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The threshold infection level for Wolbachia invasion in a two-sex mosquito population model. (English) Zbl 1417.92182

Summary: In this paper, we formulate a new Wolbachia infection model in a two-sex mosquito population with stage structure. Some key factors of Wolbachia infection, including cytoplasmic incompatibility (CI), male killing (MK) effect, maternal transmission, fecundity cost due to fitness effect and different mortality rates for infected individuals, are captured. Dynamical analysis has been carried out, and the basic reproduction number \(R_0\) for Wolbachia infection has been calculated. Our analysis shows that Wolbachia can establish in a mosquito population if \(R_0\) is greater than unity. If \(R_0\) is less than unity, Wolbachia establishment still can be achieved if backward bifurcation occurs. Under this circumstance, the initial values lying in the basin of attraction of the stable Wolbachia-established equilibrium are essential to guarantee Wolbachia establishment. In particular, the method to find the basin of attraction and evaluate the threshold initial values is given. Besides, according to a comparison of different releasing strategies, it is shown that, from the perspective of economy and disease control, keeping the number of infected female mosquitoes to a necessary minimum by relying on higher number of male mosquitoes released is a desirable strategy. Moreover, global and local sensitivity analysis and numerical simulation have been performed to explore the impact of model parameters to the success of population establishment. Our results suggest that low levels of MK effect and fitness costs as well as high levels of CI and maternal inheritance are in favor of Wolbachia establishment. Moreover, not considering MK effect and incomplete CI effect may result in the underestimation of the number of infected mosquitoes needed to be released.

MSC:

92D30 Epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
Full Text: DOI

References:

[1] Becker N, Petric D, Boase C, Lane J, Zgomba M, Dahl C, Kaiser A (2003) Mosquitoes and their control, vol 2. Springer, Berlin · doi:10.1007/978-1-4757-5897-9
[2] Bian G, Xu Y, Lu P, Xie Y, Xi Z (2010) The endosymbiotic bacterium \[{{\mathit{Wolbachia}}}\] Wolbachia induces resistance to dengue virus in aedes aegypti. PLoS Pathog 6(4):e1000833 · doi:10.1371/journal.ppat.1000833
[3] Bliman PA, Aronna MS, Coelho FC, Da Silva MA (2018) Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control. J Math Biol 76(5):1269-1300 · Zbl 1392.92096 · doi:10.1007/s00285-017-1174-x
[4] Campo-Duarte DE, Vasilieva O, Cardona-Salgado D, Svinin M (2018) Optimal control approach for establishing wMelPop Wolbachia infection among wild Aedes aegypti populations. J Math Biol 76(7):1907-1950 · Zbl 1390.92133 · doi:10.1007/s00285-018-1213-2
[5] Cannavó F (2012) Sensitivity analysis for volcanic source modeling quality assessment and model selection. Comput Geosci 44:52-59 · doi:10.1016/j.cageo.2012.03.008
[6] Caspari E, Watson G (1959) On the evolutionary importance of cytoplasmic sterility in mosquitoes. Evolution 13(4):568-570 · doi:10.1111/j.1558-5646.1959.tb03045.x
[7] Chan MH, Kim PS (2013) Modelling a Wolbachia invasion using a slow – fast dispersal reaction – diffusion approach. Bull Math Biol 75(9):1501-1523 · Zbl 1311.92173 · doi:10.1007/s11538-013-9857-y
[8] Delatte H, Gimonneau G, Triboire A, Fontenille D (2009) Influence of temperature on immature development, survival, longevity, fecundity, and gonotrophic cycles of Aedes albopictus, vector of Chikungunya and Dengue in the Indian Ocean. J Med Entomol 46(1):33-41 · doi:10.1603/033.046.0105
[9] Dorigatti I, McCormack C, Nedjati-Gilani G, Ferguson NM (2018) Using Wolbachia for dengue control: insights from modelling. Trends Parasitol 34(2):102-113 · doi:10.1016/j.pt.2017.11.002
[10] Egas M, Vala F, Breeuwer JAJ (2002) On the evolution of cytoplasmic incompatibility in haplodiploid species. Evolution 56(6):1101-1109 · doi:10.1111/j.0014-3820.2002.tb01424.x
[11] Engelstädter J, Telschow A, Hammerstein P (2004) Infection dynamics of different Wolbachia-types within one host population. J Theor Biol 231(3):345-355 · Zbl 1447.92420
[12] Fang J, Gourley SA, Lou Y (2016) Stage-structured models of intra-and inter-specific competition within age classes. J Differ Equ 260(2):1918-1953 · Zbl 1382.34089 · doi:10.1016/j.jde.2015.09.048
[13] Farkas JZ, Gourley SA, Liu R, Yakubu A-A (2017) Modelling Wolbachia infection in a sex-structured mosquito population carrying west nile virus. J Math Biol 75(3):621-647 · Zbl 1387.92082 · doi:10.1007/s00285-017-1096-7
[14] Farkas JZ, Hinow P (2010) Structured and unstructured continuous models for Wolbachia infections. Bull Math Biol 72(8):2067-2088 · Zbl 1201.92044 · doi:10.1007/s11538-010-9528-1
[15] Fenton A, Johnson KN, Brownlie JC, Hurst GD (2011) Solving the Wolbachia paradox: modeling the tripartite interaction between host, Wolbachia, and a natural enemy. Am Nat 178(3):333-342 · doi:10.1086/661247
[16] Hancock PA, Sinkins SP, Godfray HCJ (2011) Population dynamic models of the spread of Wolbachia. Am Nat 177(3):323-333 · doi:10.1086/658121
[17] Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42(4):599-653 · Zbl 0993.92033 · doi:10.1137/S0036144500371907
[18] Hilgenboecker K, Hammerstein P, Schlattmann P, Telschow A, Werren JH (2008) How many species are infected with Wolbachia? A statistical analysis of current data. FEMS Microbiol Lett 281(2):215-220 · doi:10.1111/j.1574-6968.2008.01110.x
[19] Hu L, Tang M, Wu Z, Xi Z, Yu J (2018) The threshold infection level for Wolbachia invasion in random environments. J Differ Equ. https://doi.org/10.1016/j.jde.2018.09.035 · Zbl 1406.34072
[20] Hughes H, Britton NF (2013) Modelling the use of Wolbachia to control dengue fever transmission. Bull Math Biol 75(5):796-818 · Zbl 1273.92034 · doi:10.1007/s11538-013-9835-4
[21] Jiggins FM (2017) The spread of Wolbachia through mosquito populations. PLoS Biol 15(6):e2002780. https://doi.org/10.1371/journal.pbio.2002780 · doi:10.1371/journal.pbio.2002780
[22] Joshi D, McFadden MJ, Bevins D, Zhang F, Xi Z (2014) Wolbachia strain w AlbB confers both fitness costs and benefit on Anopheles stephensi. Parasites Vectors 7(1):336 · doi:10.1186/1756-3305-7-336
[23] Keeling MJ, Jiggins F, Read JM (2003) The invasion and coexistence of competing Wolbachia strains. Heredity 91(4):382 · doi:10.1038/sj.hdy.6800343
[24] Koiller J, Silva MD, Souza M, Codeço C, Iggidr A et al. (2014) Aedes, Wolbachia and Dengue. [Research Report] RR-8462, Inria Nancy - Grand Est (Villers-lès-Nancy, France). 47. https://hal.inria.fr/hal-00939411/document
[25] Laven H (1951) Crossing experiments with culex strains. Evolution 5(4):370-375 · doi:10.1111/j.1558-5646.1951.tb02795.x
[26] Li J (2017) New revised simple models for interactive wild and sterile mosquito populations and their dynamics. J Biol Dyn 11(sup2):316-333 · Zbl 1448.92217 · doi:10.1080/17513758.2016.1216613
[27] Li MT, Sun GQ, Yakob L, Zhu HP, Jin Z, Zhang WY (2016) The driving force for 2014 dengue outbreak in Guangdong, China. PloS one 11(11):e0166211 · doi:10.1371/journal.pone.0166211
[28] McMeniman CJ, Lane RV, Cass BN, Fong AW, Sidhu M, Wang Y-F, Oneill SL (2009) Stable introduction of a life-shortening Wolbachia infection into the mosquito aedes aegypti. Science 323(5910):141-144 · doi:10.1126/science.1165326
[29] Munga S, Minakawa N, Zhou G, Githeko AK, Yan G (2007) Survivorship of immature stages of anopheles gambiae sl (Diptera: Culicidae) in natural habitats in Western Kenya highlands. J Med Entomol 44(5):758-764 · doi:10.1093/jmedent/44.5.758
[30] Ndii MZ, Hickson RI, Allingham D, Mercer G (2015) Modelling the transmission dynamics of dengue in the presence of Wolbachia. Math Biosci 262:157-166 · Zbl 1315.92083 · doi:10.1016/j.mbs.2014.12.011
[31] Ndii MZ, Hickson RI, Mercer GN (2012) Modelling the introduction of Wolbachia into Aedes aegypti mosquitoes to reduce dengue transmission. ANZIAM J 53(3):213-227 · Zbl 1316.93104 · doi:10.1017/S1446181112000132
[32] O’Neill SL, Giordano R, Colbert A, Karr TL, Robertson HM (1992) 16S rRNA phylogenetic analysis of the bacterial endosymbionts associated with cytoplasmic incompatibility in insects. Proc Natl Acad Sci 89(7):2699-2702 · doi:10.1073/pnas.89.7.2699
[33] Sallet G, Silva Moacyr AHB (2015) Monotone dynamical systems and some models of Wolbachia in Aedes aegypti populations. Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, INRIA, 20:145-176. https://hal.inria.fr/hal-01320616/document
[34] Saltelli A, Annoni P, Azzini I, Campolongo F, Ratto M, Tarantola S (2010) Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Comput Phys Commun 181(2):259-270 · Zbl 1219.93116 · doi:10.1016/j.cpc.2009.09.018
[35] Sarrazin F, Pianosi F, Wagener T (2016) Global sensitivity analysis of environmental models: convergence and validation. Environ Model Softw 79:135-152 · doi:10.1016/j.envsoft.2016.02.005
[36] Schofield P (2002) Spatially explicit models of turelli-hoffmann Wolbachia invasive wave fronts. J Theor Biol 215(1):121-131 · doi:10.1006/jtbi.2001.2493
[37] Schraiber JG, Kaczmarczyk AN, Kwok R, Park M, Silverstein R, Rutaganira FU, Aggarwal T, Schwemmer MA, Hom CL, Grosberg RK et al (2012) Constraints on the use of lifespan-shortening Wolbachia to control dengue fever. J Theor Biol 297:26-32 · Zbl 1336.92085 · doi:10.1016/j.jtbi.2011.12.006
[38] Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Math Comput Simul 55(1):271-280 · Zbl 1005.65004 · doi:10.1016/S0378-4754(00)00270-6
[39] Turelli M, Barton NH (2017) Deploying dengue-suppressing Wolbachia: robust models predict slow but effective spatial spread in Aedes aegypti. Theor Popul Biol 115:45-60 · Zbl 1381.92099 · doi:10.1016/j.tpb.2017.03.003
[40] Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1):29-48 · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[41] Walker T, Johnson PH, Moreira LA, Iturbe-Ormaetxe I, Frentiu FD et al (2011) The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations. Nature 476:450-453 · doi:10.1038/nature10355
[42] Wang L, Zhao H, Oliva SM, Zhu H (2017) Modeling the transmission and control of Zika in Brazil. Sci Rep 7(1):7721 · doi:10.1038/s41598-017-07264-y
[43] Werren JH (1997) Biology of Wolbachia. Annu Rev Entomol 42(1):587-609 · doi:10.1146/annurev.ento.42.1.587
[44] Werren JH, Windsor DM (2000) Wolbachia infection frequencies in insects: evidence of a global equilibrium? Proc R Soc Lond B: Biol Sci 267(1450):1277-1285 · doi:10.1098/rspb.2000.1139
[45] Xue L, Manore CA, Thongsripong P, Hyman JM (2017) Two-sex mosquito model for the persistence of Wolbachia. J Biol Dyn 11(sup1):216-237 · Zbl 1447.92493 · doi:10.1080/17513758.2016.1229051
[46] Yen JH, Barr AR (1971) New hypothesis of the cause of cytoplasmic incompatibility in Culex pipiens L. Nature 232(5313):657-658 · doi:10.1038/232657a0
[47] Zhang X, Tang S, Cheke RA (2015) Birth-pulse models of Wolbachia-induced cytoplasmic incompatibility in mosquitoes for dengue virus control. Nonlinear Anal: Real World Appl 22:236-258 · Zbl 1327.92071 · doi:10.1016/j.nonrwa.2014.09.004
[48] Zhang X, Tang S, Cheke RA (2015) Models to assess how best to replace dengue virus vectors with Wolbachia-infected mosquito populations. Math Biosci 269:164-177 · Zbl 1335.92106 · doi:10.1016/j.mbs.2015.09.004
[49] Zhang X, Tang S, Cheke RA, Zhu H (2016) Modeling the effects of augmentation strategies on the control of dengue fever with an impulsive differential equation. Bull Math Biol 78(10):1968-2010 · Zbl 1361.92075 · doi:10.1007/s11538-016-0208-7
[50] Zheng B, Tang M, Yu J (2014) Modeling \[{{\mathit{Wolbachia}}}\] Wolbachia spread in mosquitoes through delay differential equations. SIAM J Appl Math 74(3):743-770 · Zbl 1303.92124 · doi:10.1137/13093354X
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