes and interfering with their reproductive processes, and genetically inheriting and further transmitting a diminished potential to carry Plasmodium. To get insight into the possible outcomes, we formulate an ODE model for the interactions between wild and transgenic mosquito populations, which is subject to state-dependent impulsive perturbations. By first investigating the dynamics of the unperturbed system, we determine certain sufficient conditions for the existence and orbital stability of positive order-1 solution of the model system with state-dependent impulsive perturbations. Their feasibility is then illustrated by means of numerical simulations. In addition, to adequately control the wild mosquito population, we use a multi-target approach which, in addition to accounting for the total costs, keeps track of the total size of the wild mosquito population. To trade off these objectives, we consider the concept of Pareto efficiency to determine suitable control strategies which are near-optimal. Finally, we carry out numerical simulations to illustrate the Pareto frontier and then characterize the detailed Pareto efficient control regime.