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.

中文翻译：

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出生脉冲模型评估携带沃尔巴克氏体的蚊子释放对控制登革热的影响


登革热是一种由登革热病毒引起的蚊媒疾病，给世界带来了巨大的疾病负担。沃尔巴克氏体作为一种创新技术已被批准用于抑制蚊子中登革热病毒的复制或减少野生蚊子的数量。我们首先建立具有性别结构的出生脉冲模型来描述蚊子种群的非线性动态，然后研究沃尔巴克氏体如何抑制或替代野生蚊子。通过分析系统的频闪图，证明了系统周期解的存在性和稳定性。理论结果表明，在完全母本传播的情况下，沃尔巴克氏体将成功建立并完全取代野生蚊子。在不完全母体传播下，可能存在两对双稳态周期解，包括置换失败和部分置换周期解并存，或者置换失败和灭蚊周期解并存。对于这些双稳态情况，如果携带沃尔巴克氏体的蚊子的初始比例大于临界阈值，则部分替换或消灭蚊子的策略可以成功。由于现场Wolbachia蚊子的初始比例并不总是足够大，因此我们将蚊子释放引入到出生脉冲模型中，并获得了完全替代周期解稳定的条件。最后，我们给出了丰富的数值模拟来展示系统的多个吸引子、三种常见释放策略的比较以及沃尔巴克氏体诱导的参数对控制策略的影响。最后一节，我们总结了在实践中控制蚊子和登革热的一些关键建议。 这项工作将有助于公共卫生当局设计适当的释放策略来控制登革热的传播。