Qualitative analysis of a Filippov wild-sterile mosquito population model with immigration

Effectively combating mosquito-borne diseases necessitates innovative strategies beyond traditional methods like insecticide spraying and bed nets. Among these strategies, the sterile insect technique (SIT) emerges as a promising approach. Previous studies have utilized ordinary differential equations to simulate the release of sterile mosquitoes, aiming to reduce or eradicate wild mosquito populations. However, these models assume immediate release, leading to escalated costs. Inspired by this, we propose a non-smooth Filippov model that examines the interaction between wild and sterile mosquitoes. In our model, the release of sterile mosquitoes occurs when the population density of wild mosquitoes surpasses a specified threshold. We incorporate a density-dependent birth rate for wild mosquitoes and consider the impact of immigration. This paper unveils the complex dynamics exhibited by the proposed model, encompassing local sliding bifurcation and the presence of bistability, which entails the coexistence of regular equilibria and pseudo-equilibria, as crucial model parameters, including the threshold value, are varied. Moreover, the system exhibits hysteresis phenomena when manipulating the rate of sterile mosquito release. The existence of three types of limit cycles in the Filippov system is ruled out. Our main findings indicate that reducing the threshold value to an appropriate level can enhance the effectiveness of controlling wild insects. This highlights the economic benefits of employing SIT with a threshold policy control to impede the spread of disease-carrying insects while bolstering economic outcomes.