2018, 15(2): 523-541. doi: 10.3934/mbe.2018024

Complex wolbachia infection dynamics in mosquitoes with imperfect maternal transmission

1. 

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong 510006, China

2. 

Department of Parasitology of Zhongshan School of Medicine, Sun Yat-sen University, Guangzhou 510080, China

3. 

Key laboratory for Tropical Diseases Control, (SYSU) Ministry of Education, Guangzhou, Guangdong 510080, China

* Corresponding author: jsyu@gzhu.edu.cn

Received  December 27, 2016 Accepted  May 2, 2017 Published  June 2017

Dengue, malaria, and Zika are dangerous diseases primarily transmitted by Aedes aegypti, Aedes albopictus, and Anopheles stephensi. In the last few years, a new disease control method, besides pesticide spraying to kill mosquitoes, has been developed by releasing mosquitoes carrying bacterium Wolbachia into the natural areas to infect the wild population of mosquitoes and block disease transmission. The bacterium is transmitted by infected mothers and the maternal transmission was assumed to be perfect in virtually all previous models. However, recent experiments on Aedes aegypti and Anopheles stephensi showed that the transmission can be imperfect. In this work, we develop a model to describe how the imperfect maternal transmission affects the dynamics of Wolbachia spread. We establish two useful identities and employ them to find sufficient and necessary conditions under which the system exhibits monomorphic, bistable, and polymorphic dynamics. These analytical results may help find a plausible explanation for the recent observation that the Wolbachia strain wMelPop failed to establish in the natural populations in Australia and Vietnam.

Citation: Bo Zheng, Wenliang Guo, Linchao Hu, Mugen Huang, Jianshe Yu. Complex wolbachia infection dynamics in mosquitoes with imperfect maternal transmission. Mathematical Biosciences & Engineering, 2018, 15 (2) : 523-541. doi: 10.3934/mbe.2018024
References:
[1]

G. BianY. XuP. Lu and Z. Xi, The endosymbiotic bacterium Wolbachia induces resistance to dengue virus in Aedes aegypti, PLoS Pathogens, 6 (2010), e1000833. doi: 10.1371/journal.ppat.1000833.

[2]

G. BianD. JoshiY. DongP. LuG. ZhouX. PanY. XuG. Dimopoulos and Z. Xi, Wolbachia invades Anopheles stephensi populations and induces refractoriness to plasmodium infection, Science, 340 (2013), 748-751. doi: 10.1126/science.1236192.

[3]

L. B. CarringtonJ. R. LipkowitzA. A. Hoffmann and M. Turelli, A re-examination of Wolbachia-induced cytoplasmic incompatibility in California Drosophila simulans, PLoS One, 6 (2011), e22565.

[4]

E. Caspari and G. S. Watson, On the evolutionary importance of cytoplasmic sterility in mosquitoes, Evolution, 13 (1959), 568-570. doi: 10.2307/2406138.

[5]

H. L. DutraL. M Dos SantosE. P. CaragataJ. B. SilvaD. A. Villela and R. Maciel-De-Freitas, From lab to field: The influence of urban landscapes on the invasive potential of Wolbachia in Brazilian Aedes aegypti mosquitoes, PLoS Neglected Tropical Diseases, 9 (2015), e0003689. doi: 10.1371/journal.pntd.0003689.

[6]

H. L. DutraM. N. RochaF. B. DiasS. B. MansurE. P. Caragata and L. A. Moreira, Wolbachia, blocks currently circulating Zika virus isolates in Brazilian Aedes aegypti, Cell Host Microbe, 19 (2016), 771-774. doi: 10.1016/j.chom.2016.04.021.

[7]

S. L. DobsonC. W. Fox and F. M. Jiggins, The effect of Wolbachia-induced cytoplasmic incompatibility on host population size in natural and manipulated systems, Proceedings of the Royal Society B Biological Sciences, 269 (2002), 437-445. doi: 10.1098/rspb.2001.1876.

[8]

J. Z. Farkas and P. Hinow, Structured and unstructured continuous models for Wolbachia infections, Bulletin of Mathematical Biology, 72 (2010), 2067-2088. doi: 10.1007/s11538-010-9528-1.

[9]

P. E. Fine, On the dynamics of symbiote-dependent cytoplasmic incompatibility in culicine mosquitoes, Journal of Invertebrate Pathology, 31 (1978), 10-18. doi: 10.1016/0022-2011(78)90102-7.

[10]

F. D. FrentiuT. ZakirT. WalkerJ. PopoviciA. T. Pyke and D. H. A. Van, Limited dengue virus replication in field-collected Aedes aegypti mosquitoes infected with Wolbachia, PLoS Neglected Tropical Diseases, 8 (2014), e2688. doi: 10.1371/journal.pntd.0002688.

[11]

C. A. HammD. J. BegunA. VoC. C. SmithP. SaelaoA. O. ShaverJ. Jaenike and M. Turelli, Wolbachia do not live by reproductive manipulation alone: Infection polymorphism in Drosophila suzukii and D. subpulchrella, Molecular Ecology, 23 (2014), 4871-4885.

[12]

R. Haygood and M. Turelli, Evolution of incompatibility-inducing microbes in subdivided host populations, Evolution, 63 (2009), 432-447. doi: 10.1111/j.1558-5646.2008.00550.x.

[13]

M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, And An Introduction to Chaos Second edition. Pure and Applied Mathematics (Amsterdam), 60. Elsevier/Academic Press, Amsterdam, 2004.

[14]

A. A. Hoffmann and M. Turelli, Unidirectional incompatibility in Drosophila simulans: Inheritance, geographic variation and fitness effects, Genetics, 119 (1988), 435-444.

[15]

A. A. Hoffmann, M. Turelli and L. G. Harshman, Factors affecting the distribution of cytoplasmic incompatibility in Drosophila simulans, Genetics, 126 (1991), 933–948.

[16]

A. A. Hoffmann and M. Turelli, Cytoplasmic incompatibility in insects, In Influential Passengers: Inherited Microorganisms and Arthropod Reproduction, S. L. O'Neill et al. , eds, Oxford University Press, Oxford, (1997), 42–80.

[17]

A. A. HoffmannB. L. MontgomeryJ. PopoviciI. IturbeormaetxeP. H. JohnsonF. MuzziM. GreenfieldM. DurkanY. S. LeongY. DongH. Cook and J. Axford, J. Axford, Gallahan, N. Kenny, C. Omodei, E. A. McGraw, P. A. Ryan, S. A. Ritchie, M. Turelli and S. L. O'Neill, Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission, Nature, 476 (2011), 454-459.

[18]

A. A. HoffmannI. IturbeormaetxeA. G. CallahanB. L. PhillipsK. BillingtonJ. K. AxfordB. MontgomeryA. P. Turley and S. L. O'Neill, Stability of the wMel Wolbachia infection following Invasion into Aedes aegypti populations, PLoS Neglected Tropical Diseases, 8 (2014), e3115-e3115. doi: 10.1371/journal.pntd.0003115.

[19]

L. HuM. HuangM. TangJ. Yu and B. Zheng, Wolbachia spread dynamics in stochastic environments, Theoretical Population Biology, 106 (2015), 32-44. doi: 10.1016/j.tpb.2015.09.003.

[20]

M. HuangM. Tang and Y. Yu, Wolbachia infection dynamics by reaction-diffusion equations, Science China Mathematics, 58 (2015), 77-96. doi: 10.1007/s11425-014-4934-8.

[21]

M. HuangJ. YuL. Hu and B. Zheng, Qualitative analysis for a Wolbachia infection model with diffusion, Science China Mathematics, 59 (2016), 1249-1266. doi: 10.1007/s11425-016-5149-y.

[22]

M. J. KeelingF. M. Jiggins and J. M. Read, The invasion and coexistence of competing Wolbachia strains, Heredity, 91 (2003), 382-388. doi: 10.1038/sj.hdy.6800343.

[23]

P. KriesnerA. A. HoffmannS. F. LeeM. Turelli and A. R. Weeks, Rapid sequential spread of two Wolbachia variants in Drosophila simulan, PLoS Pathogens, 9 (2013), e1003607. doi: 10.1371/journal.ppat.1003607.

[24]

P. KriesnerW. R. ConnerA. R. WeeksM. Turelli and A. A. Hoffmann, Wolbachia infection frequency cline in Drosophila melanogaster and the possible role of reproductive dormancy, Evolution, 70 (2016), 979-997.

[25]

C. J. McmenimanR. V. LaneB. N. CassA. W. FongM. SidhuY. F. Wang and S. L. O'Neill, Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141-144. doi: 10.1126/science.1165326.

[26]

T. H. NguyenH. L. NguyenT. Y. NguyenS. N. VuN. D. TranT. N. LeS. KutcherT. P. HurstT. T. DuongJ. A. JefferyJ. M. DarbroB. H. KayI. Iturbe-OrmaetxeJ. PopoviciB. L. MontgomeryA. P. TurleyF. ZigtermanH. CookP. E. CookP. H. JohnsonP. A. RyanC. J. PatonS. A. RitchieC. P. SimmonsL. O'Neill and A. A. Hoffmann, Field evaluation of the establishment potential of wMelPop Wolbachia in Australia and Vietnam for dengue control, Parasites Vectors, 8 (2015), p563. doi: 10.1186/s13071-015-1174-x.

[27]

J. L. RasgonL. M. Styer and T. W. Scott, Wolbachia-induced mortality as a mechanism to modulate pathogen transmission by vector arthropods, Journal of Medical Entomology, 40 (2003), 125-132. doi: 10.1603/0022-2585-40.2.125.

[28]

P. A. RossI. WiwatanaratanabutrJ. K. AxfordV. L. WhiteN. M. Endersby-Harshman and A. A. Hoffmann, Wolbachia infections in Aedes aegypti differ markedly in their response to cyclical heat stress, PLoS Pathogens, 13 (2017), e1006006. doi: 10.1371/journal.ppat.1006006.

[29]

J. G. SchraiberA. N. KaczmarczykR. KwokM. ParkR. SilversteinF. U. RutaganiraT. AggarwalM. A. SchwemmerC. L. HomR. K. Grosberg and S. J. Schreiber, Constraints on the use of lifespan-shortening Wolbachia to control dengue fever, Journal of Theoretical Biology, 297 (2012), 26-32. doi: 10.1016/j.jtbi.2011.12.006.

[30]

M. Turelli and A. A. Hoffmann, Rapid spread of an inherited incompatibility factor in California Drosophila, Nature, 353 (1991), 440-442. doi: 10.1038/353440a0.

[31]

M. Turelli, Evolution of incompatibility-inducing microbes and their hosts, Evolution, 48 (1994), 1500-1513. doi: 10.2307/2410244.

[32]

M. Turelli and A. A. Hoffmann, Cytoplasmic incompatibility in Drosophila simulans: Dynamics and parameter estimates from natural populations, Genetics, 140 (1995), 1319-1338.

[33]

M. Turelli, Microbe-induced cytoplasmic incompatibility as a mechanism for introducing transgenes into arthropod populations, Insect Molecular Biology, 8 (1999), 243-255. doi: 10.1046/j.1365-2583.1999.820243.x.

[34]

M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232-241. doi: 10.1111/j.1558-5646.2009.00822.x.

[35]

T. WalkerP. H. JohnsonL. A. MoreiraI. IturbeormaetxeF. D. FrentiuC. J. McMenimanY. S. LeongY. DongJ. AxfordP. KriesnerA. L. LloydS. A. RitchieS. L. O'Neill and A. A. Hoffmann, The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450-453. doi: 10.1038/nature10355.

[36]

A. R. WeeksM. TurelliW. R. HarcombeK. T. Reynolds and A. A. Hoffmann, From parasite to mutualist: Rapid evolution of Wolbachia in natural populations of Drosophila, PLoS Biology, 5 (2007), e114-e114. doi: 10.1371/journal.pbio.0050114.

[37]

Z. Xi and S. L. Dobson, Characterization of Wolbachia transfection efficiency by using microinjection of embryonic cytoplasm and embryo homogenate, Applied Environmental Microbiology, 71 (2005), 3199-3204. doi: 10.1128/AEM.71.6.3199-3204.2005.

[38]

Z. XiJ. L. DeanC. C. Khoo and S. L. Dobson, Generation of a novel Wolbachia infection in Aedes albopictus (Asian tiger mosquito) via embryonic microinjection, Insect Biochemistry Molecular Biology, 35 (2005), 903-910. doi: 10.1016/j.ibmb.2005.03.015.

[39]

Z. XiC. C. Khoo and S. L. Dobson, Wolbachia establishment and invasion in an Aedes aegypti laboratory population, Science, 310 (2005), 326-328. doi: 10.1126/science.1117607.

[40]

Z. XiC. C. Khoo and S. L. Dobson, Interspecific transfer of Wolbachia into the mosquito disease vector Aedes albopictus, Proceedings of the Royal Society B Biological Sciences, 273 (2006), 1317-1322. doi: 10.1098/rspb.2005.3405.

[41]

B. ZhengM. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM Journal on Applied Mathematics, 74 (2014), 743-770. doi: 10.1137/13093354X.

show all references

References:
[1]

G. BianY. XuP. Lu and Z. Xi, The endosymbiotic bacterium Wolbachia induces resistance to dengue virus in Aedes aegypti, PLoS Pathogens, 6 (2010), e1000833. doi: 10.1371/journal.ppat.1000833.

[2]

G. BianD. JoshiY. DongP. LuG. ZhouX. PanY. XuG. Dimopoulos and Z. Xi, Wolbachia invades Anopheles stephensi populations and induces refractoriness to plasmodium infection, Science, 340 (2013), 748-751. doi: 10.1126/science.1236192.

[3]

L. B. CarringtonJ. R. LipkowitzA. A. Hoffmann and M. Turelli, A re-examination of Wolbachia-induced cytoplasmic incompatibility in California Drosophila simulans, PLoS One, 6 (2011), e22565.

[4]

E. Caspari and G. S. Watson, On the evolutionary importance of cytoplasmic sterility in mosquitoes, Evolution, 13 (1959), 568-570. doi: 10.2307/2406138.

[5]

H. L. DutraL. M Dos SantosE. P. CaragataJ. B. SilvaD. A. Villela and R. Maciel-De-Freitas, From lab to field: The influence of urban landscapes on the invasive potential of Wolbachia in Brazilian Aedes aegypti mosquitoes, PLoS Neglected Tropical Diseases, 9 (2015), e0003689. doi: 10.1371/journal.pntd.0003689.

[6]

H. L. DutraM. N. RochaF. B. DiasS. B. MansurE. P. Caragata and L. A. Moreira, Wolbachia, blocks currently circulating Zika virus isolates in Brazilian Aedes aegypti, Cell Host Microbe, 19 (2016), 771-774. doi: 10.1016/j.chom.2016.04.021.

[7]

S. L. DobsonC. W. Fox and F. M. Jiggins, The effect of Wolbachia-induced cytoplasmic incompatibility on host population size in natural and manipulated systems, Proceedings of the Royal Society B Biological Sciences, 269 (2002), 437-445. doi: 10.1098/rspb.2001.1876.

[8]

J. Z. Farkas and P. Hinow, Structured and unstructured continuous models for Wolbachia infections, Bulletin of Mathematical Biology, 72 (2010), 2067-2088. doi: 10.1007/s11538-010-9528-1.

[9]

P. E. Fine, On the dynamics of symbiote-dependent cytoplasmic incompatibility in culicine mosquitoes, Journal of Invertebrate Pathology, 31 (1978), 10-18. doi: 10.1016/0022-2011(78)90102-7.

[10]

F. D. FrentiuT. ZakirT. WalkerJ. PopoviciA. T. Pyke and D. H. A. Van, Limited dengue virus replication in field-collected Aedes aegypti mosquitoes infected with Wolbachia, PLoS Neglected Tropical Diseases, 8 (2014), e2688. doi: 10.1371/journal.pntd.0002688.

[11]

C. A. HammD. J. BegunA. VoC. C. SmithP. SaelaoA. O. ShaverJ. Jaenike and M. Turelli, Wolbachia do not live by reproductive manipulation alone: Infection polymorphism in Drosophila suzukii and D. subpulchrella, Molecular Ecology, 23 (2014), 4871-4885.

[12]

R. Haygood and M. Turelli, Evolution of incompatibility-inducing microbes in subdivided host populations, Evolution, 63 (2009), 432-447. doi: 10.1111/j.1558-5646.2008.00550.x.

[13]

M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, And An Introduction to Chaos Second edition. Pure and Applied Mathematics (Amsterdam), 60. Elsevier/Academic Press, Amsterdam, 2004.

[14]

A. A. Hoffmann and M. Turelli, Unidirectional incompatibility in Drosophila simulans: Inheritance, geographic variation and fitness effects, Genetics, 119 (1988), 435-444.

[15]

A. A. Hoffmann, M. Turelli and L. G. Harshman, Factors affecting the distribution of cytoplasmic incompatibility in Drosophila simulans, Genetics, 126 (1991), 933–948.

[16]

A. A. Hoffmann and M. Turelli, Cytoplasmic incompatibility in insects, In Influential Passengers: Inherited Microorganisms and Arthropod Reproduction, S. L. O'Neill et al. , eds, Oxford University Press, Oxford, (1997), 42–80.

[17]

A. A. HoffmannB. L. MontgomeryJ. PopoviciI. IturbeormaetxeP. H. JohnsonF. MuzziM. GreenfieldM. DurkanY. S. LeongY. DongH. Cook and J. Axford, J. Axford, Gallahan, N. Kenny, C. Omodei, E. A. McGraw, P. A. Ryan, S. A. Ritchie, M. Turelli and S. L. O'Neill, Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission, Nature, 476 (2011), 454-459.

[18]

A. A. HoffmannI. IturbeormaetxeA. G. CallahanB. L. PhillipsK. BillingtonJ. K. AxfordB. MontgomeryA. P. Turley and S. L. O'Neill, Stability of the wMel Wolbachia infection following Invasion into Aedes aegypti populations, PLoS Neglected Tropical Diseases, 8 (2014), e3115-e3115. doi: 10.1371/journal.pntd.0003115.

[19]

L. HuM. HuangM. TangJ. Yu and B. Zheng, Wolbachia spread dynamics in stochastic environments, Theoretical Population Biology, 106 (2015), 32-44. doi: 10.1016/j.tpb.2015.09.003.

[20]

M. HuangM. Tang and Y. Yu, Wolbachia infection dynamics by reaction-diffusion equations, Science China Mathematics, 58 (2015), 77-96. doi: 10.1007/s11425-014-4934-8.

[21]

M. HuangJ. YuL. Hu and B. Zheng, Qualitative analysis for a Wolbachia infection model with diffusion, Science China Mathematics, 59 (2016), 1249-1266. doi: 10.1007/s11425-016-5149-y.

[22]

M. J. KeelingF. M. Jiggins and J. M. Read, The invasion and coexistence of competing Wolbachia strains, Heredity, 91 (2003), 382-388. doi: 10.1038/sj.hdy.6800343.

[23]

P. KriesnerA. A. HoffmannS. F. LeeM. Turelli and A. R. Weeks, Rapid sequential spread of two Wolbachia variants in Drosophila simulan, PLoS Pathogens, 9 (2013), e1003607. doi: 10.1371/journal.ppat.1003607.

[24]

P. KriesnerW. R. ConnerA. R. WeeksM. Turelli and A. A. Hoffmann, Wolbachia infection frequency cline in Drosophila melanogaster and the possible role of reproductive dormancy, Evolution, 70 (2016), 979-997.

[25]

C. J. McmenimanR. V. LaneB. N. CassA. W. FongM. SidhuY. F. Wang and S. L. O'Neill, Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141-144. doi: 10.1126/science.1165326.

[26]

T. H. NguyenH. L. NguyenT. Y. NguyenS. N. VuN. D. TranT. N. LeS. KutcherT. P. HurstT. T. DuongJ. A. JefferyJ. M. DarbroB. H. KayI. Iturbe-OrmaetxeJ. PopoviciB. L. MontgomeryA. P. TurleyF. ZigtermanH. CookP. E. CookP. H. JohnsonP. A. RyanC. J. PatonS. A. RitchieC. P. SimmonsL. O'Neill and A. A. Hoffmann, Field evaluation of the establishment potential of wMelPop Wolbachia in Australia and Vietnam for dengue control, Parasites Vectors, 8 (2015), p563. doi: 10.1186/s13071-015-1174-x.

[27]

J. L. RasgonL. M. Styer and T. W. Scott, Wolbachia-induced mortality as a mechanism to modulate pathogen transmission by vector arthropods, Journal of Medical Entomology, 40 (2003), 125-132. doi: 10.1603/0022-2585-40.2.125.

[28]

P. A. RossI. WiwatanaratanabutrJ. K. AxfordV. L. WhiteN. M. Endersby-Harshman and A. A. Hoffmann, Wolbachia infections in Aedes aegypti differ markedly in their response to cyclical heat stress, PLoS Pathogens, 13 (2017), e1006006. doi: 10.1371/journal.ppat.1006006.

[29]

J. G. SchraiberA. N. KaczmarczykR. KwokM. ParkR. SilversteinF. U. RutaganiraT. AggarwalM. A. SchwemmerC. L. HomR. K. Grosberg and S. J. Schreiber, Constraints on the use of lifespan-shortening Wolbachia to control dengue fever, Journal of Theoretical Biology, 297 (2012), 26-32. doi: 10.1016/j.jtbi.2011.12.006.

[30]

M. Turelli and A. A. Hoffmann, Rapid spread of an inherited incompatibility factor in California Drosophila, Nature, 353 (1991), 440-442. doi: 10.1038/353440a0.

[31]

M. Turelli, Evolution of incompatibility-inducing microbes and their hosts, Evolution, 48 (1994), 1500-1513. doi: 10.2307/2410244.

[32]

M. Turelli and A. A. Hoffmann, Cytoplasmic incompatibility in Drosophila simulans: Dynamics and parameter estimates from natural populations, Genetics, 140 (1995), 1319-1338.

[33]

M. Turelli, Microbe-induced cytoplasmic incompatibility as a mechanism for introducing transgenes into arthropod populations, Insect Molecular Biology, 8 (1999), 243-255. doi: 10.1046/j.1365-2583.1999.820243.x.

[34]

M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232-241. doi: 10.1111/j.1558-5646.2009.00822.x.

[35]

T. WalkerP. H. JohnsonL. A. MoreiraI. IturbeormaetxeF. D. FrentiuC. J. McMenimanY. S. LeongY. DongJ. AxfordP. KriesnerA. L. LloydS. A. RitchieS. L. O'Neill and A. A. Hoffmann, The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450-453. doi: 10.1038/nature10355.

[36]

A. R. WeeksM. TurelliW. R. HarcombeK. T. Reynolds and A. A. Hoffmann, From parasite to mutualist: Rapid evolution of Wolbachia in natural populations of Drosophila, PLoS Biology, 5 (2007), e114-e114. doi: 10.1371/journal.pbio.0050114.

[37]

Z. Xi and S. L. Dobson, Characterization of Wolbachia transfection efficiency by using microinjection of embryonic cytoplasm and embryo homogenate, Applied Environmental Microbiology, 71 (2005), 3199-3204. doi: 10.1128/AEM.71.6.3199-3204.2005.

[38]

Z. XiJ. L. DeanC. C. Khoo and S. L. Dobson, Generation of a novel Wolbachia infection in Aedes albopictus (Asian tiger mosquito) via embryonic microinjection, Insect Biochemistry Molecular Biology, 35 (2005), 903-910. doi: 10.1016/j.ibmb.2005.03.015.

[39]

Z. XiC. C. Khoo and S. L. Dobson, Wolbachia establishment and invasion in an Aedes aegypti laboratory population, Science, 310 (2005), 326-328. doi: 10.1126/science.1117607.

[40]

Z. XiC. C. Khoo and S. L. Dobson, Interspecific transfer of Wolbachia into the mosquito disease vector Aedes albopictus, Proceedings of the Royal Society B Biological Sciences, 273 (2006), 1317-1322. doi: 10.1098/rspb.2005.3405.

[41]

B. ZhengM. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM Journal on Applied Mathematics, 74 (2014), 743-770. doi: 10.1137/13093354X.

Figure 1.  The phase portrait in the degenerate case. (A) When $\beta\mu\leq 2$, the curve $\Gamma_y$ stays in the domain $x+y>\beta\mu/2$ as a decreasing curve connecting $E_1$ and $E_2$. (B) When $\beta\mu>2$, the curve $\Gamma_y$ starts from $E_2$ in the domain $1/2<x+y<\beta\mu/2$, increases first and then intersects the line $x+y=\beta\mu/2$ to enter the domain $x+y>\beta\mu/2$, and remains in this domain as a decreasing curve until reaching $E_1$. In both cases, given the initial value $(x_0, y_0)\in \mathbb{R}_{+0}^2\setminus \Gamma_y$, solution of (4)-(5) tends to the unique intersecting point (the hollow point) of the curve (21) and the nullcline $\Gamma_y$
Figure 2.  The separatrices $y=h(x)$ are sandwiched by $y=h_0(x)$ and $y=h_1(x)$
Figure 3.  Dynamics complexity induced by imperfect maternal transmission. (A) When $\beta\leq\delta$, small $\mu$ corresponds to bistable case (C3). Increasing $\mu$ leads to (C9) and (C7), and the positive solutions converge to the monomorphic state $E_2$. (B) When $\beta>\delta$, we consider three subclasses. (B1): $\nu_1<\nu_2$. As $\mu$ increases, (C4), (C6), (C2), (C8) and (C7) occur consecutively, and the system transits from the monomorphic state $E_1$ to the polymorphic state $E^*$ and finally the monomorphic state $E_2$. (B2): When $\nu_1=\nu_2$, only (C1), (C2) and (C7) can occur. The system is polymorphic at $E^*$ when $\mu<\nu_1$, and is monomorphic at $E_2$ when $\mu>\nu_1$. (B3): $\nu_1>\nu_2$. As $\mu$ increases, (C4), (C5), (C3), (C9) and (C7) occur one after another, and the system transits from the monomorphic state $E_1$ to the bistable state and finally the monomorphic state $E_2$
Figure 4.  The maternal transmission leakage hinders the Wolbachia invasion. (Left) Fix $y_0=0.02$, we plot the ratio of $x_0$ to $y_0$ against the leakage rate $\mu=0.01, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.45$ and $0.5$ for the leakage case (4)-(5). (Right) The combination of negative effects of imperfect maternal transmission and incomplete CI could make Wolbachia invasion impossible. Again, fix $y_0=0.02$ and $s_h=0.8369$. When the leakage rate is 0.15, the incomplete CI mechanism leads the threshold ratio increase from 1.95 to 7. Worse than that, when the leakage rate is 0.3, the ratio as high as 200 is still insufficient to ensure successful Wolbachia invasion
[1]

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