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Quantifying the survival uncertainty of Wolbachia-infected mosquitoes in a spatial model. (English) Zbl 1406.92522
Summary: Artificial releases of Wolbachia-infected Aedes mosquitoes have been under study in the past years for fighting vector-borne diseases such as dengue, chikungunya and zika. Several strains of this bacterium cause cytoplasmic incompatibility (CI) and can also affect their host’s fecundity or lifespan, while highly reducing vector competence for the main arboviruses.
We consider and answer the following questions: 1) what should be the initial condition (i.e. size of the initial mosquito population) to have invasion with one mosquito release source? We note that it is hard to have an invasion in such case. 2) How many release points does one need to have sufficiently high probability of invasion? 3) What happens if one accounts for uncertainty in the release protocol (e.g. unequal spacing among release points)?
We build a framework based on existing reaction-diffusion models for the uncertainty quantification in this context, obtain both theoretical and numerical lower bounds for the probability of release success and give new quantitative results on the one dimensional case.

##### MSC:
 92D25 Population dynamics (general) 92C60 Medical epidemiology 35K57 Reaction-diffusion equations
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##### References:
 [1] http://www.cdc.gov/zika/transmission/index.html, 2016. [2] L. Alphey, Genetic control of mosquitoes, Annual Review of Entomology, 59, 205, (2014) [3] L. Alphey; A. McKemey; D. Nimmo; O. M. Neira; R. Lacroix; K. Matzen; C. Beech, Genetic control of aedes mosquitoes, Pathogens and Global Health, 107, 170, (2013) [4] N. H. Barton; M. Turelli, Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of allee effects, The American Naturalist, 178, E48, (2011) [5] N. Barton; G. Hewitt, Adaptation, speciation and hybrid zones, Nature, 341, 497, (1989) [6] N. Barton; S. Rouhani, The probability of fixation of a new karyotype in a continuous population, Evolution, 45, 499, (1991) [7] S. Bhatt; P. W. Gething; O. J. Brady; J. P. Messina; A. W. Farlow; C. L. Moyes; J. M. Drake; J. S. Brownstein; A. G. Hoen; O. Sankoh; M. F. Myers; D. B. George; T. Jaenisch; G. R. W. Wint; C. P. Simmons; T. W. Scott; J. J. Farrar; S. I. Hay, The global distribution and burden of dengue, Nature, 496, 504, (2013) [8] M. S. C. Blagrove, C. Arias-Goeta, C. Di Genua, A.-B. Failloux and S. P. Sinkins, A WolbachiawMel transinfection in Aedes albopictus is not detrimental to host fitness and inhibits Chikungunya virus, PLoS Neglected Tropical Diseases, 7 (2013), e2152. [9] M. H. T. Chan; P. S. Kim, Modeling a wolbachia invasion using a slow-fast dispersal reaction-diffusion approach, Bulletin of Mathematical Biology, 75, 1501, (2013) · Zbl 1311.92173 [10] P. R. Crain, J. W. Mains, E. Suh, Y. Huang, P. H. Crowley and S. L. Dobson, Wolbachia infections that reduce immature insect survival: Predicted impacts on population replacement, BMC Evolutionary Biology, 11 (2011), p290. [11] Y. Du; H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, Journal of the European Mathematical Society, 12, 279, (2010) · Zbl 1207.35061 [12] G. L. C. Dutra, L. M. B. dos Santos, E. P. Caragata, J. B. L. Silva, D. A. M. Villela, R. Maciel-de Freitas and L. A. Moreira, 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. [13] P. Erdos; A. Rényi, On a classical problem of probability theory, Magyar. Tud. Akad. Mat. Kutato Int. Kozl., 6, 215, (1961) · Zbl 0102.35201 [14] A. Fenton; K. N. Johnson; J. C. Brownlie; G. D. D. Hurst, Solving the wolbachia paradox: modeling the tripartite interaction between host, wolbachia, and a natural enemy, The American Naturalist, 178, 333, (2011) [15] P. A. Hancock and H. C. J. Godfray, Modelling the spread of Wolbachia in spatially heterogeneous environments, Journal of The Royal Society Interface, 9 (2012), p253. [16] P. A. Hancock; S. P. Sinkins; H. C. J. Godfray, Population dynamic models of the spread of wolbachia, The American Naturalist, 177, 323, (2011) [17] P. A. Hancock, S. P. Sinkins and H. C. J. Godfray, Strategies for introducing Wolbachia to reduce transmission of mosquito-borne diseases, PLoS Neglected Tropical Diseases, 5 (2011), e1024. [18] A. A. Hoffmann, I. Iturbe-Ormaetxe, A. G. Callahan, B. L. Phillips, K. Billington, J. K. Axford, B. Montgomery, A. 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. [19] A. A. Hoffmann; B. L. Montgomery; J. Popovici; I. Iturbe-Ormaetxe; P. H. Johnson; F. Muzzi; M. Greenfield; M. Durkan; Y. S. Leong; Y. Dong; H. Cook; J. Axford; A. G. Callahan; N. Kenny; C. Omodei; E. A. McGraw; P. A. Ryan; S. A. Ritchie; M. Turelli; S. L. O’Neill, Successful establishment of wolbachia in aedes populations to suppress dengue transmission, Nature, 476, 454, (2011) [20] H. Hughes; N. F. Britton, Modeling the use of wolbachia to control dengue fever transmission, Bulletin of Mathematical Biology, 75, 796, (2013) · Zbl 1273.92034 [21] V. A. Jansen; M. Turelli; H. C. J. Godfray, Stochastic spread of wolbachia, Proceedings of the Royal Society of London B: Biological Sciences, 275, 2769, (2008) [22] K. N. Johnson, The impact of wolbachia on virus infection in mosquitoes, Viruses, 7, 5705, (2015) [23] R. Maciel-de Freitas; R. Souza-Santos; C. T. Codeço; R. Lourenço-de Oliveira, Influence of the spatial distribution of human hosts and large size containers on the dispersal of the mosquito aedes aegypti within the first gonotrophic cycle, Medical and Veterinary Entomology, 24, 74, (2010) [24] H. Matano; P. Poláčik, Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. part i: A general quasiconvergence theorem and its consequences, Communications in Partial Differential Equations, 41, 785, (2016) · Zbl 1345.35052 [25] C. B. Muratov; X. Zhong, Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations, Discrete and Continuous Dynamical Systems, 37, 915, (2017) · Zbl 1364.35145 [26] T. H. Nguyen, H. L. Nguyen, T. Y. Nguyen, S. N. Vu, N. D. Tran, T. N. Le, Q. M. Vien, T. C. Bui, H. T. Le, S. Kutcher, T. P. Hurst, T. T. H. Duong, J. A. L. Jeffery, J. M. Darbro, B. H. Kay, I. Iturbe-Ormaetxe, J. Popovici, B. L. Montgomery, A. P. Turley, F. Zigterman, H. Cook, P. E. Cook, P. H. Johnson, P. A. Ryan, C. J. Paton, S. A. Ritchie, C. P. Simmons, S. L. 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. [27] M. Otero; N. Schweigmann; H. G. Solari, A stochastic spatial dynamical model for aedes aegypti, Bulletin of Mathematical Biology, 70, 1297, (2008) · Zbl 1142.92028 [28] T. Ouyang; J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, Journal of Differential Equations, 146, 121, (1998) · Zbl 0918.35049 [29] T. Ouyang; J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem, ⅱ, Journal of Differential Equations, 158, 94, (1999) · Zbl 0947.35067 [30] P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on $$\mathbb{R}^N$$, Archive for Rational Mechanics and Analysis, 199, 69, (2011) · Zbl 1262.35130 [31] S. Rouhani; N. Barton, Speciation and the ”Shifting balance” in a continuous population, Theoretical Population Biology, 31, 465, (1987) · Zbl 0614.92011 [32] M. Strugarek; N. Vauchelet, Reduction to a single closed equation for 2 by 2 reaction-diffusion systems of Lotka-Volterra type, SIAM Journal on Applied Mathematics, 76, 2060, (2016) · Zbl 1355.35108 [33] M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64, 232, (2010) [34] F. Vavre; S. Charlat, Making (good) use of wolbachia: what the models say, Current Opinion in Microbiology, 15, 263, (2012) [35] D. A. M. Villela, C. T. Codeço, F. Figueiredo, G. A. Garcia, R. Maciel-de Freitas and C. J. Struchiner, A Bayesian hierarchical model for estimation of abundance and spatial density of Aedes aegypti, PLoS ONE, 10 (2015), e0123794. [36] T. Walker; P. H. Johnson; L. A. Moreira; I. Iturbe-Ormaetxe; F. D. Frentiu; C. J. McMeniman; Y. S. Leong; Y. Dong; J. Axford; P. Kriesner; A. L. Lloyd; S. A. Ritchie; S. L. O’Neill; A. A. Hoffmann, The wmel wolbachia strain blocks dengue and invades caged aedes aegypti populations, Nature, 476, 450, (2011) [37] H. L. Yeap; P. Mee; T. Walker; A. R. Weeks; S. L. O’Neill; P. Johnson; S. A. Ritchie; K. M. Richardson; C. Doig; N. M. Endersby; A. A. Hoffmann, Dynamics of the “Popcorn” wolbachia infection in outbred aedes aegypti informs prospects for mosquito vector control, Genetics, 187, 583, (2011) [38] H. L. Yeap; G. Rasic; N. M. Endersby-Harshman; S. F. Lee; E. Arguni; H. Le Nguyen; A. A. Hoffmann, Mitochondrial DNA variants help monitor the dynamics of wolbachia invasion into host populations, Heredity, 116, 265, (2016) [39] B. Zheng; M. Tang; J. Yu; J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, Journal of Mathematical Biology, 76, 235, (2018) · Zbl 1392.92113 [40] A. Zlatos, Sharp transition between extinction and propagation of reaction, Journal of the American Mathematical Society, 19, 251, (2006) · Zbl 1081.35011
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