Journal of Mathematical Biology

, Volume 76, Issue 1–2, pp 235–263 | Cite as

Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission

  • Bo Zheng
  • Moxun Tang
  • Jianshe Yu
  • Junxiong Qiu


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.


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

Mathematics Subject Classification

92B05 37N25 34D25 34D23 92D30 



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.


  1. Bian G, Xu Y, Lu P, Xie Y, Xi Z (2010) The endosymbiotic bacterium Wolbachia induces resistance to dengue virus in Aedes aegypti. PLoS Pathog 6(4):e1000833CrossRefGoogle Scholar
  2. Bian G, Joshi D, Dong Y, Lu P, Zhou G, Pan X, Xu Y, Dimopoulos G, Xi Z (2013) Wolbachia invades Anopheles stephensi populations and induces refractoriness to plasmodium infection. Science 340:748–751CrossRefGoogle Scholar
  3. Caspari E, Watson GS (1959) On the evolutionary importance of cytoplasmic sterility in mosquitoes. Evolution 13:568–570CrossRefGoogle Scholar
  4. Dutra HLC, Rocha MN, Dias FBS, Mansur SB, Caragata EP, Moreira LA (2016) Wolbachia blocks currently circulating Zika virus isolates in Brazilian Aedes aegypti mosquitoes. Cell Host Microbe 19:771–774CrossRefGoogle Scholar
  5. Dobson SL, Fox CW, Jiggins FM (2002) The effect of Wolbachia-induced cytoplasmic incompatibility on host population size in natural and manipulated systems. Proc R Soc Lond B Biol Sci 269:437–445CrossRefGoogle Scholar
  6. Farkas JZ, Hinow P (2010) Structured and unstructured continuous models for Wolbachia infections. Bull Math Biol 72:2067–2088MathSciNetCrossRefzbMATHGoogle Scholar
  7. Hamm CA, Begun DJ, Vo A, Smith CC, Saelao P, Shaver AO, Jaenike J, Turelli M (2014) Wolbachia do not live by reproductive manipulation alone: infection polymorphism in Drosophila suzukii and D. subpulchrella. Mol Ecol 23:4871–4885CrossRefGoogle Scholar
  8. Haygood R, Turelli M (2009) Evolution of incompatibility inducing microbes in subdivided host populations. Evolution 63:432–447CrossRefGoogle Scholar
  9. Hirsch MW, Smale S, Devaney RL (2003) Differential equations, dynamical systems, and an introduction to chaos, 2nd edn. Academic Press, San DiegozbMATHGoogle Scholar
  10. Hu L, Huang M, Tang M, Yu J, Zheng B (2015) Wolbachia spread dynamics in stochastic environments. Theor Popul Biol 106:32–44CrossRefzbMATHGoogle Scholar
  11. Huang M, Tang M, Yu J (2015) Wolbachia infection dynamics by reaction-diffusion equations. Sci China Math 58:77–96MathSciNetCrossRefzbMATHGoogle Scholar
  12. Huang M, Yu J, Hu L, Zheng B (2016) Qualitative analysis for a Wolbachia infection model with diffusion. Sci China Math 59:1249–1266MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hoffmann AA, Turelli M (1988) Unidirectional incompatibility in Drosophila simulans: inheritance, geographic variation and fitness effects. Genetics 119:435–444Google Scholar
  14. Hoffmann AA, Iturbeormaetxe I, Callahan AG, Phillips BL, Billington K, Axford JK, Montgomery B, Turley AP, O’Neill SL (2014) Stability of the \(w\)Mel Wolbachia infection following Invasion into Aedes aegypti populations. PLoS Negl Trop Dis 8(9):e3115–e3115CrossRefGoogle Scholar
  15. Hoffmann AA, Montgomery BL, Popovici J, Iturbe-Ormaetxe I, Johnson PH, Muzzi F, Greenfield M, Durkan M, Leong YS, Dong Y, Cook H, Axford J, Callahan AG, Kenny N, Omodei C, McGraw EA, Ryan PA, Ritchie SA, Turelli M, O’Neill SL (2011) Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission. Nature 476:454–457CrossRefGoogle Scholar
  16. Keeling MJ, Jiggins FM, Read JM (2003) The invasion and coexistence of competing Wolbachia strains. Heredity 91:382–388CrossRefGoogle Scholar
  17. Kriesner P, Hoffmann AA, Lee SF, Turelli M, Weeks AR (2013) Rapid sequential spread of two Wolbachia variants in Drosophila simulans. PLoS Pathog 9(9):e1003607CrossRefGoogle Scholar
  18. Mcmeniman CJ, Lane RV, Cass BN, Fong AWC, Sidhu M, Wang YF, O’Neill SL (2009) Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypi. Science 323:141–144CrossRefGoogle Scholar
  19. Nisbet RM, Gurney WSC (1983) The systematic formulation of population models for insects with dynamically varying instar duration. Theor Popul Biol 23:114–135MathSciNetCrossRefzbMATHGoogle Scholar
  20. Rasgon JL, Styer LM, Scott TW (2003) Wolbachia-induced mortality as a mechanism to modulate pathogen transmission by vector arthropods. J Med Entomol 40:125–132CrossRefGoogle Scholar
  21. Turelli M, Hoffmann AA (1991) Rapid spread of an inherited incompatibility factor in California Drosophila. Nature 353:440–442CrossRefGoogle Scholar
  22. Turelli M, Hoffmann AA (1995) Cytoplasmic incompatibility in Drosophila simulans: dynamics and parameter estimates from natural populations. Genetics 140:1319–1338Google Scholar
  23. Turelli M (2010) Cytoplasmic incompatibility in populations with overlapping generations. Evolution 64:232–241CrossRefGoogle Scholar
  24. Waltz E (2016) US reviews plan to infect mosquitoes with bacteria to stop disease. Nature 533:450–451CrossRefGoogle Scholar
  25. Walker T, Johnson PH, Moreira LA, Iturbe-Ormaetxe I, Frentiu FD, McMeniman CJ, Leong YS, Dong Y, Axford J, Kriesner P, Lloyd AL, Ritchie SA, O’Neill SL, Hoffmann AA (2011) The \(w\)Mel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations. Nature 476:450–453CrossRefGoogle Scholar
  26. Weeks AR, Turelli M, Harcombe WR, Reynolds KT, Hoffmann AA (2007) From parasite to mutualist: rapid evolution of Wolbachia in natural populations of Drosophila. PLoS Biol 5(5):e114CrossRefGoogle Scholar
  27. Xi Z, Khoo CC, Dobson SL (2005) Wolbachia establishment and invasion in an Aedes aegypti laboratory population. Science 310:326–328CrossRefGoogle Scholar
  28. Zhang Z, Ding T, Huang W, Dong Z (2006) Qualitative theory of differential equations, vol 101. American Mathematical Soc, ProvidenceGoogle Scholar
  29. Zheng B, Tang M, Yu J (2014) Modeling Wolbachia spread in mosquitoes through delay differential equations. SIAM J Appl Math 74:743–770MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Bo Zheng
    • 1
    • 2
  • Moxun Tang
    • 2
  • Jianshe Yu
    • 1
  • Junxiong Qiu
    • 1
  1. 1.College of Mathematics and Information SciencesGuangzhou UniversityGuangzhouPeople’s Republic of China
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

Personalised recommendations