Birth-pulse models to assess the effects of Wolbachia-carrying mosquito releases on the control of dengue fever

Dengue fever as a mosquito -borne disease caused by dengue virus is responsible for a substantial disease burden to the world. Wolbachia as an innovative technique has been approved to inhibit the replication of dengue viruses in mosquitoes or reduce the number of wild mosquitoes. We firstly establish a birth -pulse model with sex structures to depict the nonlinear dynamics of mosquito population and then to investigate how Wolbachia can suppress or replace wild mosquitoes. The existence and stability of periodic solutions of the system are proved by analyzing its stroboscopic map. Theoretical results show that under complete maternal transmission, Wolbachia will successfully establish and completely replace wild mosquitoes. Under incomplete maternal transmission, there may be two pairs of bistable periodic solutions, including the coexistence of replacement failure and partial replacement periodic solutions, or the coexistence of replacement failure and mosquito eradication periodic solutions. For these bistable situations, if the initial proportion of Wolbachia-carrying mosquitoes is greater than the critical threshold, then the strategy of partial replacement or mosquito eradication can be succeeded. Since the initial proportion of Wolbachia mosquitoes is not always greater enough in the field, we introduce mosquito releases into the birth -pulse model, and obtain the condition for the stability of complete replacement periodic solution. Finally, we give rich numerical simulations to show the multiple attractors of the systems, the comparison of three common release strategies and the effects of Wolbachia-induced parameters on control strategies. At last section, we summarize some key suggestions on the control of mosquitoes and dengue fever in practice. This work will be helpful for public health authorities in designing proper release strategies to control the spread of dengue fever.