The threshold infection level for Wolbachia invasion in random environments

Abstract

Dengue fever and Zika are mosquito-borne diseases threatening human health. A novel strategy for mosquito-borne disease control uses the bacterium Wolbachia to block virus transmission. It requires releasing Wolbachia infected mosquitoes to exceed a threshold level. Since an accurate forecast for temperature and rainfall, the major environmental conditions regulating the mosquito dynamics, is often not available over a long time period, it is important to explore how the threshold releasing level changes in random environments. In this work, we estimate the threshold level in a stochastic system of differential equations where the reproduction rates of mosquitoes change randomly. We prove that the threshold level is, surprisingly, defined by a deterministic curve that does not fluctuate with environmental conditions. The major difficulty in the proof is to construct various auxiliary curves to limit the dynamic behaviors of the whole family of innumerable solutions satisfying a given initial condition.