Optimal Dirac controls for time-periodic bistable ODEs, application to population replacement
Grégoire Nadin, David Nahmani, Nicolas Vauchelet
TL;DR
This study analyzes optimal Dirac-like controls for time-periodic bistable ODEs modeling population replacement, focusing on the proportion $p(t)$ of a targeted type and an instantaneous release of individuals. By treating releases as scaled impulses, the authors derive an asymptotic limit in which the jump at the intervention time satisfies $G(p(t_0^+))-G(p(t_0^-))=\frac{C}{K(t_0)}$, and show that the optimal timing minimizes $\underline{C}(t)=K(t)G(p^T_M(t))$, with the minimizers of this limit problem governing the asymptotic optimal strategy. They prove convergence results for the impulse sequence and the associated optimal times, and they establish uniqueness results for the periodic solutions in several regimes, including small periods, perturbed nonlinearities, and separated-variable nonlinearities. The separated-variables setting yields a particularly transparent reduction: the optimal single release occurs at the time of minimal carrying capacity $K(t)$, and two releases do not outperform this single impulse. The framework is then specialized to population replacement in Wolbachia-based mosquito control, deriving a scalar reduced model and illustrating impulse-optimal timing and total releases via numerical simulations, thereby linking abstract theory to a practical biocontrol application.
Abstract
This work addresses an optimal control problem on a dynamics governed by a nonlinear differential equation with a bistable time-periodic nonlinearity. This problem, relevant in population dynamics, models the strategy of replacing a population of A-type individuals by a population of B-type individuals in a time-varying environment, focusing on the evolution of the proportion of B-type individuals among the whole population. The control term accounts for the instant release of B-type individuals. Our main goal, after noting some interesting properties on the differential equation, is to determine the optimal time at which this release should be operated to ensure population replacement while minimizing the release effort. The results establish that the optimal release time appears to be the minimizer of a function involving the carrying capacity of the environment and the threshold periodic solution of the dynamics; they also describe the convergence of the whole optimal release strategy. An application to the biocontrol of mosquito populations using Wolbachia-infected individuals illustrates the relevance of the theoretical results. Wolbachia is a bacterium that helps preventing the transmission of some viruses from mosquitoes to humans, making the optimization of Wolbachia propagation in a mosquito population a crucial issue.