Finding Strategies to Regulate Propagation and Containment of Dengue via Invariant Manifold Analysis

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.