# Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission

## Abstract

Mosquitoes are primary vectors of life-threatening diseases such as dengue, malaria, and Zika. A new control method involves releasing mosquitoes carrying bacterium *Wolbachia* into the natural areas to infect wild mosquitoes and block disease transmission. In this work, we use differential equations to describe *Wolbachia* spreading dynamics, focusing on the poorly understood effect of imperfect maternal transmission. We establish two useful identities and employ them to prove that the system exhibits monomorphic, bistable, and polymorphic dynamics, and give sufficient and necessary conditions for each case. The results suggest that the largest maternal transmission leakage rate supporting *Wolbachia* spreading does not necessarily increase with the fitness of infected mosquitoes. The bistable dynamics is defined by the existence of two stable equilibria, whose basins of attraction are divided by the separatrix of a saddle point. By exploring the analytical property of the separatrix with some sharp estimates, we find that *Wolbachia* in a completely infected population could be wiped out ultimately if the initial population size is small. Surprisingly, when the infection shortens the lifespan of infected females that would impede *Wolbachia* spreading, such a reversion phenomenon does not occur.

## Keywords

Population dynamics*Wolbachia*Imperfect maternal transmission Cytoplasmic incompatibility Monomorphism, polymorphism, bistability

## Mathematics Subject Classification

92B05 37N25 34D25 34D23 92D30## Notes

### Acknowledgements

This work was supported by China Scholarship Council (No. 201409945004), National Natural Science Foundation of China (11301103, 11631005, 11626246), Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16), and Guangdong Innovative Research Team program (2011S009). We thank Glenn Webb, Michael Turelli, and Zhiyong Xi for their suggestions and encouragements. We are also indebted to the two anonymous reviewers for their careful reading of the manuscript and constructive criticism.

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