---
abstract: 'Scientists have been seeking ways to use [*Wolbachia*]{} to eliminate the mosquitoes that spread human diseases. Could [*Wolbachia*]{} be the determining factor in controlling the mosquito-borne infectious diseases? To answer this question mathematically, we develop a reaction-diffusion model with free boundary in a one-dimensional environment. We divide the female mosquito population into two groups: one is the uninfected mosquito population that grows in the whole region while the other is the mosquito population infected with [*Wolbachia*]{} that occupies a finite small region and invades the environment with a spreading front governed by a free boundary satisfying the well-known one-phase Stefan condition. For the resulting free boundary problem, we establish criteria under which spreading and vanishing occur. Our results provide useful insights on designing a feasible mosquito releasing strategy to invade the whole mosquito population with [*Wolbachia*]{} infection and thus eventually eradicate the mosquito-borne diseases.'
author:
- 'Yunfeng Liu[^1], Zhiming Guo[^2], Mohammad El Smaily[^3]  and Lin Wang[^4]'
date: October 2019
title: A Wolbachia infection model with free boundary
---

[**Keywords.**]{} [*Wolbachia*]{} infection; reaction-diffusion systems; free boundary; spreading-vanishing dichotomy.

Introduction {#intro}
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Recently, several public health projects have been launched, in China [@Xinhua], USA [@Google] and France [@Naturenews], with an 