Threshold dynamics of a stochastic mathematical model for Wolbachia infections.

Abstract

A stochastic mathematical model is proposed to study how environmental heterogeneity and the augmentation of mosquitoes with $Wolbachia$ bacteria affect the outcomes of dengue disease. The existence and uniqueness of the positive solutions of the system are studied. Then the V-geometrically ergodicity and stochastic ultimate boundedness are investigated. Further, threshold conditions for successful population replacement are derived and the existence of a unique ergodic steady-state distribution of the system is explored. The results show that the ratio of infected to uninfected mosquitoes has a great influence on population replacement. Moreover, environmental noise plays a significant role in control of dengue fever.