Figure 1.
Profile of $f$ defined in (2) (left) and of its anti-derivative $F$ (right) with parameters given by (5).
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Optimal design for dynamical modeling of pest populations
August 2018, 15(4): 961-991.
doi: 10.3934/mbe.2018043
Quantifying the survival uncertainty of Wolbachia-infected mosquitoes in a spatial model
| 1. | AgroParisTech, 16 rue Claude Bernard, 75231 Paris Cedex 05, France |
| 2. | Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, équipe Mamba, F-75005 Paris, France |
| 3. | LAGA - UMR 7539 Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément 93430 Villetaneuse, France |
| 4. | IMPA, Estrada Dona Castorina, 110 Jardim Botânico 22460-320, Rio de Janeiro, RJ, Brazil |
Artificial releases of Wolbachia-infected Aedes mosquitoes have been under study in the past yearsfor 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 successand give new quantitative results on the one dimensional case.
Keywords:
Reaction-diffusion equation,
Wolbachia,
uncertainty quantification,
population replacement,
mosquito release protocol.
Mathematics Subject Classification:
Primary: 35K57, 35B40, 92D25; Secondary: 60H30.
Citation:
Martin Strugarek, Nicolas Vauchelet, Jorge P. Zubelli. Quantifying the survival uncertainty of Wolbachia-infected mosquitoes in a spatial model. Mathematical Biosciences & Engineering,
2018, 15
(4)
: 961-991.
doi: 10.3934/mbe.2018043
![]()
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Genetic control of Aedes mosquitoes, Pathogens and Global Health, 107 (2013), 170-179.
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The probability of fixation of a new karyotype in a continuous population, Evolution, 45 (1991), 499-517.
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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.
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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.
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A. Fenton, K. N. Johnson, J. C. Brownlie and G. D. D. Hurst,
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doi: 10.1086/661247. |
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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.
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P. A. Hancock, S. P. Sinkins and H. C. J. Godfray,
Population dynamic models of the spread of Wolbachia, The American Naturalist, 177 (2011), 323-333.
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P. A. Hancock, S. P. Sinkins and H. C. J. Godfray,
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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.
doi: 10.1371/journal.pntd.0003115. |
| [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 and S. L. O'Neill,
Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission, Nature, 476 (2011), 454-457.
doi: 10.1038/nature10356. |
| [20] |
H. Hughes and N. F. Britton,
Modeling the use of Wolbachia to control dengue fever transmission, Bulletin of Mathematical Biology, 75 (2013), 796-818.
doi: 10.1007/s11538-013-9835-4. |
| [21] |
V. A. Jansen, M. Turelli and H. C. J. Godfray,
Stochastic spread of Wolbachia, Proceedings of the Royal Society of London B: Biological Sciences, 275 (2008), 2769-2776.
doi: 10.1098/rspb.2008.0914. |
| [22] |
K. N. Johnson,
The impact of Wolbachia on virus infection in mosquitoes, Viruses, 7 (2015), 5705-5717.
doi: 10.3390/v7112903. |
| [23] |
R. Maciel-de Freitas, R. Souza-Santos, C. T. Codeço and 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 (2010), 74-82.
doi: 10.1111/j.1365-2915.2009.00851.x. |
| [24] |
H. Matano and 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 (2016), 785-811.
doi: 10.1080/03605302.2016.1156697. |
| [25] |
C. B. Muratov and X. Zhong,
Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations, Discrete and Continuous Dynamical Systems, 37 (2017), 915-944.
doi: 10.3934/dcds.2017038. |
| [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.
doi: 10.1186/s13071-015-1174-x. |
| [27] |
M. Otero, N. Schweigmann and H. G. Solari,
A stochastic spatial dynamical model for Aedes aegypti, Bulletin of Mathematical Biology, 70 (2008), 1297-1325.
doi: 10.1007/s11538-008-9300-y. |
| [28] |
T. Ouyang and J. Shi,
Exact multiplicity of positive solutions for a class of semilinear problems, Journal of Differential Equations, 146 (1998), 121-156.
doi: 10.1006/jdeq.1998.3414. |
| [29] |
T. Ouyang and J. Shi,
Exact multiplicity of positive solutions for a class of semilinear problem, Ⅱ, Journal of Differential Equations, 158 (1999), 94-151.
doi: 10.1016/S0022-0396(99)80020-5. |
| [30] |
P. Poláčik,
Threshold solutions and sharp transitions for nonautonomous parabolic equations on $\mathbb{R}^N$, Archive for Rational Mechanics and Analysis, 199 (2011), 69-97.
doi: 10.1007/s00205-010-0316-8. |
| [31] |
S. Rouhani and N. Barton,
Speciation and the ''Shifting Balance" in a continuous population, Theoretical Population Biology, 31 (1987), 465-492.
doi: 10.1016/0040-5809(87)90016-5. |
| [32] |
M. Strugarek and 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 (2016), 2060-2080.
doi: 10.1137/16M1059217. |
| [33] |
M. Turelli,
Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232-241.
doi: 10.1111/j.1558-5646.2009.00822.x. |
| [34] |
F. Vavre and S. Charlat,
Making (good) use of Wolbachia: What the models say, Current Opinion in Microbiology, 15 (2012), 263-268.
doi: 10.1016/j.mib.2012.03.005. |
| [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.
doi: 10.1371/journal.pone.0123794. |
| [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 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. |
| [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 and A. A. Hoffmann,
Dynamics of the "Popcorn" Wolbachia infection in outbred Aedes aegypti informs prospects for mosquito vector control, Genetics, 187 (2011), 583-595.
doi: 10.1534/genetics.110.122390. |
| [38] |
H. L. Yeap, G. Rasic, N. M. Endersby-Harshman, S. F. Lee, E. Arguni, H. Le Nguyen and A. A. Hoffmann,
Mitochondrial DNA variants help monitor the dynamics of Wolbachia invasion into host populations, Heredity, 116 (2016), 265-276.
doi: 10.1038/hdy.2015.97. |
| [39] |
B. Zheng, M. Tang, J. 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. |
| [40] |
A. Zlatos,
Sharp transition between extinction and propagation of reaction, Journal of the American Mathematical Society, 19 (2006), 251-263.
doi: 10.1090/S0894-0347-05-00504-7. |
show all references
References:
| [1] | |
| [2] |
L. Alphey,
Genetic control of mosquitoes, Annual Review of Entomology, 59 (2014), 205-224.
doi: 10.1146/annurev-ento-011613-162002. |
| [3] |
L. Alphey, A. McKemey, D. Nimmo, O. M. Neira, R. Lacroix, K. Matzen and C. Beech,
Genetic control of Aedes mosquitoes, Pathogens and Global Health, 107 (2013), 170-179.
doi: 10.1179/2047773213Y.0000000095. |
| [4] |
N. H. Barton and M. Turelli,
Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of Allee effects, The American Naturalist, 178 (2011), E48-E75.
doi: 10.1086/661246. |
| [5] |
N. Barton and G. Hewitt,
Adaptation, speciation and hybrid zones, Nature, 341 (1989), 497-503.
doi: 10.1038/341497a0. |
| [6] |
N. Barton and S. Rouhani,
The probability of fixation of a new karyotype in a continuous population, Evolution, 45 (1991), 499-517.
doi: 10.1111/j.1558-5646.1991.tb04326.x. |
| [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 and S. I. Hay,
The global distribution and burden of dengue, Nature, 496 (2013), 504-507.
doi: 10.1038/nature12060. |
| [8] |
M. S. C. Blagrove, C. Arias-Goeta, C. Di Genua, A.-B. Failloux and S. P. Sinkins,
A Wolbachia wMel transinfection in Aedes albopictus is not detrimental to host fitness and inhibits Chikungunya virus,
PLoS Neglected Tropical Diseases, 7 (2013), e2152.
doi: 10.1371/journal.pntd.0002152. |
| [9] |
M. H. T. Chan and P. S. Kim,
Modeling 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. |
| [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.
doi: 10.1186/1471-2148-11-290. |
| [11] |
Y. Du and H. Matano,
Convergence and sharp thresholds for propagation in nonlinear diffusion problems, Journal of the European Mathematical Society, 12 (2010), 279-312.
doi: 10.4171/JEMS/198. |
| [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.
doi: 10.1371/journal.pntd.0003689. |
| [13] |
P. Erdos and A. Rényi,
On a classical problem of probability theory, Magyar. Tud. Akad. Mat. Kutato Int. Kozl., 6 (1961), 215-220.
|
| [14] |
A. Fenton, K. N. Johnson, J. C. Brownlie and G. D. D. Hurst,
Solving the Wolbachia paradox: Modeling the tripartite interaction between host, Wolbachia, and a natural enemy, The American Naturalist, 178 (2011), 333-342.
doi: 10.1086/661247. |
| [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.
doi: 10.1098/rsif.2012.0253. |
| [16] |
P. A. Hancock, S. 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. |
| [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.
doi: 10.1371/journal.pntd.0001024. |
| [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.
doi: 10.1371/journal.pntd.0003115. |
| [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 and S. L. O'Neill,
Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission, Nature, 476 (2011), 454-457.
doi: 10.1038/nature10356. |
| [20] |
H. Hughes and N. F. Britton,
Modeling the use of Wolbachia to control dengue fever transmission, Bulletin of Mathematical Biology, 75 (2013), 796-818.
doi: 10.1007/s11538-013-9835-4. |
| [21] |
V. A. Jansen, M. Turelli and H. C. J. Godfray,
Stochastic spread of Wolbachia, Proceedings of the Royal Society of London B: Biological Sciences, 275 (2008), 2769-2776.
doi: 10.1098/rspb.2008.0914. |
| [22] |
K. N. Johnson,
The impact of Wolbachia on virus infection in mosquitoes, Viruses, 7 (2015), 5705-5717.
doi: 10.3390/v7112903. |
| [23] |
R. Maciel-de Freitas, R. Souza-Santos, C. T. Codeço and 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 (2010), 74-82.
doi: 10.1111/j.1365-2915.2009.00851.x. |
| [24] |
H. Matano and 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 (2016), 785-811.
doi: 10.1080/03605302.2016.1156697. |
| [25] |
C. B. Muratov and X. Zhong,
Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations, Discrete and Continuous Dynamical Systems, 37 (2017), 915-944.
doi: 10.3934/dcds.2017038. |
| [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.
doi: 10.1186/s13071-015-1174-x. |
| [27] |
M. Otero, N. Schweigmann and H. G. Solari,
A stochastic spatial dynamical model for Aedes aegypti, Bulletin of Mathematical Biology, 70 (2008), 1297-1325.
doi: 10.1007/s11538-008-9300-y. |
| [28] |
T. Ouyang and J. Shi,
Exact multiplicity of positive solutions for a class of semilinear problems, Journal of Differential Equations, 146 (1998), 121-156.
doi: 10.1006/jdeq.1998.3414. |
| [29] |
T. Ouyang and J. Shi,
Exact multiplicity of positive solutions for a class of semilinear problem, Ⅱ, Journal of Differential Equations, 158 (1999), 94-151.
doi: 10.1016/S0022-0396(99)80020-5. |
| [30] |
P. Poláčik,
Threshold solutions and sharp transitions for nonautonomous parabolic equations on $\mathbb{R}^N$, Archive for Rational Mechanics and Analysis, 199 (2011), 69-97.
doi: 10.1007/s00205-010-0316-8. |
| [31] |
S. Rouhani and N. Barton,
Speciation and the ''Shifting Balance" in a continuous population, Theoretical Population Biology, 31 (1987), 465-492.
doi: 10.1016/0040-5809(87)90016-5. |
| [32] |
M. Strugarek and 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 (2016), 2060-2080.
doi: 10.1137/16M1059217. |
| [33] |
M. Turelli,
Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232-241.
doi: 10.1111/j.1558-5646.2009.00822.x. |
| [34] |
F. Vavre and S. Charlat,
Making (good) use of Wolbachia: What the models say, Current Opinion in Microbiology, 15 (2012), 263-268.
doi: 10.1016/j.mib.2012.03.005. |
| [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.
doi: 10.1371/journal.pone.0123794. |
| [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 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. |
| [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 and A. A. Hoffmann,
Dynamics of the "Popcorn" Wolbachia infection in outbred Aedes aegypti informs prospects for mosquito vector control, Genetics, 187 (2011), 583-595.
doi: 10.1534/genetics.110.122390. |
| [38] |
H. L. Yeap, G. Rasic, N. M. Endersby-Harshman, S. F. Lee, E. Arguni, H. Le Nguyen and A. A. Hoffmann,
Mitochondrial DNA variants help monitor the dynamics of Wolbachia invasion into host populations, Heredity, 116 (2016), 265-276.
doi: 10.1038/hdy.2015.97. |
| [39] |
B. Zheng, M. Tang, J. 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. |
| [40] |
A. Zlatos,
Sharp transition between extinction and propagation of reaction, Journal of the American Mathematical Society, 19 (2006), 251-263.
doi: 10.1090/S0894-0347-05-00504-7. |
Figure 2.
Time dynamics with three different initial releases belonging to the set $RP_{50}^2(N)$ of (10), with $N/(N+N_0) = 0.75$ . Integration is performed on the domain $[-L, L]$ with $L = 50 \textrm{km}$ . The release box is plotted in dashed red on the first picture of each configuration. Left: Release box $[-2 L/3, 2 L/3]^2$ . Center: Release box $[-L/2, L/2]^2$ . Right: Release box $[-L/12.5, L/12.5]^2$ . From top to bottom: increasing time $t \in \{0, 1, 25, 50, 75\}$ , in days. The color indicates the value of $p$ (with the scale on the right).
Figure 3.
Comparison of minimal invasion radii $R_{\alpha}$ (obtained by energy) in dashed line and $L_{\alpha}$ (obtained by critical bubbles)
in solid line, varying with the maximal infection frequency level $\alpha$ . The scale is such that $\sigma=1$ .
Figure 4.
Two $G_{\sigma}$ profiles and their sum (in thick line). The level $G_{\sigma} (0)$ is the dashed line. On the left, $h=\sqrt{2\log(2)\sigma}$ . On the right, $h>\sqrt{2\log(2)\sigma}$ .
Figure 5.
Under-estimation $\beta^{\lambda, R^*} (-L, L)$ of introduction success probability for $L$ ranging from $R^*/2 = 5.49$ to $3 R^*/2 = 16.47$ .
The seven curves correspond to increasing number of release points. (From bottom to top: $20$ to $80$ release points).
Figure 6.
Effect of losing the constant $2 \sqrt{2 \log(2)}$ in Proposition 6: under-estimation $\beta^{\lambda, R^*} (-L, L)$ of introduction success probability for $L$ ranging from $R^*/2 = 5.49$ to $3 R^*/2 = 16.47$ , with $80$ release points.
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