SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1392-1437, January 2020.
Dengue, zika, and chikungunya are viruses transmitted to humans by Aedes aegypti mosquitoes. In the absence of medical treatments and efficient vaccines, one of the control methods is to introduce Aedes aegypti mosquitoes infected by the bacterium Wolbachia into a population of wild (uninfected) mosquitoes. The goal consists in achieving population replacement in finite time by driving the population of wild females towards extinction, while keeping Wolbachia-infected mosquitoes alive and persistent. We consider a two-dimensional competition model between wild Aedes aegypti female mosquitoes and those infected with Wolbachia. Our goal is to examine the basin of attraction of a desired equilibrium state which represents the population replacement. For this, we study how the stable manifold that forms the basin boundary of interest changes under parameter variation. To achieve this, we first combine tools from dynamical systems and geometric singular perturbation theory with numerical continuation methods. This allows us to present a strategy to get the desired population replacement with a minimum number of released infected mosquitoes in a human intervention by choosing an appropriate combination of initial conditions and parameter values. Second, we characterize traveling waves in a spatiotemporal extension of our model. To this aim, we propose a new method to calculate and visualize 3D invariant manifolds of an associated 4D dynamical system. In this way, we find uncountably many heteroclinic connections between stationary states (each associated with a wave front exhibiting the desired population replacement) as intersections of global invariant manifolds in the 4D phase space.

中文翻译：

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通过不变流形分析找到调节登革热繁殖和控制的策略

SIAM应用动力系统杂志，第19卷，第2期，第1392-1437页，2020年1月。
登革热，寨卡病毒和基孔肯雅热是埃及伊蚊（Aedes aegypti）蚊子传播给人类的病毒。在没有药物和有效疫苗的情况下，控制方法之一是将被细菌Wolbachia感染的埃及伊蚊（Aedes aegypti）蚊子引入野生（未感染）蚊子种群。目标是通过驱使野生雌性种群灭绝，同时保持感染了Wolbachia的蚊子存活和持久，在有限的时间内实现种群替代。我们考虑野生埃及伊蚊和感染沃尔巴克氏菌的二维竞争模型。我们的目标是研究代表种群替代的理想平衡状态的吸引盆。为了这，我们研究了形成感兴趣盆地边界的稳定流形在参数变化下如何变化。为了实现这一点，我们首先将动力学系统和几何奇异摄动理论的工具与数值连续方法相结合。这使我们能够提出一种策略，通过选择初始条件和参数值的适当组合，在人为干预下以最少数量的已释放受感染蚊子来实现所需的种群替代。其次，我们在模型的时空扩展中表征行波。为此，我们提出了一种新的方法来计算和可视化关联的4D动力学系统的3D不变歧管。通过这种方式，