Bulletin of Mathematical Biology

, Volume 78, Issue 10, pp 1968–2010 | Cite as

Modeling the Effects of Augmentation Strategies on the Control of Dengue Fever With an Impulsive Differential Equation

  • Xianghong Zhang
  • Sanyi TangEmail author
  • Robert A. Cheke
  • Huaiping Zhu
Original Article


Dengue fever has rapidly become the world’s most common vector-borne viral disease. Use of endosymbiotic Wolbachia is an innovative technology to prevent vector mosquitoes from reproducing and so break the cycle of dengue transmission. However, strategies such as population eradication and replacement will only succeed if appropriate augmentations with Wolbachia-infected mosquitoes that take account of a variety of factors are carried out. Here, we describe the spread of Wolbachia in mosquito populations using an impulsive differential system with four state variables, incorporating the effects of cytoplasmic incompatibility and the augmentation of Wolbachia-infected mosquitoes with different sex ratios. We then evaluated (a) how each parameter value contributes to the success of population replacement; (b) how different release quantities of infected mosquitoes with different sex ratios affect the success of population suppression or replacement; and (c) how the success of these two strategies can be realized to block the transmission of dengue fever. Analysis of the system’s stability, bifurcations and sensitivity reveals the existence of forward and backward bifurcations, multiple attractors and the contribution of each parameter to the success of the strategies. The results indicate that the initial density of mosquitoes, the quantities of mosquitoes released in augmentations and their sex ratios have impacts on whether or not the strategies of population suppression or replacement can be achieved. Therefore, successful strategies rely on selecting suitable strains of Wolbachia and carefully designing the mosquito augmentation program.


Dengue fever Wolbachia-infected mosquitoes Bifurcation Mosquito augmentation 



Tang is partially supported by the National Natural Science Foundation of China (NSFCs 11171199, 11471201, 11601268) and by the Fundamental Research Funds for the Central Universities (GK201305010, GK201401004, KJ1600522). Zhu is partially supported by NSERC and CIHR of Canada. Zhang is partially supported by Excellent Doctoral Dissertation of Shaanxi Normal University (S2014YB01). We thank the anonymous referees for their careful reading and comments which helped to improve our manuscript.


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Copyright information

© Society for Mathematical Biology 2016

Authors and Affiliations

  • Xianghong Zhang
    • 1
  • Sanyi Tang
    • 1
    Email author
  • Robert A. Cheke
    • 2
  • Huaiping Zhu
    • 3
  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China
  2. 2.Natural Resources InstituteUniversity of Greenwich at MedwayChathamUK
  3. 3.LAMPS, Department of Mathematics and StatisticsYork UniversityTorontoCanada

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