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 $\mathit{W}\mathit{o}\mathit{l}\mathit{b}\mathit{a}\mathit{c}\mathit{h}\mathit{i}\mathit{a}$ 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.

Keywords:

• Dengue fever
• population replacement
• ergodicity
• Lyapunov function

Acknowledgments

The author is very grateful to the anonymous referee for a careful reading, helpful suggestions and valuable comments which led to the improvement of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under grants (11961024(Y. Tan), 11801047(J. Yang)), and by the Joint Training Base Construction Project for Graduate Students in Chongqing (JDLHPYJD2021016), and by the Group Building Scientific Innovation Project for universities in Chongqing (CXQT21021).