The population replacement method is a biological control where the mosquitoes carrying a bacterium called Wolbachia are released to replace the wild mosquitoes. This bacterium can help reduce the vectorial capacity and reproduction of mosquitoes. The migration of individuals between the treated area and the outside affects the invasion and persistence of Wolbachia-carrying mosquitoes. We derived a reaction-diffusion model with an inhomogeneous Robin boundary condition and introduced some particular parameters to consider the non-zero flux on the

boundary (see [1]). Then by using phase plane analysis and the time mapping method, we provide some results about the existence and stability of non-trivial steady-state solutions for our model. From the ecological point of view, we obtain the critical domain size to guarantee the invasion of the Wolbachia-carrying mosquitoes.

The Sterile Insect Technique (SIT) is used to eliminate wild mosquitoes by releasing sterile males to mate with wild females and prevent them from reproducing. Sterile individuals can be released in moving sets with a constant velocity. The application of this "rolling carpet" strategy faces the risk of re-invasion of wild populations in the treated zone. In studying this phenomenon, we derived a reaction-diffusion system of the Fisher-KPP type in which we have the so-called hair-trigger effect. To avoid the re-invasion, our strategy is to keep releasing a small number of sterile males in the treated zone. An exponentially decreasing source term moving with a fixed velocity was imposed in the model to characterize the release of sterile insects. 

In one-dimensional space, we succeed in constructing a "forced" traveling wave moving at the same speed as the releases and pushing back the invasive front of wild mosquitoes (see [2]). 

In two-dimensional space, we extended our strategy to a radially symmetric domain and obtained the elimination of mosquito population in both bistable and monostable cases (paper in preparation).   

To study the efficiency of the SIT in an inaccessible area with a connection with a treated zone, we developed a two-patch model that describes the dispersal between the two areas using discrete diffusion. However, a complete result on the persistence and extinction of the population depending on the dispersal has not been obtained yet for the multi-species case. We provided some results on asymptotic behaviors of the two-patch model considering releases of sterile individuals in one patch. We also obtained sufficient conditions to guarantee the elimination of mosquitoes on the inaccessible patch and approximate numerically the optimal number of mosquitoes released. The dependence of this number on diffusion between two patches and other intrinsic biological parameters was also studied (see [3]).

We worked with real data from the mark-released-recapture experiments collected by entomologists (paper in preparation). An individual-based model was derived to describe the dynamic of mosquitoes using a stochastic process that expresses the movement of a mosquito. The survival rate in terms of life duration and the capturing rate are described explicitly as random variables. The corresponding Fokker-Planck equation was developed and used for numerical simulation. Then, parameter estimations using the linkage of the mechanistic vision of the reaction-diffusion model with the stochastic vision of the observation data were made with the mechanistic-statistical approach. We estimated the parameters in both homogeneous and heterogeneous scenarios and drew a sample from the posterior distribution by a Markov Chain Monte Carlo (MCMC) algorithm.