and $b_2(x)$ in Section \[Section4\], while the natural death rate is assumed to be spatially independent.

Throughout this paper, we assume that $b_{1}(x)$ and $b_{2}(x)$ satisfy the following conditions, unless otherwise stated: $$\tag{\bf $B_{1}$}\label{B2}
\left\| \begin{array}{l}\exists \,\theta\in(0,1)\text{ such that } b_{i}\in C^{0,\theta}\left([0,+\infty)\right)\cap L^{\infty}\left([0,+\infty)\right), \\ \\
 b_{i}\geq0,~i=1,2.\end{array}\right.$$ $C^{0,\theta}\left([0,+\infty)\right)$ is the Hölder space with Hölder exponent $\theta.$ The initial conditions $u_{0}$ and $v_{0}$ are assumed to be bounded and satisfy $$\label{eq initial functions}
\left\{\begin{array}{l}
u_{0}\in C^{2}([0,h_{0}]),\vspace{7 pt}\\
u'_{0}(0)=u_{0}(h_{0})=0,\vspace{7 pt}\\

u_{0}(x)>0 \text{ for all }x\in (0,h_{0}),\vspace{7 pt}\\

v_{0}\in C^{2}[0,\infty) \cap {L}^{\infty}[0,\infty)~\text{ and }~v_{0}>0.
\end{array}\right.$$

For the free boundary problem -, the main question we are concerned about is whether the infected population can eventually occupy the whole space or not.

If the infected population eventually occupies the whole space, i.e. $$\displaystyle{\lim_{t\to\infty}h(t)=+\infty},$$ we say [*spreading*]{} occurs; otherwise, we say [*vanishing*]{} occurs.

The main goal of this work is to derive conditions under which the spreading occurs. If spreading occurs, then the whole mosquito population will become infected with [*Wolbachia*]{} bacteria and this leads to the [*extinction*]{} of the mosquito population and eventually the eradication of mosquito-borne diseases.

#### Organization of the paper. {#organization-of-the-paper. .unnumbered}

The paper is organized as follows. We first establish the 