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. The mosquito population infected with Wolbachia 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 that infects the whole mosquito population with Wolbachia and eradicates the mosquito-borne diseases eventually.

科学家一直在寻找利用沃尔巴氏菌消除传播人类疾病的蚊子的方法。沃尔巴克氏菌能否成为控制蚊媒传染病的决定因素？为了从数学上回答这个问题，我们建立了一个在一维环境中具有自由边界的反应扩散模型。我们将雌性蚊子分为两类：一类是在整个区域内生长的未感染蚊子种群，另一类是被沃尔巴赫氏菌感染的蚊子种群，该蚊子种群只占一个有限的小区域。感染了沃尔巴克氏菌的蚊子通过一个自由边界控制着扩展的前沿来入侵环境，自由边界满足了众所周知的单相Stefan条件。对于由此产生的自由边界问题，我们建立了发生扩散和消失的标准。我们的结果为设计可行的驱蚊策略提供了有用的见识，该策略可以用沃尔巴克氏菌感染整个蚊子种群，并最终消除蚊子传播的疾病。