# Establishing Traveling Wave in Bistable Reaction-Diffusion System by Feedback

## TL;DR

This paper demonstrates how a feedback control law can induce traveling wave solutions in a bistable reaction-diffusion model of Wolbachia-infected mosquitoes, aiding disease control strategies.

## Contribution

It introduces a simple feedback law that guarantees invasion and stabilization of infection in a bistable reaction-diffusion system across any spatial dimension.

## Key findings

- Feedback law induces traveling waves in the model.
- Finite-time application leads to invasion of the entire space.
- Stabilization achieved in any spatial dimension.

## Abstract

Several stains of the intracellular parasitic bacterium Wolbachia limit severely the competence of the mosquitoes Aedes aegypti as a vector of dengue fever and possibly other arboviroses. For this reason, the release of mosquitoes infected by this bacterium in natural populations is presently considered a promising tool in the control of these diseases. Following works by M. Turelli [4] and subsequently M. Strugarek et al. [21, 22], we consider a simple scalar reaction-diffusion model describing the evolution of the proportion of infected mosquitoes, sufficient to reveal the bistable nature of the Wolbachia dynamics. A simple distributed feedback law is proposed, whose application on a compact domain during finite time is shown to be sufficient to invade the whole space. The corresponding stabilization result is established for any space dimension.