December  2020, 25(12): 4755-4778. doi: 10.3934/dcdsb.2020123

Wolbachia infection dynamics in mosquito population with the CI effect suffering by uninfected ova produced by infected females

1. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

2. 

College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang 150001, China

* Corresponding author: lxue@hrbeu.edu.cn (L. Xue)

Received  November 2019 Revised  December 2019 Published  March 2020

Mosquito-borne diseases pose a great threat to humans' health. Wolbachia is a promising biological weapon to control the mosquito population, and is not harmful to humans' health, environment, and ecology. In this work, we present a stage-structure model to investigate the effective releasing strategies for Wolbachia-infected mosquitoes. Besides some key factors for Wolbachia infection, the CI effect suffering by uninfected ova produced by infected females, which is often neglected, is also incorporated. We analyze the conditions under which Wolbachia infection still can be established even if the basic reproduction number is less than unity. Numerical simulations manifest that the threshold value of infected mosquitoes required to be released at the beginning can be evaluated by the stable manifold of a saddle equilibrium, and low levels of MK effect, fitness costs, as well as high levels of CI effect and maternal inheritance all contribute to the establishment of Wolbachia-infection. Moreover, our results suggest that ignoring the CI effect suffering by uninfected ova produced by infected females may result in the overestimation of the threshold infection level for the Wolbachia invasion.

Citation: Hui Wan, Yunyan Cao, Ling Xue. Wolbachia infection dynamics in mosquito population with the CI effect suffering by uninfected ova produced by infected females. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4755-4778. doi: 10.3934/dcdsb.2020123
References:
[1]

A. I. AdekunleM. T. Meehan and E. S. McBryde, Mathematical analysis of a Wolbachia invasive model with imperfect maternal transmission and loss of Wolbachia infection, Infectious Disease Modelling, 4 (2019), 265-285.  doi: 10.1016/j.idm.2019.10.001.  Google Scholar

[2]

N. Becker, D. Petrić, C. Boase, J. Lane, M. Zgomba, C. Dahl and A. Kaiser, Mosquitoes and Their Control, Vol. 2, Springer-Verlag, Berlin-Heidelberg, 2003. Google Scholar

[3]

G. Bian, Y. Xu, P. Lu, Y. Xie 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.  Google Scholar

[4]

P.-A. BlimanM. S. AronnaF. C. CoelhoM. A. da Silva and A. H. B. Moacyr, Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control, Journal of Mathematical Biology, 76 (2018), 1269-1300.  doi: 10.1007/s00285-017-1174-x.  Google Scholar

[5]

D. E. Campo-DuarteO. VasilievaD. Cardona-Salgado and M. Svinin, Optimal control approach for establishing wMelPop Wolbachia infection among wild Aedes aegypti populations, Journal of Mathematical Biology, 76 (2018), 1907-1950.  doi: 10.1007/s00285-018-1213-2.  Google Scholar

[6]

E. Caspari and G. S. Watson, [10.1111/j.1558-5646.1959.tb03045.x] On the evolutionary importance of cytoplasmic sterility in mosquitoes, Evolution, 13 (1959), 568-570. Google Scholar

[7]

M. H. Chan and P. S. Kim, Modelling a Wolbachia invasion using a slow–fast dispersal reaction–diffusion approach, Bulletin of Mathematical Biology, 75 (2013), 1501-1523.  doi: 10.1007/s11538-013-9857-y.  Google Scholar

[8]

H. DelatteG. GimonneauA. Triboire and D. Fontenille, Influence of temperature on immature development, survival, longevity, fecundity, and gonotrophic cycles of Aedes albopictus, vector of chikungunya and dengue in the Indian Ocean, Journal of Medical Entomology, 46 (2009), 33-41.  doi: 10.1603/033.046.0105.  Google Scholar

[9]

I. DorigattiC. McCormackG. Nedjati-Gilani and N. M. Ferguson, Using Wolbachia for dengue control: Insights from modelling, Trends in Parasitology, 34 (2018), 102-113.  doi: 10.1016/j.pt.2017.11.002.  Google Scholar

[10]

M. EgasF. Vala and J. A. J. Breeuwer, On the evolution of cytoplasmic incompatibility in haplodiploid species, Evolution, 56 (2002), 1101-1109.  doi: 10.1111/j.0014-3820.2002.tb01424.x.  Google Scholar

[11]

J. FangS. A. Gourley and Y. Lou, Stage-structured models of intra- and inter-specific competetition within age classes, Journal of Differential Equations, 260 (2016), 1918-1953.  doi: 10.1016/j.jde.2015.09.048.  Google Scholar

[12]

J. FangG. Lin and H. Wan, Analysis of a stage-structured dengue model, Discrete Continuous Dynamical Systems-B, 23 (2018), 4045-4061.  doi: 10.3934/dcdsb.2018125.  Google Scholar

[13]

J. Z. FarkasS. A. GourleyR. Liu and A.-A. Yakubu, Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus, Journal of Mathematical Biology, 75 (2017), 621-647.  doi: 10.1007/s00285-017-1096-7.  Google Scholar

[14]

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.  Google Scholar

[15]

L. M. FieldA. A. JamesM. Turelli and A. Hoffmann, 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.  Google Scholar

[16]

P. A. HancockS. P. Sinkins and H. C. J. Godfray, Population dynamic models of the spread of Wolbachia, The American Naturalist, 177 (2011), 323-333.  doi: 10.1086/658121.  Google Scholar

[17]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[18]

H. Hughes and N. F. Britton, Modelling the use of Wolbachia to control dengue fever transmission, Bulletin of Mathematical Biology, 75 (2013), 796-818.  doi: 10.1007/s11538-013-9835-4.  Google Scholar

[19]

F. M. Jiggins, The spread of Wolbachia through mosquito populations, PLoS Biology, 15 (2017), e2002780. doi: 10.1371/journal.pbio.2002780.  Google Scholar

[20]

D. Joshi, M. J. McFadden, D. Bevins, F. Zhang and Z. Xi, Wolbachia strain wAlbB confers both fitness costs and benefit on Anopheles stephensi, Parasites & Vectors, 7 (2014), 336. doi: 10.1186/1756-3305-7-336.  Google Scholar

[21]

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

[22]

H. Laven, Crossing experiments with culex strains, Evolution, 5 (1951), 370-375.  doi: 10.2307/2405682.  Google Scholar

[23]

D. Li and H. Wan, The threshold infection level for Wolbachia invasion in a two-sex mosquito population model, Bulletin of Mathematical Biology, 81 (2019), 2596-2624.  doi: 10.1007/s11538-019-00620-1.  Google Scholar

[24]

J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, Journal of Biological Dynamics, 11 (2017), 316-333.  doi: 10.1080/17513758.2016.1216613.  Google Scholar

[25]

M.-T. Li, G.-Q. Sun, L. Yakob, H.-P. Zhu, Z. Jin and W.-Y. Zhang, The driving force for 2014 dengue outbreak in Guangdong, China, PloS One, 11 (2016), e0166211. doi: 10.1371/journal.pone.0166211.  Google Scholar

[26]

Y. Li and X. Liu, An impulsive model for Wolbachia infection control of mosquito-borne diseases with general birth and death rate functions, Nonlinear Analysis: Real World Applications, 37 (2017), 412-432.  doi: 10.1016/j.nonrwa.2017.03.003.  Google Scholar

[27]

C. J. McMenimanR. V. LaneB. N. CassA. W. C. 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.  Google Scholar

[28]

S. MungaN. MinakawaG. ZhouA. K. Githeko and G. Yan, Survivorship of immature stages of Anopheles gambiae s.l. (Diptera: Culicidae) in natural habitats in western Kenya highlands, Journal of Medical Entomology, 44 (2007), 758-764.   Google Scholar

[29]

M. Z. NdiiR. I. HicksonD. Allingham and G. Mercer, Modelling the transmission dynamics of dengue in the presence of Wolbachia, Mathematical Biosciences, 262 (2015), 157-166.  doi: 10.1016/j.mbs.2014.12.011.  Google Scholar

[30]

M. Z. NdiiR. I. Hickson and G. N. Mercer, Modelling the introduction of Wolbachia into Aedes aegypti mosquitoes to reduce dengue transmission, The ANZIAM Journal, 53 (2012), 213-227.  doi: 10.1017/S1446181112000132.  Google Scholar

[31]

S. L. O'NeillR. GiordanoA. ColbertT. L. Karr and H. M. Robertson, 16S rRNA phylogenetic analysis of the bacterial endosymbionts associated with cytoplasmic incompatibility in insects, Proceedings of the National Academy of Sciences of the United States of America, 89 (1992), 2699-2702.  doi: 10.1073/pnas.89.7.2699.  Google Scholar

[32]

Z. QuL. Xue and J. M. Hyman, Modeling the transmission of Wolbachia in mosquitoes for controlling mosquito-borne diseases, SIAM Journal on Applied Mathematics, 78 (2018), 826-852.  doi: 10.1137/17M1130800.  Google Scholar

[33]

J. L. Rasgon and T. W. Scott, Wolbachia and cytoplasmic incompatibility in the California Culex pipiens mosquito species complex: Parameter estimates and infection dynamics in natural populations, Genetics, 165 (2003), 2029-2038.   Google Scholar

[34]

A. SaltelliP. AnnoniI. AzziniF. CampolongoM. Ratto and S. Tarantola, Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications, 181 (2010), 259-270.  doi: 10.1016/j.cpc.2009.09.018.  Google Scholar

[35]

F. SarrazinF. Pianosi and T. Wagener, Global Sensitivity Analysis of environmental models: Convergence and validation, Environmental Modelling & Software, 79 (2016), 135-152.  doi: 10.1016/j.envsoft.2016.02.005.  Google Scholar

[36]

P. Schofield, Spatially explicit models of Turelli-Hoffmann Wolbachia invasive wave fronts, Journal of Theoretical Biology, 215 (2002), 121-131.  doi: 10.1006/jtbi.2001.2493.  Google Scholar

[37]

J. G. Schraiber, A. N. Kaczmarczyk, R. Kwok, et al., Constraints on the use of lifespanshortening Wolbachia to control dengue fever, Journal of Theoretical Biology, 297 (2012), 26-32. doi: 10.1016/j.jtbi.2011.12.006.  Google Scholar

[38]

S. P. Sinkins, Wolbachia and cytoplasmic incompatibility in mosquitoes, Insect Biochemistry and Molecular Biology, 34 (2004), 723-729.  doi: 10.1016/j.ibmb.2004.03.025.  Google Scholar

[39]

I. M. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Mathematics and Computers in Simulation, 55 (2001), 271-280.  doi: 10.1016/S0378-4754(00)00270-6.  Google Scholar

[40]

M. Turelli and N. H. Barton, Deploying dengue-suppressing Wolbachia: Robust models predict slow but effective spatial spread in Aedes aegypti, Theoretical Population Biology, 115 (2017), 45-60.  doi: 10.1016/j.tpb.2017.03.003.  Google Scholar

[41]

P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[42]

T. Walker, P. H. Johnson, L. A. Moreira, et al., The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450-453. doi: 10.1038/nature10355.  Google Scholar

[43]

L. Wang, H. Zhao, S. M. Oliva and H. Zhu, Modeling the transmission and control of Zika in Brazil, Scientific Reports, 7 (2017), Art. 7721. doi: 10.1038/s41598-017-07264-y.  Google Scholar

[44]

J. H. Werren, Biology of Wolbachia, Annual Review of Entomology, 42 (1997), 587-609.  doi: 10.1146/annurev.ento.42.1.587.  Google Scholar

[45]

L. Xue, X. Fang and J. M. Hyman, Comparing the effectiveness of different strains of Wolbachia for controlling chikungunya, dengue fever, and zika, PLOS Neglected Tropical Diseases, 12 (2018), e0006666. doi: 10.1371/journal.pntd.0006666.  Google Scholar

[46]

L. XueC. A. ManoreP. Thongsripong and J. M. Hyman, Two-sex mosquito model for the persistence of Wolbachia, Journal of Biological Dynamics, 11 (2017), 216-237.  doi: 10.1080/17513758.2016.1229051.  Google Scholar

[47]

J. H. Yen and A. R. Barr, A new hypothesis of the cause of cytoplasmic incompatibility in Culex pipiens L., Nature, 232 (1971), 657-658.  doi: 10.1038/232657a0.  Google Scholar

[48]

X. ZhangS. Tang and R. A. Cheke, Birth-pulse models of Wolbachia-induced cytoplasmic incompatibility in mosquitoes for dengue virus control, Nonlinear Analysis: Real World Applications, 22 (2015), 236-258.  doi: 10.1016/j.nonrwa.2014.09.004.  Google Scholar

[49]

X. ZhangS. Tang and R. A. Cheke, Models to assess how best to replace dengue virus vectors with Wolbachia-infected mosquito populations, Mathematical Biosciences, 269 (2015), 164-177.  doi: 10.1016/j.mbs.2015.09.004.  Google Scholar

[50]

X. ZhangS. TangR. A. Cheke and H. Zhu, Modeling the effects of augmentation strategies on the control of dengue fever with an impulsive differential equation, Bulletin of Mathematical Biology, 78 (2016), 1968-2010.  doi: 10.1007/s11538-016-0208-7.  Google Scholar

[51]

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.  Google Scholar

[52]

B. ZhengM. TangJ. Yu and J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, Journal of Mathematical Biology, 76 (2018), 235-263.  doi: 10.1007/s00285-017-1142-5.  Google Scholar

[53]

X. Zheng, D. Zhang, Y. Li, C. Yang, Y. Wu, X. Liang, Y. Liang, X. Pan, L. Hu, Q. Sun et al., Incompatible and sterile insect techniques combined eliminate mosquitoes, Nature, 572 (2019), 56-61. doi: 10.1038/s41586-019-1407-9.  Google Scholar

show all references

References:
[1]

A. I. AdekunleM. T. Meehan and E. S. McBryde, Mathematical analysis of a Wolbachia invasive model with imperfect maternal transmission and loss of Wolbachia infection, Infectious Disease Modelling, 4 (2019), 265-285.  doi: 10.1016/j.idm.2019.10.001.  Google Scholar

[2]

N. Becker, D. Petrić, C. Boase, J. Lane, M. Zgomba, C. Dahl and A. Kaiser, Mosquitoes and Their Control, Vol. 2, Springer-Verlag, Berlin-Heidelberg, 2003. Google Scholar

[3]

G. Bian, Y. Xu, P. Lu, Y. Xie 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.  Google Scholar

[4]

P.-A. BlimanM. S. AronnaF. C. CoelhoM. A. da Silva and A. H. B. Moacyr, Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control, Journal of Mathematical Biology, 76 (2018), 1269-1300.  doi: 10.1007/s00285-017-1174-x.  Google Scholar

[5]

D. E. Campo-DuarteO. VasilievaD. Cardona-Salgado and M. Svinin, Optimal control approach for establishing wMelPop Wolbachia infection among wild Aedes aegypti populations, Journal of Mathematical Biology, 76 (2018), 1907-1950.  doi: 10.1007/s00285-018-1213-2.  Google Scholar

[6]

E. Caspari and G. S. Watson, [10.1111/j.1558-5646.1959.tb03045.x] On the evolutionary importance of cytoplasmic sterility in mosquitoes, Evolution, 13 (1959), 568-570. Google Scholar

[7]

M. H. Chan and P. S. Kim, Modelling a Wolbachia invasion using a slow–fast dispersal reaction–diffusion approach, Bulletin of Mathematical Biology, 75 (2013), 1501-1523.  doi: 10.1007/s11538-013-9857-y.  Google Scholar

[8]

H. DelatteG. GimonneauA. Triboire and D. Fontenille, Influence of temperature on immature development, survival, longevity, fecundity, and gonotrophic cycles of Aedes albopictus, vector of chikungunya and dengue in the Indian Ocean, Journal of Medical Entomology, 46 (2009), 33-41.  doi: 10.1603/033.046.0105.  Google Scholar

[9]

I. DorigattiC. McCormackG. Nedjati-Gilani and N. M. Ferguson, Using Wolbachia for dengue control: Insights from modelling, Trends in Parasitology, 34 (2018), 102-113.  doi: 10.1016/j.pt.2017.11.002.  Google Scholar

[10]

M. EgasF. Vala and J. A. J. Breeuwer, On the evolution of cytoplasmic incompatibility in haplodiploid species, Evolution, 56 (2002), 1101-1109.  doi: 10.1111/j.0014-3820.2002.tb01424.x.  Google Scholar

[11]

J. FangS. A. Gourley and Y. Lou, Stage-structured models of intra- and inter-specific competetition within age classes, Journal of Differential Equations, 260 (2016), 1918-1953.  doi: 10.1016/j.jde.2015.09.048.  Google Scholar

[12]

J. FangG. Lin and H. Wan, Analysis of a stage-structured dengue model, Discrete Continuous Dynamical Systems-B, 23 (2018), 4045-4061.  doi: 10.3934/dcdsb.2018125.  Google Scholar

[13]

J. Z. FarkasS. A. GourleyR. Liu and A.-A. Yakubu, Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus, Journal of Mathematical Biology, 75 (2017), 621-647.  doi: 10.1007/s00285-017-1096-7.  Google Scholar

[14]

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.  Google Scholar

[15]

L. M. FieldA. A. JamesM. Turelli and A. Hoffmann, 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.  Google Scholar

[16]

P. A. HancockS. P. Sinkins and H. C. J. Godfray, Population dynamic models of the spread of Wolbachia, The American Naturalist, 177 (2011), 323-333.  doi: 10.1086/658121.  Google Scholar

[17]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[18]

H. Hughes and N. F. Britton, Modelling the use of Wolbachia to control dengue fever transmission, Bulletin of Mathematical Biology, 75 (2013), 796-818.  doi: 10.1007/s11538-013-9835-4.  Google Scholar

[19]

F. M. Jiggins, The spread of Wolbachia through mosquito populations, PLoS Biology, 15 (2017), e2002780. doi: 10.1371/journal.pbio.2002780.  Google Scholar

[20]

D. Joshi, M. J. McFadden, D. Bevins, F. Zhang and Z. Xi, Wolbachia strain wAlbB confers both fitness costs and benefit on Anopheles stephensi, Parasites & Vectors, 7 (2014), 336. doi: 10.1186/1756-3305-7-336.  Google Scholar

[21]

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

[22]

H. Laven, Crossing experiments with culex strains, Evolution, 5 (1951), 370-375.  doi: 10.2307/2405682.  Google Scholar

[23]

D. Li and H. Wan, The threshold infection level for Wolbachia invasion in a two-sex mosquito population model, Bulletin of Mathematical Biology, 81 (2019), 2596-2624.  doi: 10.1007/s11538-019-00620-1.  Google Scholar

[24]

J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, Journal of Biological Dynamics, 11 (2017), 316-333.  doi: 10.1080/17513758.2016.1216613.  Google Scholar

[25]

M.-T. Li, G.-Q. Sun, L. Yakob, H.-P. Zhu, Z. Jin and W.-Y. Zhang, The driving force for 2014 dengue outbreak in Guangdong, China, PloS One, 11 (2016), e0166211. doi: 10.1371/journal.pone.0166211.  Google Scholar

[26]

Y. Li and X. Liu, An impulsive model for Wolbachia infection control of mosquito-borne diseases with general birth and death rate functions, Nonlinear Analysis: Real World Applications, 37 (2017), 412-432.  doi: 10.1016/j.nonrwa.2017.03.003.  Google Scholar

[27]

C. J. McMenimanR. V. LaneB. N. CassA. W. C. 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.  Google Scholar

[28]

S. MungaN. MinakawaG. ZhouA. K. Githeko and G. Yan, Survivorship of immature stages of Anopheles gambiae s.l. (Diptera: Culicidae) in natural habitats in western Kenya highlands, Journal of Medical Entomology, 44 (2007), 758-764.   Google Scholar

[29]

M. Z. NdiiR. I. HicksonD. Allingham and G. Mercer, Modelling the transmission dynamics of dengue in the presence of Wolbachia, Mathematical Biosciences, 262 (2015), 157-166.  doi: 10.1016/j.mbs.2014.12.011.  Google Scholar

[30]

M. Z. NdiiR. I. Hickson and G. N. Mercer, Modelling the introduction of Wolbachia into Aedes aegypti mosquitoes to reduce dengue transmission, The ANZIAM Journal, 53 (2012), 213-227.  doi: 10.1017/S1446181112000132.  Google Scholar

[31]

S. L. O'NeillR. GiordanoA. ColbertT. L. Karr and H. M. Robertson, 16S rRNA phylogenetic analysis of the bacterial endosymbionts associated with cytoplasmic incompatibility in insects, Proceedings of the National Academy of Sciences of the United States of America, 89 (1992), 2699-2702.  doi: 10.1073/pnas.89.7.2699.  Google Scholar

[32]

Z. QuL. Xue and J. M. Hyman, Modeling the transmission of Wolbachia in mosquitoes for controlling mosquito-borne diseases, SIAM Journal on Applied Mathematics, 78 (2018), 826-852.  doi: 10.1137/17M1130800.  Google Scholar

[33]

J. L. Rasgon and T. W. Scott, Wolbachia and cytoplasmic incompatibility in the California Culex pipiens mosquito species complex: Parameter estimates and infection dynamics in natural populations, Genetics, 165 (2003), 2029-2038.   Google Scholar

[34]

A. SaltelliP. AnnoniI. AzziniF. CampolongoM. Ratto and S. Tarantola, Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications, 181 (2010), 259-270.  doi: 10.1016/j.cpc.2009.09.018.  Google Scholar

[35]

F. SarrazinF. Pianosi and T. Wagener, Global Sensitivity Analysis of environmental models: Convergence and validation, Environmental Modelling & Software, 79 (2016), 135-152.  doi: 10.1016/j.envsoft.2016.02.005.  Google Scholar

[36]

P. Schofield, Spatially explicit models of Turelli-Hoffmann Wolbachia invasive wave fronts, Journal of Theoretical Biology, 215 (2002), 121-131.  doi: 10.1006/jtbi.2001.2493.  Google Scholar

[37]

J. G. Schraiber, A. N. Kaczmarczyk, R. Kwok, et al., Constraints on the use of lifespanshortening Wolbachia to control dengue fever, Journal of Theoretical Biology, 297 (2012), 26-32. doi: 10.1016/j.jtbi.2011.12.006.  Google Scholar

[38]

S. P. Sinkins, Wolbachia and cytoplasmic incompatibility in mosquitoes, Insect Biochemistry and Molecular Biology, 34 (2004), 723-729.  doi: 10.1016/j.ibmb.2004.03.025.  Google Scholar

[39]

I. M. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Mathematics and Computers in Simulation, 55 (2001), 271-280.  doi: 10.1016/S0378-4754(00)00270-6.  Google Scholar

[40]

M. Turelli and N. H. Barton, Deploying dengue-suppressing Wolbachia: Robust models predict slow but effective spatial spread in Aedes aegypti, Theoretical Population Biology, 115 (2017), 45-60.  doi: 10.1016/j.tpb.2017.03.003.  Google Scholar

[41]

P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[42]

T. Walker, P. H. Johnson, L. A. Moreira, et al., The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450-453. doi: 10.1038/nature10355.  Google Scholar

[43]

L. Wang, H. Zhao, S. M. Oliva and H. Zhu, Modeling the transmission and control of Zika in Brazil, Scientific Reports, 7 (2017), Art. 7721. doi: 10.1038/s41598-017-07264-y.  Google Scholar

[44]

J. H. Werren, Biology of Wolbachia, Annual Review of Entomology, 42 (1997), 587-609.  doi: 10.1146/annurev.ento.42.1.587.  Google Scholar

[45]

L. Xue, X. Fang and J. M. Hyman, Comparing the effectiveness of different strains of Wolbachia for controlling chikungunya, dengue fever, and zika, PLOS Neglected Tropical Diseases, 12 (2018), e0006666. doi: 10.1371/journal.pntd.0006666.  Google Scholar

[46]

L. XueC. A. ManoreP. Thongsripong and J. M. Hyman, Two-sex mosquito model for the persistence of Wolbachia, Journal of Biological Dynamics, 11 (2017), 216-237.  doi: 10.1080/17513758.2016.1229051.  Google Scholar

[47]

J. H. Yen and A. R. Barr, A new hypothesis of the cause of cytoplasmic incompatibility in Culex pipiens L., Nature, 232 (1971), 657-658.  doi: 10.1038/232657a0.  Google Scholar

[48]

X. ZhangS. Tang and R. A. Cheke, Birth-pulse models of Wolbachia-induced cytoplasmic incompatibility in mosquitoes for dengue virus control, Nonlinear Analysis: Real World Applications, 22 (2015), 236-258.  doi: 10.1016/j.nonrwa.2014.09.004.  Google Scholar

[49]

X. ZhangS. Tang and R. A. Cheke, Models to assess how best to replace dengue virus vectors with Wolbachia-infected mosquito populations, Mathematical Biosciences, 269 (2015), 164-177.  doi: 10.1016/j.mbs.2015.09.004.  Google Scholar

[50]

X. ZhangS. TangR. A. Cheke and H. Zhu, Modeling the effects of augmentation strategies on the control of dengue fever with an impulsive differential equation, Bulletin of Mathematical Biology, 78 (2016), 1968-2010.  doi: 10.1007/s11538-016-0208-7.  Google Scholar

[51]

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.  Google Scholar

[52]

B. ZhengM. TangJ. Yu and J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, Journal of Mathematical Biology, 76 (2018), 235-263.  doi: 10.1007/s00285-017-1142-5.  Google Scholar

[53]

X. Zheng, D. Zhang, Y. Li, C. Yang, Y. Wu, X. Liang, Y. Liang, X. Pan, L. Hu, Q. Sun et al., Incompatible and sterile insect techniques combined eliminate mosquitoes, Nature, 572 (2019), 56-61. doi: 10.1038/s41586-019-1407-9.  Google Scholar

Figure 1.  The distribution of Wolbachia-infected equilibria (WE) in the plane of $ (\xi,q) $ when $ 0<v<1 $, $ (1-v)(1-f_u)d_{mw} < v(1-\tau)(1-f_w)d_m $ and $ R_{0w}>1 $. In $ D_1 $, $ R_0>1 $, there exists a unique Wolbachia-infected equilibrium $ E^* $. There are two coexistence equilibria in $ D_2 $ where $ \hat{R_0}<R_0<1 $, and the two equilibria coalesce when the parameters approach the curve segment of $ L_3 $ between $ K_0 $ and $ K_3 $ where $ R_0 = \hat{R_0} $. There is a unique coexistence equilibrium on the segment of $ L_2 $ between $ K_3 $ and $ K_4 $, where $ R_0 = 1 $. In the other regions of the first quadrant, no WE exists
Figure 2.  The backward bifurcation occurs at $ R_0 = 1 $ when $ v = 0.9 $. Parameter values are taken from Table 4 except that $ d_{fw} $ varies
Figure 3.  The distribution of Wolbachia-infected equilibria (WE) in the plane of $ (\xi,q) $ when $ v = 1 $ and $ R_{0w}>1 $. In the region $ D_{11} $, where $ R_0 < 1-q $, there exists an unstable WE $ E_{02} $. In the region $ D_{22} $, where $ 1-q<R_0<1 $, There are two WE. One is a coexistence equilibria $ E^* $ and the other is the boundary equilibrium, $ E_{02} $, which is stable. In the region $ D_{12} $, there also exists a unique stable WE $ E_{02} $
Figure 4.  The distribution of real parts of eigenvalues of Jacobian matrix at $ E^*_{1} $
Figure 5.  The distribution of product of all six eigenvalues of Jacobian matrix at $ E^*_{2} $ and $ E_c^* $
Figure 6.  Stable manifold of the saddle $ E_2^* $ and $ E_c^* $. Parameter values used are given in Table 4
Figure 7.  The threshold values of Wolbachia-infected male and female mosquitoes. Parameter values are given in Table 4 except for $ v $ being varied. Other initial values of state variables are $ U_A(0) = 10000 $, $ U_F(0) = 5335 $, $ U_M(0) = 5991 $, $ W_A(0) = 0 $
Figure 8.  Time series of female mosquito for different releasing strategies. The parameter values are given in Table 4. (Ⅰ) $ W_F(0) = W_M(0) = 0 $; (Ⅱ) $ W_F(0) = 0, W_M(0) = 1,500,000 $; (Ⅲ) $ W_F(0) = 1, W_M(0) = 150,000 $. Other initial values of state variables are $ U_A(0) = 4068, U_F(0) = 3335, U_M(0) = 2991, W_A(0) = 0 $
Figure 9.  Global sensitivity analysis
Figure 10.  Bifurcation diagram with respect to $ \tau $, $ q $, $ v $, $ \alpha $. The parameter values are given in Table 4
Figure 11.  Threshold values of $ W_M(0) $ for Wolbachia establishment with respect to $ \tau $, $ q $, $ v $, $ \alpha $. Parameter values used are given in Table 4. Other initial values of state variables are: $ U_A(0) = 4068,U_F(0) = 3335,U_M(0) = 2991,W_A(0) = 0,W_F(0) = 1 $
Figure 12.  The impact of the CI effect suffering by uninfected ova produced by infected females when the maternal transmission is imperfect ($ v = 0.9 $). The parameter values are given in Table 4 except for $ \lambda $. Other initial values of state variables are: $ U_A(0) = 4068 $, $ U_F(0) = 3335 $, $ U_M(0) = 2991 $, $ W_A(0) = 0 $. The dash line corresponds to the case $ \lambda = 0 $, and the solid line corresponds to the case $ \lambda = q $, respectively
Table 1.  State variables in the Model (1) for the transmission dynamics of Wolbachia
State variables Uninfected Infected
Aquatic stage $ U_A $ $ W_A $
Adult females $ U_F $ $ W_F $
Adult males $ U_M $ $ W_M $
State variables Uninfected Infected
Aquatic stage $ U_A $ $ W_A $
Adult females $ U_F $ $ W_F $
Adult males $ U_M $ $ W_M $
Table 2.  Definition of the parameters
Interpretation Parameter
Natural birth rate of mosquitoes $ \theta $
Natural death rate of mosquitoes in aquatic stages $ d_a $
Natural death rate of adult Wolbachia-uninfected male mosquitoes $ d_{m} $
Natural death rate of adult Wolbachia-infected male mosquitoes $ d_{mw} $
Natural death rate of adult Wolbachia-uninfected female mosquitoes $ d_{f} $
Natural death rate of adult Wolbachia-infected female mosquitoes $ d_{fw} $
Density-dependent mortality of immature mosquitoes $ \kappa $
Per capita development rate of immature mosquitoes $ \varphi $
Fraction of births that are female mosquitoes for Wolbachia-free mosquitoes $ f_u $
Fraction of births that are female mosquitoes for Wolbachia-infected mosquitoes $ f_w $
Probability of cytoplasmic incompatibility (CI) $ q $
The CI parameter of uninfected ova produced by infected females $ \lambda $
Fertility cost coefficient of the Wolbachia infection to reproductive output $ \alpha $
Maternal transmission rate for Wolbachia infection $ v $
Probability of male killing (MK) induced by Wolbachia infection $ \tau $
Interpretation Parameter
Natural birth rate of mosquitoes $ \theta $
Natural death rate of mosquitoes in aquatic stages $ d_a $
Natural death rate of adult Wolbachia-uninfected male mosquitoes $ d_{m} $
Natural death rate of adult Wolbachia-infected male mosquitoes $ d_{mw} $
Natural death rate of adult Wolbachia-uninfected female mosquitoes $ d_{f} $
Natural death rate of adult Wolbachia-infected female mosquitoes $ d_{fw} $
Density-dependent mortality of immature mosquitoes $ \kappa $
Per capita development rate of immature mosquitoes $ \varphi $
Fraction of births that are female mosquitoes for Wolbachia-free mosquitoes $ f_u $
Fraction of births that are female mosquitoes for Wolbachia-infected mosquitoes $ f_w $
Probability of cytoplasmic incompatibility (CI) $ q $
The CI parameter of uninfected ova produced by infected females $ \lambda $
Fertility cost coefficient of the Wolbachia infection to reproductive output $ \alpha $
Maternal transmission rate for Wolbachia infection $ v $
Probability of male killing (MK) induced by Wolbachia infection $ \tau $
Table 3.  Existence and stability of equilibria of Model 1 ($ \checkmark $, Wolbachia is established; $ \times $, Wolbachia is extinct; $ ? $, Depending on initial values; E, exist; N, does not exist; S, locally asymptotically stable; U, unstable), where $ R_{0u} = \frac{f_u\varphi\theta}{d_f(d_a+\varphi)} $, $ R_0 = \frac{(1-\alpha)vd_ff_w}{d_{fw}f_u} $, $ R_{0w} = \frac{(1-\alpha)\theta vf_w\varphi}{d_{fw}(d_a+\varphi)} = R_{0u}R_0 $
$ v $ Threshold condition $ E_{01} $ $ E_{02} $ $ E^{*} $
Case 1 $ v=1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, U E, S N
$ \checkmark $ $ R_0>1 $
Case 2 $ v=1 $ $ R_{0w}>1 $, $ R_{0u}\leq1 $, N E, S N
$ \checkmark $ ($ R_0>1 $)
Case 3 $ v=1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, S E, S E(one)
$ ? $ $ 1-q<R_0<1 $
Case 4 $ v=1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, S E, U N
$ \times $ $ R_0< 1-q $
Case 5 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, U N E(one)
$ \checkmark $ $ R_0>1 $
Case 6 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, S N N
$ \times $ $ R_0<1 $
$ (1-v)(1-f_u)d_{mw} \geq v(1-\tau)(1-f_w)d_m $
Case 7 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, S N E(two)
$ ? $ $ \hat{R_0}<R_0<1 $
$ (1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m $
Case 8 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E N E(one)
Singularity $ R_0=\hat{R_0} $ or $ R_0=1 $,
$ (1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m $
Case 9 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, , E, S N N
$ \times $ $ R_0<\hat{R_0} $
$ (1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m $
Case 10 $ 0<v\leq1 $ $ R_{0w}\leq1 $, $ R_{0u}>1 $, E, S N N
$ \times $ ($ R_0<1 $)
Case 11 $ 0<v\leq1 $ $ R_{0w}\leq1 $, $ R_{0u}\leq1 $ N N N
$ \times $
$ v $ Threshold condition $ E_{01} $ $ E_{02} $ $ E^{*} $
Case 1 $ v=1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, U E, S N
$ \checkmark $ $ R_0>1 $
Case 2 $ v=1 $ $ R_{0w}>1 $, $ R_{0u}\leq1 $, N E, S N
$ \checkmark $ ($ R_0>1 $)
Case 3 $ v=1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, S E, S E(one)
$ ? $ $ 1-q<R_0<1 $
Case 4 $ v=1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, S E, U N
$ \times $ $ R_0< 1-q $
Case 5 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, U N E(one)
$ \checkmark $ $ R_0>1 $
Case 6 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, S N N
$ \times $ $ R_0<1 $
$ (1-v)(1-f_u)d_{mw} \geq v(1-\tau)(1-f_w)d_m $
Case 7 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E, S N E(two)
$ ? $ $ \hat{R_0}<R_0<1 $
$ (1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m $
Case 8 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, E N E(one)
Singularity $ R_0=\hat{R_0} $ or $ R_0=1 $,
$ (1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m $
Case 9 $ 0<v<1 $ $ R_{0w}>1 $, $ R_{0u}>1 $, , E, S N N
$ \times $ $ R_0<\hat{R_0} $
$ (1-v)(1-f_u)d_{mw}< v(1-\tau)(1-f_w)d_m $
Case 10 $ 0<v\leq1 $ $ R_{0w}\leq1 $, $ R_{0u}>1 $, E, S N N
$ \times $ ($ R_0<1 $)
Case 11 $ 0<v\leq1 $ $ R_{0w}\leq1 $, $ R_{0u}\leq1 $ N N N
$ \times $
Table 4.  Parameter values
Parameter Value Range Reference
$ \theta $ 50 [0, 75] [8]
$ q $ 0.9 [0.5, 1] [21,50]
$ v $ 0.9, 0.1 (0.85, 1] [29,42]
$ d_a $ 0.2 [0.2, 0.75] [11,25]
$ d_f $ 0.061 [1/55, 1/11] [46]
$ d_{fw} $ 0.068 [1/55, 1/11] [46]
$ d_m $ 0.068 [1/31, 1/7] [46]
$ d_{mw} $ 0.068 [1/31, 1/7] [46]
$ \varphi $ 0.1 [1/20, 1/7] [28]
$ \alpha $ 0.1 (0, 1] [49]
$ f_u $ 0.5 [0.34, 0.6] [46,8]
$ f_w $ 0.5 [0.34, 0.6] [46,8]
$ \tau $ 0.01 [0, 1) [13]
$ \kappa $ 0.01 [0.01, 0.01] Assumed
Parameter Value Range Reference
$ \theta $ 50 [0, 75] [8]
$ q $ 0.9 [0.5, 1] [21,50]
$ v $ 0.9, 0.1 (0.85, 1] [29,42]
$ d_a $ 0.2 [0.2, 0.75] [11,25]
$ d_f $ 0.061 [1/55, 1/11] [46]
$ d_{fw} $ 0.068 [1/55, 1/11] [46]
$ d_m $ 0.068 [1/31, 1/7] [46]
$ d_{mw} $ 0.068 [1/31, 1/7] [46]
$ \varphi $ 0.1 [1/20, 1/7] [28]
$ \alpha $ 0.1 (0, 1] [49]
$ f_u $ 0.5 [0.34, 0.6] [46,8]
$ f_w $ 0.5 [0.34, 0.6] [46,8]
$ \tau $ 0.01 [0, 1) [13]
$ \kappa $ 0.01 [0.01, 0.01] Assumed
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