Nonlinear dynamics of propagation and containment of dengue
Autor: Pablo Aguirre Olea
Coautor(es): Dana Contreras-Julio, José Mujica, Olga Vasilieva
Arboviruses such as dengue, zyka and chikungunya are viruses transmitted to humans by mosquitoes. In particular, Aedes Aegypti mosquito is the responsible for dengue transmission. In the absence of medical treatments and 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 mosquitoes towards extinction while keeping Wolbachia-infected mosquitoes alive. This strategy has several advantages for control of dengue: Wolbachia decreases the virulence of the dengue infection and it reduces the lifespan of the mosquito. Moreover, mating of a female uninfected by Wolbachia and an infected male leads to sterile eggs.
We consider a competition model between wild Aedes Aegypti female mosquitoes and those infected with the bacteria Wolbachia in the form of a system of nonlinear differential equations. Our goal is to examine the basin of attraction of a desired equilibrium state. For this purpose, we study how the stable manifold that forms the basin boundary of interest changes under parameter variation. To achieve this, we combine analytical 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 amount of released infected mosquitoes in a human, external intervention by choosing an appropriate combination of initial conditions and parameter values.